This is a multi-part message in MIME format. ---------------0001101301876 Content-Type: text/plain; name="00-11.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-11.keywords" vector fields on tori, simultaneous resonances, pseudodifferential operators, simultaneous Diophantine conditions, resonant normal forms, global hypoellipticity, inhomogeneous Diophantine conditions, generalized Hausdorff dimension ---------------0001101301876 Content-Type: application/postscript; name="dgyams1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dgyams1.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.66a Copyright 1986-97 Radical Eye Software (www.radicaleye.com) %%Title: dgyams1.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips dgyams1 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.11.27:1631 %%BeginProcSet: texc.pro %! /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind 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b(58G35,)g(11J13;)j (Secondary)386 4710 y(58G15,)28 b(11J15.)486 4801 y Fn(Key)e(wor)l(ds)h (and)g(phr)l(ases.)40 b FD(v)n(ector)24 b(\014elds)h(on)f(tori,)h(sim)n (ultaneous)f(resonances,)i(pseudo)r(di\013eren-)386 4892 y(tial)f(op)r(erators,)h(sim)n(ultaneous)e(Diophan)n(tine)g (conditions,)h(resonan)n(t)g(normal)e(forms,)i(global)h(h)n(yp)r(o-)386 4984 y(ellipticit)n(y)-6 b(,)26 b(inhomogeneous)g(Diophan)n(tine)f (conditions,)i(generalized)g(Hausdor\013)e(dimension.)486 5075 y(Researc)n(h)g(partially)i(supp)r(orted)e(b)n(y)g(the)g(EPSR)n (C.)486 5166 y(P)n(artially)k(supp)r(orted)e(b)n(y)g(V)-6 b(olksw)n(agen-Stiftung)28 b(\(RiP-program)g(at)g(Ob)r(erw)n(olfac)n (h\),)i(b)n(y)d(a)h(Co-)386 5257 y(ordinated)d(Researc)n(h)h(Pro)t (ject)h(of)f(the)f(Univ)n(ersit)n(y)f(of)i(Cagliari,)j(b)n(y)24 b(MURST,)h(Italy)-6 b(,)26 b(and)f(b)n(y)f(NFS,)386 5349 y(Bulgaria.)486 5440 y(P)n(artially)i(supp)r(orted)f(b)n(y)f(V)-6 b(olksw)n(agen-Stiftung)25 b(\(RiP-program)g(at)h(Ob)r(erw)n(olfac)n (h\),)g(Gran)n(t-in-)386 5531 y(Aid)h(for)h(Scien)n(ti\014c)f(Researc)n (h)h(\(No.)40 b(11640183\),)31 b(Ministry)d(of)g(Education,)g(Science)g (and)f(Culture,)386 5623 y(Japan)f(and)f(b)n(y)g(Ch)n(uo)h(Univ)n (ersit)n(y)-6 b(,)24 b(T)-6 b(oky)n(o,)26 b(Japan.)1863 5722 y Fq(1)p eop %%Page: 2 2 2 1 bop 386 239 a Fq(2)728 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g(YOSHINO) 486 434 y Fw(W)-8 b(e)33 b(p)s(erturb)d(the)j Fu(X)1211 449 y Ft(k)1286 434 y Fw(with)e(zero'th)j(order)d(classical)h(pseudo)s (di\013eren)m(tial)d(op)s(era-)386 542 y(tors)h(so)h(that)g(the)g(op)s (erators)1036 695 y Fu(b)1075 657 y Ft(k)1118 695 y Fw(\()p Fu(x;)15 b(D)s Fw(\))26 b(:=)f Fp(\000)p Fu(iX)1682 710 y Ft(k)1745 695 y Fw(+)20 b Fu(a)1884 657 y Ft(k)1927 695 y Fw(\()p Fu(x;)15 b(D)s Fw(\))p Fu(;)107 b Fw(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d;)-2323 b Fw(\(1.2\))386 851 y(are)31 b(comm)m(uting,)g(where)f Fu(D)f Fw(=)c(\()p Fu(D)1609 865 y Ft(x)1649 874 y Fo(1)1688 851 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(D)1980 865 y 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Fw(\(1.3\))386 1736 y(W)-8 b(e)35 b(will)c(actually)i (consider)f(p)s(erturbations)g(with)g(functions)g(in)h(the)h(Gevrey)g (class)386 1844 y Fu(G)458 1811 y Ft(\033)504 1844 y Fw(\()p Fs(T)598 1811 y Ft(n)647 1844 y Fw(\).)56 b(Recall)35 b(that)h Fu(G)1316 1811 y Ft(\033)1362 1844 y Fw(\()p Fs(T)1456 1811 y Ft(n)1505 1844 y Fw(\))f(is)g(the)g(set)h(of)f(all)f Fu(f)43 b Fp(2)33 b Fu(C)2473 1811 y Fm(1)2547 1844 y Fw(\()p Fs(T)2641 1811 y Ft(n)2690 1844 y Fw(\))j(suc)m(h)f(that)g (there)386 1958 y(exists)i Fu(C)44 b(>)37 b Fw(0)h(satisfying)e(max) 1525 1972 y Ft(x)p Fm(2)p Fl(T)1662 1953 y FA(n)1718 1958 y Fp(j)p Fu(@)1796 1925 y Ft(\014)1844 1958 y Fu(f)10 b Fw(\()p Fu(x)p Fw(\))p Fp(j)37 b(\024)g Fu(C)2263 1925 y Fm(j)p Ft(\014)s Fm(j)p Fq(+1)2439 1958 y Fw(\()p Fu(\014)5 b Fw(!\))2590 1925 y Ft(\033)2638 1958 y Fw(,)40 b Fu(\014)i Fp(2)37 b Fs(Z)2959 1925 y Ft(n)3002 1958 y Fw(.)62 b(Clearly)386 2066 y Fu(G)458 2033 y Fq(1)497 2066 y Fw(\()p Fs(T)591 2033 y Ft(n)640 2066 y Fw(\))30 b(coincides)g(with)f(the)h(set)h(of)g (all)e(analytic)h(functions)f(on)h Fs(T)2748 2033 y Ft(n)2797 2066 y Fw(.)486 2173 y(The)f(study)h(of)g(normal)g(forms)g(is)f(a)i (classical)e(sub)5 b(ject,)31 b(originating)d(with)386 2281 y(P)m(oincar)m(\023)-43 b(e's)29 b(celebrated)g(w)m(orks)f(on)g (the)g(reduction)f(of)h(di\013eren)m(tial)f(equations)h(in)e(the)386 2389 y(neigh)m(b)s(ourho)s(o)s(d)37 b(of)j(an)g(equilibrium)35 b(p)s(oin)m(t.)69 b(It)40 b(is)f(still)f(one)j(of)f(the)g(most)g (useful)386 2497 y(to)s(ols)34 b(in)g(Dynamical)g(Systems)g(and)g(PDE.) h(Recall)f(that)h(for)g(a)g(single)e(v)m(ector)j(\014eld)386 2605 y(\(resp)s(ectiv)m(ely)-8 b(,)34 b(map\))f(with)e(an)i(isolated)f (singular)f(\(resp)s(ectiv)m(ely)-8 b(,)34 b(\014xed\))f(p)s(oin)m(t)e (the)386 2713 y(formal)d(reduction)f(to)j(its)e(linear)f(part)h (requires)f(a)i(nonresonance)g(condition)e(on)h(the)386 2821 y(eigen)m(v)-5 b(alues)41 b(of)g(the)h(linear)d(part,)44 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(ery)g(p)s(erturb)s(ed)d(system)386 5623 y(under)j(sim)m(ultaneous)g (Diophan)m(tine)g(conditions)g(for)h Fu(!)2308 5590 y Fq(1)2347 5623 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(!)2624 5590 y Ft(k)2691 5623 y Fw(on)25 b(the)f(set)h Fs(Z)3163 5590 y Ft(n)3214 5623 y Fp(n)8 b Fw(\000)3324 5590 y Ft(!)3324 5650 y Fl(Z)p eop %%Page: 3 3 3 2 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f(FIELDS) h(ON)g(TORI)594 b(3)386 434 y Fw(in)31 b(the)h(class)f Fu(C)938 401 y Fm(1)1044 434 y Fw(or)h(Gevrey)h Fu(G)1545 401 y Ft(\033)1623 434 y Fw(will)c(b)s(e)i(exhibited.)44 b(In)31 b(particular,)g(if)g(\000)3062 401 y Ft(!)3062 461 y Fl(Z)3140 434 y Fw(=)c Fp(f)p Fw(0)p Fp(g)386 549 y Fw(it)i(is)g(p)s(ossible)e(to)k(transform)e Fp(f)p Fu(b)1520 516 y Ft(k)1563 549 y Fp(g)1608 516 y Ft(d)1608 573 y Fq(1)1679 549 y Fw(\(resp)s(ectiv)m(ely)-8 b(,)30 b Fp(fL)2335 564 y Ft(k)2378 549 y Fp(g)2423 516 y Ft(d)2423 573 y Fq(1)2464 549 y Fw(\))g(sim)m(ultaneously)d(in)m(to)j(a)386 656 y(constan)m(t)39 b(p.d.o.)62 b(\(resp)s(ectiv)m(ely)-8 b(,)40 b(di\013eren)m(tial)c(op)s(erator\).)64 b(It)38 b(will)d(also)i(b)s(e)g(sho)m(wn)386 764 y(that)27 b(in)f(general)h (the)g(Diophan)m(tine)f(condition)f(is)h(necessary)i(to)f(transform)f (the)h(p)s(er-)386 872 y(turb)s(ed)32 b(systems)i(to)h(their)e(normal)g (forms.)52 b(W)-8 b(e)35 b(then)f(consider)f(the)h(sim)m(ultaneous)386 980 y(reduction)e(of)i(nonconstan)m(t)h(v)m(ector)g(\014elds)d(to)j (constan)m(t)f(ones)g(for)g(the)g(case)g Fu(n)d Fw(=)f(2.)386 1088 y(There)k(are)g(v)m(ery)h(few)e(results)g(regarding)h(the)g (normal)f(forms)h(of)g(v)m(ector)i(\014elds)c(and)386 1196 y(di\013eren)m(tial)d(op)s(erators)h(on)g Fs(T)1428 1163 y Ft(n)1507 1196 y Fw(except)h(for)f(those)h(v)m(ector)g(\014elds) e(on)h Fs(T)2859 1163 y Fq(2)2930 1196 y Fw(of)h(the)f(form)386 1304 y Fu(@)434 1318 y Ft(x)474 1327 y Fo(1)540 1304 y Fw(+)e Fu(\025)p Fw(\()p Fu(x)p Fw(\))p Fu(@)862 1318 y Ft(x)902 1327 y Fo(2)941 1304 y Fw(,)45 b(where)c(the)h(rotation)g(n) 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2060 y Fw(is)f(globally)g Fu(C)1372 2027 y Fm(1)1472 2060 y Fw(h)m(yp)s(o)s(elliptic.)36 b(When)26 b(the)g(rotation)h(n)m(um)m(b)s (ers)d(satisfy)386 2167 y(sim)m(ultaneous)32 b(Diophan)m(tine)h (conditions)f(a)i(sim)m(ultaneous)e(reduction)g(to)i(constan)m(t)386 2275 y(v)m(ector)e(\014elds)d(will)e(b)s(e)j(sho)m(wn.)486 2383 y(The)i(second)h(part)g(of)h(this)e(pap)s(er)g(is)g(related)h(to)h (the)f(regularit)m(y)f(of)i(conjugating)386 2491 y(p.d.o's.)49 b(if)32 b(the)i(lo)m(w)m(er)f(order)g(p)s(erturbations)e(are)i(Gevrey)h Fu(G)2560 2458 y Ft(\033)2640 2491 y Fw(p.d.o's)f(.)49 b(In)32 b(fact,)j(a)386 2599 y(loss)28 b(of)i(Gevrey)g(regularit)m(y)e (due)h(to)h(the)f(sim)m(ultaneous)f(Diophan)m(tine)g(conditions)g(is) 386 2707 y(encoun)m(tered.)43 b(Although)29 b(w)m(e)j(are)f(not)g(able) g(to)g(pro)m(v)m(e)h(that)f(our)g(Gevrey)g(estimates)386 2815 y(are)24 b(b)s(est)g(p)s(ossible)d(there)j(is)f(some)h(evidence)g (to)h(suggest)f(this)f(from)g(the)i(sharp)d(loss)h(of)386 2923 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y(ptional)e(sets)h(in)f(the)i(Gevrey)f(category) -8 b(,)46 b(the)40 b(standard)f(Hausdor\013)h(dimension)e(is)386 5412 y(insu\016cien)m(t,)d(hence)h(w)m(e)g(need)f(logarithmic)g (Hausdor\013)g(dimension;)h(secondly)-8 b(,)36 b(the)386 5520 y(inhomogeneit)m(y)30 b(of)h(the)f(Diophan)m(tine)f(conditions)g (causes)i(some)g(di\016culties.)p eop %%Page: 4 4 4 3 bop 386 239 a Fq(4)728 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g(YOSHINO) 486 434 y Fw(The)k(pap)s(er)g(is)g(organized)h(as)g(follo)m(ws:)49 b(section)35 b(2)h(deals)e(with)g(the)h(geometry)h(of)386 542 y(the)k(resonan)m(t)g(set.)70 b(Section)40 b(3)g(is)f(dev)m(oted)i (to)g(the)f(sim)m(ultaneous)e(normal)h(forms)386 650 y(and)h(the)i(Nekhoroshev)f(t)m(yp)s(e)g(estimates)h(are)g(sho)m(wn)e (in)g(section)h(4.)74 b(The)40 b(global)386 758 y(h)m(yp)s(o)s (elliptict)m(y)23 b(and)j(solv)-5 b(abilit)m(y)24 b(is)h(discussed)f (in)h(section)h(5.)40 b(Finally)-8 b(,)25 b(estimates)i(for)386 866 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b(We)38 b(c)-5 b(an)39 b(always)g(\014nd)g Fu(\024)3078 3335 y Ft(j)3153 3368 y Fk(\(after)386 3476 y(changes)33 b(of)g(indic)-5 b(es)33 b Fu(x)1186 3490 y Ft(\026)1258 3476 y Fp(!)25 b Fu(x)1426 3490 y Ft(\027)1469 3476 y Fk(\))33 b(to)g(satisfy)763 3669 y Fu(\024)815 3632 y Ft(j)877 3669 y Fw(=)25 b(\()p Fu(\024)1060 3625 y Ft(j)1060 3695 y Fq(1)1100 3669 y Fu(;)15 b(\024)1192 3625 y Ft(j)1192 3695 y Fq(2)1233 3669 y Fu(;)g(:)g(:)g(:)31 b(;)15 b(\024)1501 3632 y Ft(j)1501 3692 y(n)1549 3669 y Fw(\))p Fu(;)202 b(\024)1863 3625 y Ft(j)1863 3696 y(j)1925 3669 y Fp(2)25 b Fs(N)6 b Fu(;)37 b(\024)2184 3632 y Ft(j)2184 3692 y(\027)2253 3669 y Fw(=)25 b(0)p Fu(;)15 b Fw(1)26 b Fp(\024)f Fu(\027)31 b Fp(\024)25 b Fu(j)h Fp(\000)20 b Fw(1)p Fu(:)-2611 b Fw(\(2.2\))386 3856 y Fk(Every)32 b(other)i(b)-5 b(asis)33 b Fp(f)s Fw(~)-48 b Fu(\024)1197 3823 y Ft(\027)1241 3856 y Fp(g)1286 3823 y Ft(r)1286 3880 y(\027)t Fq(=1)1453 3856 y Fk(is)32 b(r)-5 b(epr)g(esente)g(d)35 b(in)e(a)g(unique)e(way)j(in)f(the)g(form) 386 3965 y Fw(\()s(~)-48 b Fu(\024)473 3932 y Fq(1)513 3965 y Fu(;)15 b Fp(\001)g(\001)g(\001)32 b Fu(;)19 b Fw(~)-49 b Fu(\024)782 3932 y Ft(r)821 3965 y Fw(\))25 b(=)g(\()p Fu(\024)1064 3932 y Fq(1)1104 3965 y Fu(;)15 b Fp(\001)g(\001)g(\001)32 b Fu(;)15 b(\024)1373 3932 y Ft(r)1412 3965 y Fw(\))p Fu(Q)33 b Fk(for)g(some)g Fu(Q)25 b Fp(2)g Fu(S)5 b(L)2235 3932 y Fm(\006)2294 3965 y Fw(\()p Fu(r)s Fw(;)15 b Fs(Z)p Fw(\))p Fk(.)486 4072 y(ii\))43 b(L)-5 b(et)44 b Fu(\024)842 4039 y Fq(1)881 4072 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(\024)1150 4039 y Ft(r)1232 4072 y Fk(b)-5 b(e)43 b(a)h(b)-5 b(asis)44 b(of)g Fw(\000)1848 4039 y Ft(!)1848 4100 y Fl(Z)1898 4072 y Fk(.)74 b(Then)44 b(ther)-5 b(e)45 b(exists)e Fu(P)58 b Fp(2)45 b Fu(S)5 b(L)3090 4039 y Fq(+)3148 4072 y Fw(\()p Fu(n)p Fw(;)15 b Fs(Z)p Fw(\))386 4184 y Fk(such)42 b(that)i(its)e(\014rst)h Fu(r)h Fk(c)-5 b(olumns)43 b(c)-5 b(oincide)43 b(with)g Fu(\024)2201 4151 y Fq(1)2241 4184 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(\024)2510 4151 y Ft(r)2548 4184 y Fk(,)45 b(i.e.)69 b(we)43 b(extend)f(the)386 4292 y(b)-5 b(asis)43 b(of)g Fw(\000)788 4259 y Ft(!)788 4319 y Fl(Z)881 4292 y Fk(into)g(a)g(b)-5 b(asis)44 b(of)f Fs(Z)1579 4259 y Ft(n)1621 4292 y Fk(.)72 b(In)43 b(p)-5 b(articular,)47 b(if)42 b Fu(y)k Fw(=)2610 4259 y Ft(tr)2673 4292 y Fu(P)13 b(x)p Fk(,)45 b Fu(y)h Fw(=)d(\()p Fu(y)3157 4259 y Fm(0)3181 4292 y Fu(;)15 b(y)3269 4259 y Fm(00)3311 4292 y Fw(\))p Fk(,)386 4400 y Fu(y)434 4367 y Fm(0)482 4400 y Fw(=)25 b(\()p Fu(y)658 4414 y Fq(1)698 4400 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(y)959 4414 y Ft(r)997 4400 y Fw(\))p Fk(,)33 b Fu(y)1141 4367 y Fm(00)1209 4400 y Fw(=)25 b(\()p Fu(y)1385 4414 y Ft(r)r Fq(+1)1513 4400 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(y)1774 4414 y Ft(n)1821 4400 y Fw(\))p Fk(,)33 b(then)1073 4585 y Fu(P)13 b(!)1204 4547 y Ft(k)1272 4585 y Fw(=)25 b(\(0)1448 4547 y Fm(0)1472 4585 y Fu(;)15 b(\022)1558 4547 y Ft(k)1600 4585 y Fw(\))p Fu(;)202 b(\022)1908 4547 y Ft(k)1975 4585 y Fp(2)25 b Fs(R)2120 4547 y Ft(n)p Fm(\000)p Ft(r)2262 4585 y Fu(;)15 b Fw(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d)-2301 b Fw(\(2.3\))386 4779 y Fk(with)35 b Fw(\000)642 4746 y Ft(\022)642 4806 y Fl(Z)718 4779 y Fw(=)28 b Fp(f)p Fw(0)p Fp(g)p Fk(,)35 b(i.e.)46 b Fu(\022)1233 4746 y Fq(1)1271 4779 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(\022)1534 4746 y Ft(k)1610 4779 y Fk(ar)-5 b(e)35 b(nonr)-5 b(esonant)37 b(in)d Fs(R)2460 4746 y Ft(n)p Fm(\000)p Ft(r)2601 4779 y Fk(.)46 b(In)34 b(the)h Fu(y)i Fk(variables)386 4893 y(the)42 b(r)-5 b(esonant)43 b(set)f Fw(\000)1133 4860 y Ft(P)10 b(!)1133 4920 y Fl(Z)1279 4893 y Fk(c)-5 b(oincides)42 b(with)g Fs(Z)1949 4860 y Ft(r)1949 4921 y(y)1986 4902 y Fx(0)2009 4893 y Fk(.)68 b(The)42 b(set)f(of)h(al)5 b(l)42 b(matric)-5 b(es)43 b Fu(P)54 b Fk(fr)-5 b(om)386 5010 y Fu(S)5 b(L)p Fw(\()p Fu(n)p Fw(;)15 b Fs(Z)p Fw(\))28 b Fk(with)33 b(such)f(a)g(pr)-5 b(op)g(erty)35 b(is)d(isomorphic)i(to)e Fp(M)2382 5024 y Ft(n)p Fm(\000)p Ft(r)r Fm(\002)p Ft(r)2607 5010 y Fw(\()p Fs(Z)p Fw(\))15 b Fp(\010)j Fu(S)5 b(L)p Fw(\()p Fu(n)19 b Fp(\000)f Fu(r)s Fw(;)d Fs(Z)p Fw(\))p Fk(,)386 5118 y(wher)-5 b(e)37 b Fp(M)755 5132 y Ft(n)p Fm(\000)p Ft(r)r Fm(\002)p Ft(r)980 5118 y Fw(\()p Fs(Z)p Fw(\))c Fk(stands)38 b(for)f(the)f(gr)-5 b(oup)38 b(of)f Fu(n)22 b Fp(\000)h Fu(r)j Fp(\002)c Fu(r)39 b Fk(matric)-5 b(es)38 b(with)f(inte)-5 b(gr)g(al)386 5226 y(entries.)486 5407 y Fw(In)34 b(order)h(to)h(pro)m(v)m(e)g(the)f(theorem)h(w)m(e)g(recall) e(the)i(follo)m(wing)d(w)m(ell)h(kno)m(wn)h(asser-)386 5515 y(tion)d(\(cf.)47 b([30)q(])33 b(for)f(more)g(general)h(statemen)m (ts)h(on)e(sequences)h(of)f(maps)g(for)g(\014nitely)386 5623 y(generated)f(ab)s(elian)e(groups\))p eop %%Page: 5 5 5 4 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f(FIELDS) h(ON)g(TORI)594 b(5)386 434 y FE(Lemma)33 b(2.2.)46 b Fk(L)-5 b(et)38 b Fu(A)d Fp(\032)f Fu(B)40 b Fw(:=)35 b Fs(Z)1640 401 y Ft(n)1720 434 y Fk(b)-5 b(e)38 b(a)g(sub)-5 b(gr)g(oup.)59 b(Set)38 b Fu(C)j Fw(=)35 b Fu(A)2788 401 y Fm(?)2788 461 y Fl(Z)2882 434 y Fw(=)f Fp(f)p Fu(\030)40 b Fp(2)34 b Fs(Z)3272 401 y Ft(n)3349 434 y Fw(:)386 542 y Fu(\030)29 b Fp(?)c Fu(A)p Fp(g)p Fk(.)42 b(Then)1658 695 y Fu(B)29 b Fw(=)c Fu(A)c Fp(\010)f Fu(C)-1711 b Fw(\(2.4\))386 848 y Fk(holds)34 b(if)f(and)g(only)h(if)e(the)h(se)-5 b(quenc)g(e)1445 1025 y Fw(0)26 b Fp(!)f Fu(A)1741 974 y Ft(id)1726 1025 y Fp(!)g Fu(B)1965 974 y Ft(\031)1941 1025 y Fp(!)g Fu(C)32 b Fp(!)25 b Fw(0)-1929 b(\(2.5\))386 1184 y Fk(is)29 b(exact)h(\(i.e.)41 b Fu(K)7 b(er)s(\031)28 b Fw(=)c Fu(id)p Fw(\()p Fu(A)p Fw(\))p Fk(\))32 b(wher)-5 b(e)30 b Fu(\031)j Fk(\(r)-5 b(esp)g(e)g(ctively,)31 b Fu(id)p Fk(\))f(stands)h(for)f(the)g(ortho-)386 1291 y(gonal)k(pr)-5 b(oje)g(ction)34 b(on)f Fu(C)39 b Fk(\(r)-5 b(esp)g(e)g(ctively,)34 b(the)f(identity)h(map\).)386 1458 y(Pr)-5 b(o)g(of.)43 b Fw(The)36 b(existence)h(of)f(a)h(basis)e (follo)m(ws)g(from)h(the)h(w)m(ell)e(kno)m(wn)h(fact)h(that)g(an)m(y) 386 1565 y(subgroup)e Fu(A)h Fw(\(in)f(our)h(case)i Fu(A)d Fw(=)g(\000)1671 1532 y Ft(!)1671 1593 y Fl(Z)1721 1565 y Fw(\))i(of)f(a)h(\014nitely)d(generated)k(ab)s(elian)c(group)i Fu(B)386 1673 y Fw(\(in)k(our)g(case)i Fu(B)47 b Fw(=)c Fs(Z)1208 1640 y Ft(n)1250 1673 y Fw(\))e(is)f(also)h(a)g(\014nitely)e (generated)j(ab)s(elian)d(group)i(cf.)72 b([30)q(].)386 1781 y(De\014ne)40 b Fu(\031)729 1795 y Ft(j)806 1781 y Fw(:)i Fs(R)932 1748 y Ft(n)1026 1781 y Fp(!)f Fs(R)1218 1748 y Ft(n)1311 1781 y Fw(b)m(y)f Fu(\031)1499 1795 y Ft(j)1535 1781 y Fw(\()p Fu(\030)t Fw(\))i(=)p Fu(<)e(\030)t(;)25 b(~)-55 b(e)1999 1795 y Ft(j)2078 1781 y Fu(>)50 b(~)-55 b(e)2231 1795 y Ft(j)2309 1781 y Fw(=)41 b Fu(\030)2461 1795 y Ft(j)2507 1781 y Fu(~)-55 b(e)2539 1795 y Ft(j)2576 1781 y Fw(,)43 b(where)49 b Fu(~)-55 b(e)2958 1795 y Ft(j)3035 1781 y Fw(is)39 b(the)h Fu(j)5 b Fw(-)386 1889 y(th)41 b(unit)f(v)m(ector.)76 b(Without)41 b(loss)g(of)g(generalit)m (y)h(w)m(e)g(ma)m(y)g(assume)f(that)h(for)f(eac)m(h)386 1997 y Fu(j)50 b Fp(2)44 b(f)p Fw(1)p Fu(;)15 b(:)g(:)g(:)33 b(;)15 b(n)p Fp(g)42 b Fw(w)m(e)g(ha)m(v)m(e)h Fu(\031)1446 2011 y Ft(j)1482 1997 y Fu(!)1542 1964 y Ft(k)1629 1997 y Fp(6)p Fw(=)h(0)f(for)e(some)h Fu(k)s Fw(,)k(otherwise)41 b(w)m(e)h(are)g(reduced)386 2105 y(to)j(a)f(space)g(of)g(dimension)d Fu(n)29 b Fp(\000)g Fw(1.)82 b(Because)45 b(\000)2170 2072 y Ft(!)2170 2132 y Fl(Z)2264 2105 y Fw(is)e(a)h(linear)e Fs(Z)p Fw({submo)s(dul)o(e)c(of)386 2213 y Fs(Z)451 2180 y Ft(n)493 2213 y Fw(,)47 b Fu(\031)617 2227 y Fq(1)657 2213 y Fw(\000)714 2180 y Ft(!)714 2240 y Fl(Z)807 2213 y Fw(is)c(closed)g(under)f(summation,)j(and)e(therefore)h(there)f (exists)g Fu(p)3115 2227 y Fq(1)3202 2213 y Fp(2)j Fs(N)386 2324 y Fw(suc)m(h)39 b(that)h Fu(\031)858 2338 y Fq(1)897 2324 y Fw(\(\000)989 2291 y Ft(!)989 2352 y Fl(Z)1040 2324 y Fw(\))g(=)g Fu(p)1272 2338 y Fq(1)1311 2324 y Fs(Z)11 b Fu(~)-56 b(e)1418 2338 y Fq(1)1453 2324 y Fw(.)68 b(Let)39 b Fu(\024)1769 2291 y Fq(1)1849 2324 y Fp(2)h Fw(\000)2007 2291 y Ft(!)2007 2352 y Fl(Z)2096 2324 y Fw(b)s(e)f(suc)m(h)g(that)h Fu(\031)2701 2338 y Fq(1)2740 2324 y Fu(\024)2792 2291 y Fq(1)2872 2324 y Fw(=)g Fu(p)3029 2338 y Fq(1)3079 2324 y Fu(~)-56 b(e)3110 2338 y Fq(1)3150 2324 y Fw(.)67 b(Let)386 2432 y(\003)449 2446 y Fq(2)514 2432 y Fp(\032)25 b Fw(\000)667 2399 y Ft(!)667 2460 y Fl(Z)746 2432 y Fw(b)s(e)j(the)h Fs(Z)p Fw({submo)s(du)o(le)23 b(suc)m(h)28 b(that)h Fu(\031)2035 2446 y Fq(1)2075 2432 y Fw(\003)2138 2446 y Fq(2)2203 2432 y Fw(=)c Fp(f)p Fw(0)p Fp(g)p Fw(.)41 b(In)28 b(fact,)i(\003)2878 2446 y Fq(2)2947 2432 y Fw(is)d(uniquely)386 2540 y(determined:)55 b(\003)975 2554 y Fq(2)1053 2540 y Fw(=)38 b Fp(f)p Fw(\()p Fu(\030)1282 2554 y Fq(1)1322 2540 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(\030)1579 2554 y Ft(n)1626 2540 y Fw(\))39 b Fp(2)f Fw(\000)1856 2507 y Ft(!)1856 2568 y Fl(Z)1944 2540 y Fw(:)54 b Fu(\030)2063 2554 y Fq(1)2141 2540 y Fw(=)38 b(0)p Fp(g)p Fw(.)65 b(If)38 b(\003)2592 2554 y Fq(2)2670 2540 y Fw(=)g Fp(f)p Fw(0)p Fp(g)h Fw(the)g(set)f(\000)3324 2507 y Ft(!)3324 2568 y Fl(Z)386 2652 y Fw(is)f(generated)i(b)m(y)e Fu(\024)1092 2619 y Fq(1)1170 2652 y Fw(and)g Fu(r)j Fw(=)d(1.)64 b(If)37 b(\003)1838 2666 y Fq(2)1915 2652 y Fp(6)p Fw(=)h Fp(f)p Fw(0)p Fp(g)p Fw(,)j(a)d(similar)d(argumen)m(t)j (as)g(ab)s(o)m(v)m(e)386 2760 y(sho)m(ws)d(that)h Fu(\031)906 2774 y Fq(2)945 2760 y Fw(\003)1008 2774 y Fq(2)1082 2760 y Fw(=)d Fu(p)1232 2774 y Fq(2)1271 2760 y Fs(Z)11 b Fu(~)-56 b(e)1378 2774 y Fq(2)1449 2760 y Fw(for)35 b(some)h Fu(p)1872 2774 y Fq(2)1945 2760 y Fp(2)e Fs(N)6 b Fw(.)62 b(Let)36 b Fu(\024)2406 2727 y Fq(2)2480 2760 y Fp(2)d Fw(\003)2637 2774 y Fq(2)2712 2760 y Fw(b)s(e)i(a)h(v)m(ector) h(suc)m(h)386 2868 y(that)f Fu(\031)640 2882 y Fq(2)680 2868 y Fu(\024)732 2835 y Fq(2)806 2868 y Fw(=)e Fu(p)957 2882 y Fq(2)1007 2868 y Fu(~)-56 b(e)1038 2882 y Fq(2)1078 2868 y Fw(.)56 b(Clearly)35 b Fu(\024)1533 2835 y Fq(2)1607 2868 y Fw(=)f(\(0)p Fu(;)15 b(p)1878 2882 y Fq(2)1918 2868 y Fu(;)g(:)g(:)g(:)i Fw(\).)57 b(Let)36 b(\003)2428 2882 y Fq(3)2502 2868 y Fp(\032)e Fw(\003)2670 2882 y Fq(2)2745 2868 y Fw(b)s(e)h(the)h(uniquely)386 2975 y(determined)f Fs(Z)p Fw({submo)s(du)o(le)30 b(suc)m(h)36 b(that)h Fu(\031)1902 2989 y Fq(2)1941 2975 y Fw(\003)2004 2989 y Fq(3)2078 2975 y Fw(=)e Fp(f)p Fw(0)p Fp(g)p Fw(.)59 b(If)36 b(\003)2563 2989 y Fq(3)2638 2975 y Fw(=)e Fp(f)p Fw(0)p Fp(g)k Fw(w)m(e)e(see)h (that)386 3083 y(\003)449 3097 y Fq(2)530 3083 y Fw(is)j(generated)i(b) m(y)f Fu(\024)1246 3050 y Fq(2)1327 3083 y Fw(and)g Fu(r)46 b Fw(=)d(2.)73 b(Therefore)41 b(\000)2339 3050 y Ft(!)2339 3111 y Fl(Z)2431 3083 y Fw(is)f(generated)i(b)m(y)f Fu(\024)3147 3050 y Fq(1)3228 3083 y Fw(and)386 3198 y Fu(\024)438 3165 y Fq(2)478 3198 y Fw(.)77 b(If)42 b(\003)746 3212 y Fq(3)831 3198 y Fp(6)p Fw(=)k Fp(f)p Fw(0)p Fp(g)e Fw(this)d(pro)s(cedure)h(is)f(con)m(tin)m(ued)h(and)g(sequences)h Fp(f)p Fu(\024)2991 3165 y Ft(k)3035 3198 y Fp(g)j(\032)f Fw(\000)3299 3165 y Ft(!)3299 3225 y Fl(Z)3349 3198 y Fw(,)386 3306 y Fp(f)p Fu(p)477 3321 y Ft(k)520 3306 y Fp(g)36 b(\032)f Fs(N)49 b Fw(are)36 b(constructed)h(together)h(with) d(a)i(nested)f(family)f(of)i Fs(Z)p Fw({submo)s(du)o(l)o(es)386 3416 y Fp(f)p Fw(\003)494 3431 y Ft(k)537 3416 y Fp(g)p Fw(,)46 b(\003)716 3431 y Ft(k)805 3416 y Fp(\032)f Fw(\003)984 3431 y Ft(k)r Fm(\000)p Fq(1)1159 3416 y Fw(suc)m(h)d(that)i Fu(\031)1638 3431 y Ft(k)1680 3416 y Fw(\003)1743 3431 y Ft(k)1831 3416 y Fw(=)i Fu(p)1994 3431 y Ft(k)2036 3416 y Fs(Z)13 b Fu(~)-58 b(e)2143 3431 y Ft(k)2182 3416 y Fw(,)45 b Fu(\031)2304 3431 y Ft(k)2347 3416 y Fu(\024)2399 3383 y Ft(k)2487 3416 y Fw(=)g Fu(p)2649 3431 y Ft(k)2705 3416 y Fu(~)-58 b(e)2734 3431 y Ft(k)2777 3416 y Fw(.)77 b(Clearly)-8 b(,)45 b(this)386 3524 y(pro)s(cedure)27 b(ends)h(after)h(a)f(\014nite)g(n)m(um)m(b)s(er)f(of)h(steps.)40 b(Let)29 b Fu(r)f Fp(\024)d Fu(n)16 b Fp(\000)f Fw(1)29 b(b)s(e)f(the)h(smallest)386 3632 y(in)m(teger)36 b(suc)m(h)f(that)i (\003)1168 3646 y Ft(r)r Fq(+1)1330 3632 y Fw(=)d Fp(f)p Fw(0)p Fp(g)p Fw(,)39 b(then)c(the)h(v)m(ectors)h Fu(\024)2374 3599 y Ft(j)2447 3632 y Fw(form)e(a)h(basis)f(of)g(\000)3142 3599 y Ft(!)3142 3659 y Fl(Z)3228 3632 y Fw(and)386 3740 y(satisfy)30 b(\(2.2\).)486 3848 y(Let)j(no)m(w)g Fp(f)s Fw(~)-48 b Fu(\024)940 3815 y Ft(j)977 3848 y Fp(g)1022 3815 y Ft(r)1022 3872 y Fq(1)1092 3848 y Fp(\032)29 b Fw(\000)1249 3815 y Ft(!)1249 3875 y Fl(Z)1332 3848 y Fw(b)s(e)j(another)h(basis)f(of)h(\000)2183 3815 y Ft(!)2183 3875 y Fl(Z)2233 3848 y Fw(.)48 b(Then)32 b(there)h(exist)g(t)m(w)m(o)h Fu(r)25 b Fp(\002)c Fu(r)386 3967 y Fw(matrices)34 b Fu(Q)f Fw(and)1060 3944 y(~)1039 3967 y Fu(Q)h Fw(with)e(in)m(tegral)h (en)m(tries)h(suc)m(h)f(that)h(\()p Fu(\024)2479 3934 y Fq(1)2535 3967 y Fp(\001)15 b(\001)g(\001)h Fu(\024)2708 3934 y Ft(r)2746 3967 y Fw(\))p Fu(Q)31 b Fw(=)f(\()s(~)-48 b Fu(\024)3072 3934 y Fq(1)3128 3967 y Fp(\001)15 b(\001)g(\001)k Fw(~)-48 b Fu(\024)3301 3934 y Ft(r)3339 3967 y Fw(\))386 4082 y(and)36 b(\()s(~)-48 b Fu(\024)656 4049 y Fq(1)711 4082 y Fp(\001)15 b(\001)g(\001)20 b Fw(~)-48 b Fu(\024)885 4049 y Ft(r)923 4082 y Fw(\))979 4059 y(~)958 4082 y Fu(Q)36 b Fw(=)g(\()p Fu(\024)1260 4049 y Fq(1)1315 4082 y Fp(\001)15 b(\001)g(\001)i Fu(\024)1489 4049 y Ft(r)1527 4082 y Fw(\).)60 b(Hence)1945 4059 y(~)1924 4082 y Fu(Q)36 b Fw(=)f Fu(Q)2210 4049 y Fm(\000)p Fq(1)2341 4082 y Fw(and)i(det)15 b Fu(Q)g Fw(det)2915 4059 y(~)2895 4082 y Fu(Q)35 b Fw(=)h(1.)61 b(Be-)386 4197 y(cause)34 b(det)15 b Fu(Q)33 b Fw(and)g(det)1220 4174 y(~)1200 4197 y Fu(Q)g Fw(are)g(in)m(tegers)h(it)f(follo)m(ws)f(that)i(det)16 b Fu(Q)30 b Fw(=)g(det)2901 4174 y(~)2880 4197 y Fu(Q)g Fw(=)g Fp(\006)p Fw(1,)35 b(i.e.)386 4306 y Fu(Q;)519 4283 y Fw(~)498 4306 y Fu(Q)e Fp(2)f Fu(S)5 b(L)819 4273 y Fm(\006)878 4306 y Fw(\()p Fu(r)s Fw(;)15 b Fs(Z)p Fw(\).)51 b(Con)m(v)m(ersely)-8 b(,)36 b(for)f(an)m(y)g Fu(Q)e Fp(2)f Fu(S)5 b(L)2297 4273 y Fm(\006)2356 4306 y Fw(\()p Fu(r)s Fw(;)15 b Fs(Z)p Fw(\))31 b(\()p Fu(\024)2693 4273 y 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b(basis)e(in)h(the)g(v)m(ector)j(subspace)d Fu(C)1638 5367 y Fm(?)1631 5427 y Fl(R)1721 5400 y Fw(and)g Fu(\032)g Fw(is)g(a)h(linear)e(com)m(bination)h(of)g Fu(\024)3027 5367 y Fq(1)3067 5400 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(\024)3336 5367 y Ft(r)386 5515 y Fw(implying)32 b(that)k Fu(<)d(!)1133 5482 y Ft(k)1176 5515 y Fu(;)15 b(\032)34 b(>)p Fw(=)f(0)j(for)f(all)f(1)g Fp(\024)g Fu(k)i Fp(\024)e Fu(d)p Fw(.)56 b(Since)34 b Fu(\032)g Fp(2)f Fs(Z)2810 5482 y Ft(n)2852 5515 y Fw(,)k(this)d(implies)386 5623 y Fu(\032)25 b Fp(2)g Fu(A)p Fw(.)p eop %%Page: 6 6 6 5 bop 386 239 a Fq(6)728 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g(YOSHINO) 486 434 y Fw(Next,)43 b(note)e(that)g Fu(\024)1216 401 y Fq(1)1256 434 y Fu(;)15 b(:)g(:)g(:)h(\024)1469 401 y Ft(r)1548 434 y Fw(b)s(ecome)40 b(the)h(\014rst)e Fu(r)k Fw(unit)38 b(v)m(ectors)k(in)d Fs(R)3027 401 y Ft(n)3027 456 y(y)3080 434 y Fw(.)70 b(Th)m(us)386 556 y(the)40 b(in)m(v)-5 b(ariance)38 b(of)i(the)g(resonan)m(t)g(set)g(implies)d (that)j(\000)2377 523 y Ft(P)10 b(!)2377 583 y Fl(Z)2522 556 y Fw(=)40 b Fp(f)p Fu(\020)48 b Fp(2)40 b Fs(Z)2932 523 y Ft(n)3015 556 y Fw(:)56 b Fu(\020)3136 570 y Ft(r)r Fq(+1)3304 556 y Fw(=)386 670 y Fp(\001)15 b(\001)g(\001)26 b Fw(=)f Fu(\020)653 684 y Ft(n)725 670 y Fw(=)g(0)p Fp(g)j Fw(and)e(\(2.3\))j(is)c(therefore)j(true.)39 b(If)26 b Fu(\022)2151 637 y Fq(1)2190 670 y Fu(;)15 b(:)g(:)g(:)i(\022)2398 637 y Ft(n)p Fm(\000)p Ft(r)2559 670 y Fw(are)28 b(resonan)m(t)f(in)e Fs(R)3233 632 y Ft(n)p Fm(\000)p Ft(r)3233 700 y(y)3270 681 y Fx(0)7 b(0)386 800 y Fw(there)36 b(will)d(exist)j(a)g(v)m(ector)i Fu(\020)1429 767 y Fq(0)1502 800 y Fp(2)c Fs(Z)1662 767 y Ft(n)p Fm(\000)p Ft(d)1820 800 y Fp(n)24 b(f)p Fw(0)p Fp(g)38 b Fw(orthogonal)e(to)h Fu(\022)2685 767 y Fq(1)2723 800 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(\022)2986 767 y Ft(k)3028 800 y Fw(,)38 b(so)e(that)386 909 y(\(0)466 876 y Fm(0)490 909 y Fu(;)15 b(\020)577 876 y Fq(0)616 909 y Fw(\))26 b Fp(2)f Fw(\000)820 876 y Ft(P)10 b(!)820 937 y Fl(Z)925 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Fm(\002)p Ft(r)1105 1343 y Fw(\()p Fs(Z)p Fw(\).)p 3309 1343 4 62 v 3313 1285 55 4 v 3313 1343 V 3367 1343 4 62 v 486 1543 a(W)-8 b(e)22 b(call)g(the)f(basis)g Fu(\024)1206 1510 y Fq(1)1246 1543 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(\024)1515 1510 y Ft(r)1575 1543 y Fw(a)22 b(canonical)g(basis)e(of)i (\000)2399 1510 y Ft(!)2399 1570 y Fl(Z)2449 1543 y Fw(,)i(and)d Fu(r)j Fw(a)e(\()p Fs(Z)p Fw({\)dimension)386 1651 y(of)31 b(\000)547 1618 y Ft(!)547 1678 y Fl(Z)628 1651 y Fw(o)m(v)m(er)i Fs(Z)27 b Fw(and)j(write)g(dim)1478 1665 y Fl(Z)1541 1651 y Fw(\000)1598 1618 y Ft(!)1598 1678 y Fl(Z)1675 1651 y Fw(=)c Fu(r)s Fw(.)43 b(W)-8 b(e)32 b(stress)f(that)h Fs(Z)2554 1618 y Ft(n)2617 1651 y Fp(n)21 b Fw(\000)2740 1618 y Ft(!)2740 1678 y Fl(Z)2821 1651 y Fw(is)30 b(not)i(a)f(linear) 386 1759 y Fs(Z)26 b Fw(submo)s(dule)h(of)k Fs(Z)1103 1726 y Ft(n)1146 1759 y Fw(.)386 1939 y FE(Remark)j(2.3.)46 b Fk(T)-7 b(ake)44 b Fu(A)i Fw(=)f Fp(f)p Fw(\(2)p Fu(t;)15 b Fp(\000)p Fw(2)p Fu(t)p Fw(\);)32 b Fu(t)46 b Fp(2)f Fs(Z)p Fp(g)p Fk(,)d Fu(C)52 b Fw(=)45 b Fu(A)2558 1906 y Fm(?)2558 1966 y Fl(Z)2663 1939 y Fw(=)g Fp(f)p Fw(\()p Fu(t;)15 b(t)p Fw(\);)32 b Fu(t)45 b Fp(2)g Fs(Z)p Fp(g)p Fk(.)386 2050 y(Then)36 b(although)i Fu(A)p Fk(,)f Fu(B)g Fw(=)31 b Fs(Z)1398 2017 y Fq(2)1469 2050 y Fk(and)37 b Fu(C)43 b Fk(ar)-5 b(e)37 b(\014nitely)f(gener)-5 b(ate)g(d)38 b(ab)-5 b(elian)37 b(gr)-5 b(oups,)38 b(the)386 2158 y(de)-5 b(c)g(omp)g(osition)36 b(\(2.4\))e(is)e(not)i(true)e(\(the)h (se)-5 b(quenc)g(e)33 b(\(2.5\))h(is)e(not)h(exact)g(b)-5 b(e)g(c)g(ause)386 2268 y Fu(C)458 2235 y Fm(?)451 2296 y Fl(Z)542 2268 y Fp(6)p Fw(=)25 b Fu(A)p Fk(\).)486 2455 y Fw(W)-8 b(e)45 b(sa)m(y)f(that)h(v)m(ectors)g Fu(!)1419 2422 y Ft(k)1510 2455 y Fp(2)i Fs(R)1678 2422 y Ft(n)1731 2455 y Fw(,)g(\(1)i Fp(\024)e Fu(k)k Fp(\024)d Fu(d)p Fw(\))d(satisfy)e(the)h(sim)m(ultaneous)386 2563 y(resonan)m(t)31 b Fu(\033)s Fw(-Siegel)f(condition)f(if,)h(for)g(ev)m (ery)h Fu(")25 b(>)g Fw(0)1091 2791 y(lim)15 b(inf)954 2856 y Fm(j)p Ft(\030)s Fm(j!1)p Ft(;\030)s Fm(2)p Fl(Z)1319 2837 y FA(n)1355 2856 y Fm(n)p Fq(\000)1434 2833 y FA(!)1434 2879 y Fj(Z)1495 2663 y Fr(\022)1562 2791 y Fu(e)1604 2753 y Ft(")p Fm(j)p Ft(\030)s Fm(j)1711 2730 y Fo(1)p FA(=\033)1858 2791 y Fw(max)1833 2852 y Fq(1)p Fm(\024)p Ft(k)r Fm(\024)p Ft(d)2068 2791 y Fp(j)25 b Fu(<)g(!)2274 2753 y Ft(k)2317 2791 y Fu(;)15 b(\030)29 b(>)c Fp(j)2547 2663 y Fr(\023)2640 2791 y Fu(>)g Fw(0)p Fu(;)359 b Fw(\()p Fu(S)5 b(i)p Fw(\))3327 2805 y Ft(\033)386 3029 y Fw(for)30 b(1)c Fp(\024)f Fu(\033)j(<)d Fp(1)p Fw(,)31 b(and)1150 3257 y(lim)15 b(inf)1014 3322 y Fm(j)p Ft(\030)s Fm(j!1)p Ft(;\030)s Fm(2)p Fl(Z)1378 3303 y FA(n)1414 3322 y Fm(n)p Fq(\000)1493 3299 y FA(!)1493 3345 y Fj(Z)1554 3129 y Fr(\022)1621 3257 y Fp(j)p Fu(\030)t Fp(j)1715 3219 y Ft(\034)1799 3257 y Fw(max)1773 3318 y Fq(1)p Fm(\024)p Ft(k)r Fm(\024)p Ft(d)2008 3257 y Fp(j)26 b Fu(<)f(!)2215 3219 y Ft(k)2257 3257 y Fu(;)15 b(\030)30 b(>)25 b Fp(j)2488 3129 y 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Fl(Z)1013 4171 y FA(n)p Fx(\000)p FA(r)1143 3996 y Fr(\022)1210 4124 y Fu(e)1252 4086 y Ft(")p Fm(j)p Ft(\030)1339 4063 y Fx(0)o(0)1379 4086 y Fm(j)1399 4063 y Fo(1)p FA(=\033)1547 4124 y Fw(max)1521 4185 y Fq(1)p Fm(\024)p Ft(k)r Fm(\024)p Ft(d)1757 4124 y Fp(j)25 b Fu(<)g(\022)1949 4086 y Ft(k)1991 4124 y Fu(;)15 b(\030)2075 4086 y Fm(00)2143 4124 y Fu(>)25 b Fp(j)2264 3996 y Fr(\023)2356 4124 y Fu(>)g Fw(0)p Fu(;)202 b Fp(8)p Fu(")25 b(>)g Fw(0)182 b(\()p Fu(S)5 b(i)p Fw(\))3327 4091 y Fm(0)3327 4146 y Ft(\033)386 4362 y Fk(\(r)-5 b(esp)g(e)g(ctively,)1115 4594 y Fw(lim)15 b(inf)961 4660 y Fm(j)p Ft(\030)1015 4641 y Fx(0)o(0)1054 4660 y Fm(j!1)p Ft(;\030)1270 4641 y Fx(0)o(0)1309 4660 y Fm(2)p Fl(Z)1406 4641 y FA(n)p Fx(\000)p FA(r)1536 4466 y Fr(\022)1603 4594 y Fp(j)p Fu(\030)1672 4557 y Fm(0)q(0)1715 4594 y Fp(j)1740 4557 y Ft(\034)1824 4594 y Fw(max)1799 4656 y Fq(1)p Fm(\024)p Ft(k)r Fm(\024)p Ft(d)2034 4594 y Fp(j)25 b Fu(<)g(\022)2226 4557 y Ft(k)2268 4594 y Fu(;)15 b(\030)2352 4557 y Fm(00)2420 4594 y Fu(>)25 b Fp(j)2541 4466 y Fr(\023)2633 4594 y Fu(>)g Fw(0)p Fu(;)303 b Fw(\()p Fu(S)5 b(i)p Fw(\))3264 4561 y Fm(0)3264 4617 y(1)3339 4594 y Fw(\))386 4848 y Fk(for)33 b(the)g(nonr)-5 b(esonant)35 b(ve)-5 b(ctors)34 b Fu(\022)1553 4815 y Fq(1)1592 4848 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(\022)1854 4815 y Ft(k)1929 4848 y Fk(in)33 b Fs(R)2100 4810 y Ft(n)p Fm(\000)p Ft(k)2100 4878 y(y)2138 4859 y Fx(0)6 b(0)2247 4848 y Fk(\).)486 4975 y(We)35 b(p)-5 b(oint)37 b(out)f(that)i(if)d Fu(d)d Fw(=)e(1)37 b Fk(and)g Fu(!)1831 4942 y Fq(1)1901 4975 y Fp(2)31 b Fs(R)2052 4942 y Ft(n)2141 4975 y Fk(is)36 b(nonr)-5 b(esonant,)39 b(i.e.)51 b Fw(\000)3027 4942 y Ft(!)3027 5002 y Fl(Z)3109 4975 y Fw(=)30 b Fp(f)p Fw(0)p Fp(g)p Fk(,)386 5083 y(then)h Fw(\()p Fu(S)5 b(i)p Fw(\))748 5097 y Fm(1)853 5083 y Fk(\(r)-5 b(esp)g(e)g(ctively,)33 b Fw(\()p Fu(S)5 b(i)p Fw(\))1554 5097 y Ft(\033)1602 5083 y Fk(,)30 b Fw(1)c Fp(\024)f Fu(\033)j(<)d Fp(1)p Fk(\))30 b(is)g(the)h(wel)5 b(l)30 b(known)i(smal)5 b(l)31 b(divisor)386 5191 y(c)-5 b(ondition)29 b(for)e Fu(!)977 5158 y Fq(1)1043 5191 y Fk(and)h(the)g(numb)-5 b(er)27 b Fu(\034)37 b Fk(in)27 b Fw(\()p Fu(S)5 b(i)p Fw(\))2023 5205 y Fm(1)2125 5191 y Fk(is)27 b(ne)-5 b(c)g(essarily)29 b(not)f(less)f(than)h Fu(n)8 b Fp(\000)g Fw(1)386 5299 y Fk(\(cf.)62 b Fw([35)q(])p Fk(,)p Fw([31)q(])40 b Fk(and)g Fw([37)q(])p Fk(\))g(\(r)-5 b(esp)g(e)g(ctively,)42 b(c)-5 b(oincides)41 b(with)f(the)g(Gevr)-5 b(ey)39 b(typ)-5 b(e)41 b(Sie)-5 b(gel)386 5407 y(c)g(ondition)42 b(use)-5 b(d)41 b(in)f Fw([20)r(])p Fk(.)65 b(Se)-5 b(e)41 b(also)g Fw([34)r(])f Fk(and)i Fw([5])f Fk(for)g Fu(\033)i Fw(=)c(1)p Fk(\).)66 b(F)-7 b(urthermor)i(e,)45 b(if)386 5515 y Fu(n)34 b Fp(\025)g Fw(3)k Fk(and)h Fu(d)c Fw(=)f Fu(n)23 b Fp(\000)h Fw(1)p Fk(,)39 b Fw(\()p Fu(S)5 b(i)p Fw(\))1479 5529 y Fm(1)1593 5515 y Fk(in)37 b(the)h(nonr)-5 b(esonanc)g(e)40 b(c)-5 b(ase)39 b(is)e(the)i(same)f(as)g(the)386 5623 y(simultane)-5 b(ous)34 b(Diophantine)g(c)-5 b(ondition)35 b(use)-5 b(d)33 b(in)f Fw([6)q(])p Fk(.)p eop %%Page: 7 7 7 6 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f(FIELDS) h(ON)g(TORI)594 b(7)486 434 y Fw(Let)29 b Fu(C)719 401 y Fm(1)712 458 y Fq(2)p Ft(\031)794 434 y Fw(\()p Fs(R)889 401 y Ft(n)889 456 y(x)942 434 y Fw(\))g(b)s(e)f(the)g(space)i(of)e (all)g(2)p Fu(\031)k Fw(p)s(erio)s(dic)26 b Fu(C)2298 401 y Fm(1)2400 434 y Fw(functions)i(\(i.e.)40 b(the)29 b(lift)e(on)386 542 y Fs(R)446 509 y Ft(n)535 542 y Fw(of)37 b(all)f(functions)f(from)i Fu(C)1470 509 y Fm(1)1544 542 y Fw(\()p Fs(T)1638 509 y Ft(n)1687 542 y Fw(\)\).)60 b(It)37 b(is)f(readily)f(v)m(eri\014ed)h(that)i(this)d(space)j(is)386 650 y(in)m(v)-5 b(arian)m(t)29 b(under)g(the)i(action)g(of)f Fu(S)5 b(L)p Fw(\()p Fu(n)p Fw(;)15 b Fs(Z)p Fw(\),)27 b(namely)431 900 y Fu(f)10 b Fw(\()p Fu(y)s Fw(\))25 b Fp(2)g Fu(C)787 862 y Fm(1)780 922 y Fq(2)p Ft(\031)862 900 y Fw(\()p Fs(R)957 862 y Ft(n)957 922 y(y)1010 900 y Fw(\))h Fp(,)f Fu(P)1258 862 y Fm(\003)1297 900 y Fu(f)10 b Fw(\()p Fu(x)p Fw(\))25 b(:=)h Fu(f)10 b Fw(\()p Fu(P)j(x)p Fw(\))25 b Fp(2)g Fu(C)2052 862 y Fm(1)2045 922 y Fq(2)p Ft(\031)2127 900 y Fw(\()p Fs(R)2222 862 y Ft(n)2222 922 y(x)2275 900 y Fw(\))91 b(for)30 b(some)62 b Fu(P)38 b Fp(2)25 b Fu(S)5 b(L)p Fw(\()p Fu(n)p Fw(;)15 b Fs(Z)p Fw(\))386 787 y(\(2.7\))486 1051 y(W)-8 b(e)31 b(will)e(iden)m(tify)g Fu(C)1218 1018 y Fm(1)1292 1051 y Fw(\()p Fs(T)1386 1018 y Ft(n)1435 1051 y Fw(\))i(and)f Fu(G)1750 1018 y Ft(\033)1796 1051 y Fw(\()p Fs(T)1890 1018 y Ft(n)1939 1051 y Fw(\))h(with)f Fu(C)2285 1018 y Fm(1)2278 1075 y Fq(2)p Ft(\031)2360 1051 y Fw(\()p Fs(R)2455 1018 y Ft(n)2508 1051 y Fw(\))h(and)f Fu(G)2823 1018 y Ft(\033)2823 1075 y Fq(2)p Ft(\031)2905 1051 y Fw(\()p Fs(T)2999 1018 y Ft(n)3048 1051 y Fw(\),)h(resp)s(e-)386 1159 y(ctiv)m(ely)-8 b(.)754 1342 y(3.)51 b Fv(Global)33 b(canonical)g(f)n(orm)g(of)h(a)g(resonant)e(system)486 1504 y Fw(First)f(w)m(e)i(recall)e(the)i(basic)f(prop)s(erties)e(of)j Fu(G)2085 1471 y Ft(\033)2163 1504 y Fw(pseudo)s(di\013eren)m(tial)d (op)s(erators.)46 b(\()386 1612 y(cf.)60 b([7)q(],)39 b([33)q(],)g([15)q(])e(\).)60 b(In)36 b(the)h Fu(C)1557 1579 y Fm(1)1668 1612 y Fw(case,)j(the)d(constan)m(ts)h(in)d(the)i (estimates)g(b)s(elo)m(w)386 1720 y(ma)m(y)g(dep)s(end)f(on)g(the)i(m)m (ultiindices)33 b Fu(\013;)15 b(\014)42 b Fp(2)36 b Fs(Z)2075 1687 y Ft(n)2075 1743 y Fq(+)2167 1720 y Fw(and)g(the)h(index)f(of)h (homogeneit)m(y)386 1828 y Fu(j)5 b Fw(.)72 b(\(cf.)f([26)q(]\).)h(Let) 41 b Fu(F)13 b(S)1302 1795 y Ft(m)1297 1851 y(\033)1411 1828 y Fw(=)42 b Fu(F)13 b(S)1656 1795 y Ft(m)1651 1851 y(\033)1722 1828 y Fw(\()p Fs(T)1816 1795 y Ft(n)1892 1828 y Fp(\002)27 b Fs(R)2050 1795 y Ft(n)2103 1828 y Fw(\))40 b(b)s(e)g(the)h(set)g(of)g(all)e(formal)h(sums)386 1868 y Fr(P)482 1894 y Fm(1)482 1963 y Ft(j)t Fq(=0)624 1936 y Fu(a)672 1950 y Ft(m)p Fm(\000)p Ft(j)826 1936 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))31 b(suc)m(h)f(that)h Fu(a)1513 1950 y Ft(m)p Fm(\000)p Ft(j)1692 1936 y Fp(2)25 b Fu(G)1850 1903 y Ft(\033)1896 1936 y Fw(\()p Fs(T)1990 1903 y Ft(n)2059 1936 y Fp(\002)20 b Fw(\()p Fs(R)2245 1903 y Ft(n)2318 1936 y Fp(n)g(f)p Fw(0)p Fp(g)p Fw(\)\),)33 b Fu(or)s(d)2781 1951 y Ft(\030)2818 1936 y Fu(a)2866 1950 y Ft(m)p Fm(\000)p Ft(j)3046 1936 y Fw(=)24 b Fu(m)c Fp(\000)g Fu(j)386 2052 y Fw(\(i.e)31 b Fu(a)591 2066 y Ft(m)p Fm(\000)p Ft(j)775 2052 y Fw(is)f(\(p)s(ositiv)m(ely\))g (homogeneous)h(of)g(order)f Fu(m)21 b Fp(\000)f Fu(j)5 b Fw(\))32 b(and)e(that)h(there)g(exists)386 2160 y Fu(A)25 b(>)g Fw(0)31 b(satisfying)818 2311 y(sup)800 2390 y Ft(x)p Fm(2)p Fl(T)937 2371 y FA(n)989 2311 y Fp(j)p Fu(@)1067 2274 y Ft(\014)1062 2334 y(x)1114 2311 y Fu(@)1167 2267 y Ft(\015)1162 2341 y(\030)1212 2311 y Fu(a)1260 2325 y Ft(m)p Fm(\000)p Ft(j)1414 2311 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))p Fp(j)26 b(\024)f Fu(A)1835 2274 y Ft(j)t Fq(+)p Fm(j)p Ft(\015)t Fm(j)p Fq(+)p Fm(j)p Ft(\014)s Fm(j)p Fq(+1)2234 2311 y Fw(\()p Fu(\014)5 b Fw(!)p Fu(j)g Fw(!\))2452 2274 y Ft(\033)2501 2311 y Fu(\015)g Fw(!)p Fp(j)p Fu(\030)t Fp(j)2672 2274 y Ft(m)p Fm(\000)p Ft(j)t Fm(\000j)p Ft(\015)t Fm(j)386 2311 y Fw(\(3.1\))386 2512 y(for)32 b(all)e Fu(\014)j Fp(2)28 b Fs(Z)892 2480 y Ft(n)892 2535 y Fq(+)946 2512 y Fw(,)33 b Fu(\015)g Fp(2)27 b Fs(Z)1237 2480 y Ft(n)1237 2535 y Fq(+)1292 2512 y Fw(,)32 b Fu(j)i Fp(2)27 b Fs(Z)1573 2526 y Fq(+)1628 2512 y Fw(,)32 b Fu(\030)g Fp(2)27 b Fs(R)1905 2480 y Ft(n)1958 2512 y Fw(.)45 b(By)32 b(Cauc)m(h)m(y's)h(in)m(tegral)e(form) m(ula)g(and)386 2620 y(the)d(homogeneit)m(y)g(of)g Fu(a)1216 2634 y Ft(m)p Fm(\000)p Ft(j)1398 2620 y Fw(\(3.1\))h(is)e(equiv)-5 b(alen)m(t)27 b(to)h(the)g(existence)g(of)g(a)g(neigh)m(b)s(our-)386 2739 y(ho)s(o)s(d)629 2716 y(~)613 2739 y Fu(S)35 b Fw(in)30 b Fs(C)870 2706 y Ft(n)953 2739 y Fw(of)h Fu(S)1118 2706 y Ft(n)p Fm(\000)p Fq(1)1285 2739 y Fw(suc)m(h)f(that)869 2885 y(sup)777 2979 y Ft(x)p Fm(2)p Fl(T)914 2960 y FA(n)951 2979 y Ft(;\030)s Fm(2)1063 2962 y Fq(~)1052 2979 y Ft(S)1113 2885 y Fp(j)p Fu(@)1191 2847 y Ft(\014)1186 2907 y(x)1238 2885 y Fu(a)1286 2899 y Ft(m)p Fm(\000)p Ft(j)1440 2885 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))p Fp(j)27 b(\024)e Fu(A)1862 2847 y Ft(j)t Fq(+)p Fm(j)p Ft(\014)s Fm(j)p Fq(+1)2126 2885 y Fw(\()p Fu(\014)5 b Fw(!)p Fu(j)g Fw(!\))2344 2847 y Ft(\033)2393 2885 y Fu(;)76 b Fp(8)p Fu(\014)30 b Fp(2)25 b Fs(Z)2777 2847 y Ft(n)2777 2907 y Fq(+)2831 2885 y Fu(;)15 b(j)32 b Fp(2)24 b Fs(Z)3090 2899 y Fq(+)3145 2885 y Fu(:)-2784 b Fw(\(3.2\))386 3115 y(A)30 b(function)f Fu(a)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))27 b Fp(2)e Fu(G)1279 3082 y Ft(\033)1325 3115 y Fw(\()p Fs(T)1419 3082 y Ft(n)1488 3115 y Fp(\002)20 b Fs(R)1639 3082 y Ft(n)1692 3115 y Fw(\))30 b(is)g(said)f(to)i(b)s(e)f(a)h Fu(G)2419 3082 y Ft(\033)2495 3115 y Fw(realization)f(of)386 3155 y Fr(P)482 3181 y Fm(1)482 3250 y Ft(j)t Fq(=0)624 3223 y Fu(a)672 3237 y Ft(m)p Fm(\000)p Ft(j)826 3223 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))42 b Fp(2)e Fu(F)13 b(S)1307 3190 y Ft(m)1302 3246 y(\033)1374 3223 y Fw(;)45 b(w)m(e)40 b(write)f Fu(a)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))42 b Fp(\030)2237 3155 y Fr(P)2333 3181 y Fm(1)2333 3250 y Ft(j)t Fq(=0)2475 3223 y Fu(a)2523 3237 y Ft(m)p Fm(\000)p Ft(j)2677 3223 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))40 b(if)f(for)h(ev)m (ery)386 3334 y Fu(\015)30 b Fp(2)25 b Fs(Z)614 3301 y Ft(n)614 3357 y Fq(+)699 3334 y Fw(one)31 b(can)f(\014nd)f Fu(C)j Fw(=)25 b Fu(C)1472 3348 y Ft(\015)1542 3334 y Fu(>)g Fw(0)30 b(suc)m(h)g(that)637 3578 y Fp(j)p Fu(@)715 3540 y Ft(\014)710 3600 y(x)763 3578 y Fu(@)816 3534 y Ft(\015)811 3607 y(\030)861 3578 y Fw(\()p Fu(a)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))21 b Fp(\000)1296 3464 y Ft(N)1262 3492 y Fr(X)1266 3687 y Ft(j)t Fq(=0)1409 3578 y Fu(a)1457 3592 y Ft(m)p Fm(\000)p Ft(j)1610 3578 y Fw(\()p Fu(x;)15 b(\030)t Fw(\)\))p Fp(j)27 b(\024)e Fu(C)2071 3540 y Ft(N)7 b Fq(+)p Fm(j)p Ft(\015)t Fm(j)p Fq(+)p Fm(j)p Ft(\014)s Fm(j)p Fq(+1)2500 3578 y Fw(\()p Fu(\014)e Fw(!)p Fu(N)10 b Fw(!\))2759 3540 y Ft(\033)2807 3578 y Fp(j)p Fu(\030)t Fp(j)2901 3540 y Ft(m)p Fm(\000)p Ft(N)d Fm(\000j)p Ft(\015)t Fm(j\000)p Fq(1)386 3578 y Fw(\(3.3\))386 3823 y(for)37 b(all)f Fu(\014)42 b Fp(2)37 b Fs(Z)921 3790 y Ft(n)921 3846 y Fq(+)976 3823 y Fw(,)i Fu(N)47 b Fp(2)36 b Fs(Z)1322 3837 y Fq(+)1377 3823 y Fw(,)j Fu(x)e Fp(2)f Fs(T)1686 3790 y Ft(n)1735 3823 y Fw(,)j Fu(\030)i Fp(2)c Fs(R)2037 3790 y Ft(n)2090 3823 y Fw(,)j Fp(j)p Fu(\030)t Fp(j)d(\025)g Fw(1.)62 b(W)-8 b(e)38 b(denote)g(the)g(set)f(of)386 3931 y Fu(G)458 3898 y Ft(\033)533 3931 y Fw(realizations)27 b(of)i(order)f Fu(m)g Fw(b)m(y)g Fu(S)1642 3898 y Ft(m)1637 3954 y(\033)1709 3931 y Fw(\()p Fs(T)1803 3898 y Ft(n)1868 3931 y Fp(\002)16 b Fs(R)2015 3898 y Ft(n)2068 3931 y Fw(\))28 b(and)g(the)h(set)g(of)f (pseudo\014\013eren)m(tial)386 4039 y(op)s(erators)f(with)e(sym)m(b)s (ol)g(in)g Fu(S)1458 4006 y Ft(m)1453 4062 y(\033)1525 4039 y Fw(\()p Fs(T)1619 4006 y Ft(n)1680 4039 y Fp(\002)13 b Fs(R)1823 4006 y Ft(n)1876 4039 y Fw(\))27 b(b)m(y)g Fu(O)s(S)2194 4006 y Ft(m)2189 4062 y(\033)2260 4039 y Fw(\()p Fs(T)2354 4006 y Ft(n)2403 4039 y Fw(\).)40 b(W)-8 b(e)27 b(also)g(recall)f(that)h(for)386 4147 y(eac)m(h)i(~)-46 b Fu(a)25 b Fp(2)g Fu(F)13 b(S)878 4114 y Ft(m)873 4170 y(\033)972 4147 y Fw(there)27 b(exists)g(a)g(realization)f Fu(a)g Fp(2)f Fu(S)2183 4114 y Ft(m)2178 4170 y(\033)2276 4147 y Fw(suc)m(h)i(that)g Fu(a)f Fp(\030)g Fw(~)-46 b Fu(a)p Fw(.)39 b(If)27 b Fu(b)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))28 b(is)386 4255 y(another)e Fu(G)787 4222 y Ft(\033)859 4255 y Fw(realization)e(of)j(~)-46 b Fu(a)p Fw(,)27 b(then)e Fu(R)q Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b(=)f Fu(a)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))c Fp(\000)g Fu(b)p Fw(\()p Fu(x;)k(\030)t Fw(\))26 b Fp(2)f Fu(S)2862 4222 y Fm(\0001)2857 4278 y Ft(\033)2992 4255 y Fw(\()p Fs(T)3086 4222 y Ft(n)3145 4255 y Fp(\002)11 b Fs(R)3286 4222 y Ft(n)3339 4255 y Fw(\))386 4363 y(i.e.)44 b Fu(R)33 b Fw(is)d(a)i Fu(G)889 4330 y Ft(\033)967 4363 y Fw(smo)s(othing)f(sym)m(b)s(ol,)g(namely)g (for)g(some)h Fu(c)c(>)f Fw(0)32 b(the)g(next)g(estimate)386 4471 y(holds)795 4617 y(sup)777 4695 y Ft(x)p Fm(2)p Fl(T)914 4676 y FA(n)965 4617 y Fp(j)p Fu(@)1043 4579 y Ft(\014)1038 4639 y(x)1091 4617 y Fu(R)q Fw(\()p Fu(x;)15 b(\030)t Fw(\)\))p Fp(j)27 b(\024)e Fu(c)1589 4579 y Fm(\000j)p Ft(\014)s Fm(j\000)p Fq(1)1821 4617 y Fw(\()p Fu(\014)5 b Fw(!\))1972 4579 y Ft(\033)2020 4617 y Fu(e)2062 4579 y Fm(\000)p Ft(c)p Fm(j)p Ft(\030)s Fm(j)2222 4556 y Fo(1)p FA(=\033)2328 4617 y Fu(;)197 b(\030)30 b Fp(2)24 b Fs(R)2765 4579 y Ft(n)2818 4617 y Fu(;)15 b(\014)31 b Fp(2)25 b Fs(Z)3091 4579 y Ft(n)3091 4639 y Fq(+)3145 4617 y Fu(:)-2784 b Fw(\(3.4\))486 4818 y(W)-8 b(e)41 b(note)h(that)f(the)g(comp)s(osition)f(rule)f(for)i(p.d.o.)71 b(on)41 b Fs(T)2589 4785 y Ft(n)2678 4818 y Fw(is)f(v)-5 b(alid)39 b(as)i(in)e(the)386 4926 y(lo)s(cal)j(case,)k(namely)c(if)f Fu(a)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))47 b Fp(\030)1685 4858 y Fr(P)1781 4884 y Fm(1)1781 4953 y Ft(j)t Fq(=0)1923 4926 y Fu(a)1971 4940 y Ft(\026)p Fm(\000)p Ft(j)2150 4926 y Fp(2)e Fu(S)2317 4882 y Ft(\026)2312 4938 y(\033)2363 4926 y Fw(\()p Fs(T)2457 4893 y Ft(n)2535 4926 y Fp(\002)28 b Fs(R)2693 4893 y Ft(n)2746 4926 y Fw(\))43 b(and)f Fu(b)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))46 b Fp(\030)386 4976 y Fr(P)482 5002 y Fm(1)482 5071 y Ft(j)t Fq(=0)624 5044 y Fu(b)663 5058 y Ft(\027)t Fm(\000)p Ft(j)793 5044 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b Fp(2)f Fu(S)1172 5011 y Ft(\027)1167 5067 y(\033)1215 5044 y Fw(\()p Fs(T)1309 5011 y Ft(n)1379 5044 y Fp(\002)20 b Fs(R)1529 5011 y Ft(n)1582 5044 y Fw(\))31 b(then)f Fu(c)p Fw(\()p Fu(x;)15 b(D)s Fw(\))26 b(:=)g Fu(a)p Fw(\()p Fu(x;)15 b(D)s Fw(\))21 b Fp(\016)f Fu(b)p Fw(\()p Fu(x;)15 b(D)s Fw(\))386 5177 y Fp(2)40 b Fu(O)s(S)620 5133 y Ft(\027)t Fq(+)p Ft(\026)615 5189 y(\033)760 5177 y Fw(\()p Fs(T)854 5144 y Ft(n)903 5177 y Fw(\))g(with)e(sym)m(b)s(ol)g Fu(c)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))42 b(=)e Fu(a)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))28 b Fp(\016)e Fu(b)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))42 b Fp(\030)2663 5109 y Fr(P)2759 5135 y Fm(1)2759 5204 y Ft(j)t Fq(=0)2901 5177 y Fu(c)2940 5191 y Ft(\026)p Fq(+)p Ft(\027)t Fm(\000)p Ft(j)3168 5177 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))386 5293 y(giv)m(en)30 b(b)m(y)h(\(cf.)41 b([15)q(]\))685 5482 y Fu(c)724 5496 y Ft(\026)p Fq(+)p Ft(\027)t Fm(\000)p Ft(j)952 5482 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))27 b(=)1389 5396 y Fr(X)1281 5597 y Ft(p)p Fq(+)p Ft(q)r Fq(+)p Fm(j)p Ft(\014)s Fm(j)p Fq(=)p Ft(j)1672 5421 y Fw(1)p 1654 5461 82 4 v 1654 5544 a Fu(\014)5 b Fw(!)1746 5482 y Fu(D)1824 5438 y Ft(\014)1821 5511 y(\030)1871 5482 y Fu(a)1919 5496 y Ft(\026)p Fm(\000)p Ft(p)2056 5482 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))p Fu(@)2315 5444 y Ft(\014)2310 5505 y(x)2363 5482 y Fu(b)2402 5496 y Ft(\027)t Fm(\000)p Ft(q)2534 5482 y Fw(\()p Fu(x;)g(\030)t Fw(\))p Fu(;)199 b(j)30 b Fp(2)25 b Fs(Z)3182 5496 y Fq(+)3237 5482 y Fu(:)-2876 b Fw(\(3.5\))p eop %%Page: 8 8 8 7 bop 386 239 a Fq(8)728 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g(YOSHINO) 386 434 y Fw(F)-8 b(urthermore,)29 b(if)g Fu(a)p Fw(\()p Fu(x;)15 b(\030)t Fw(\))30 b(is)e(elliptic,)f(i.e.)40 b Fu(a)1920 448 y Ft(\026)1967 434 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b Fp(6)p Fw(=)f(0)30 b(for)f(\()p Fu(x;)15 b(\030)t Fw(\))26 b Fp(2)f Fs(T)2885 401 y Ft(n)2951 434 y Fp(\002)18 b Fw(\()p Fs(R)3135 401 y Ft(n)3205 434 y Fp(n)h Fw(0\),)386 544 y(then)k Fu(a)p Fw(\()p Fu(x;)15 b(D)s Fw(\))24 b(admits)e(a)i(\(formal\))f(in)m(v)m(erse)g Fu(a)1949 511 y Fm(\000)p Fq(1)2043 544 y Fw(\()p Fu(x;)15 b(D)s Fw(\))26 b Fp(2)f Fu(O)s(S)2528 511 y Fm(\000)p Ft(\026)2653 544 y Fw(satisfying)d Fu(a)6 b Fp(\016)g Fu(a)3204 511 y Fm(\000)p Fq(1)3304 544 y Fp(\000)386 652 y Fu(id)26 b Fp(2)f Fu(O)s(S)709 619 y Fm(\0001)704 675 y Ft(\033)868 652 y Fw(and)k Fu(a)1092 619 y Fm(\000)p Fq(1)1205 652 y Fp(\016)20 b Fu(a)f Fp(\000)f Fu(id)26 b Fp(2)f Fu(O)s(S)1749 619 y Fm(\0001)1744 675 y Ft(\033)1879 652 y Fw(.)40 b(Recall)29 b(that)i(the)f(theory)g(of)g(symplectic)386 760 y(maps)d(\(cf.)40 b([26)r(]\))28 b(sho)m(ws)f(that)i(for)e(a)h(giv) m(en)g Fu(P)38 b Fp(2)25 b Fu(S)5 b(L)p Fw(\()p Fu(n)p Fw(;)15 b Fs(Z)p Fw(\),)24 b(the)k(c)m(hange)h(of)f(v)-5 b(ariables)386 868 y Fu(y)29 b Fw(=)d Fu(P)13 b(x)31 b Fw(induces)e(the)i(linear)f(symplectic)g(c)m(hange)i(\()p Fu(x;)15 b(\030)t Fw(\))27 b Fp(!)f Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\),)33 b Fu(x)26 b Fw(=)g Fu(P)3055 835 y Fm(\000)p Fq(1)3150 868 y Fu(y)33 b Fw(and)386 976 y Fu(\030)c Fw(=)551 943 y Ft(tr)615 976 y Fu(P)13 b(\021)s Fw(.)41 b(W)-8 b(e)31 b(write)903 1132 y(\()p Fu(P)1009 1094 y Fm(\003)1049 1132 y Fu(a)p Fw(\)\()p Fu(y)s(;)15 b(\021)s Fw(\))27 b(=)e Fu(a)p Fw(\()p Fu(P)1615 1094 y Fm(\000)p Fq(1)1709 1132 y Fu(y)s(;)1797 1094 y Ft(tr)1861 1132 y Fu(P)13 b(\021)s Fw(\))p Fu(;)198 b(y)28 b Fp(2)d Fs(T)2456 1094 y Ft(n)2505 1132 y Fu(;)30 b(\021)f Fp(2)c Fs(R)2779 1094 y Ft(n)2832 1132 y Fu(:)-2471 b Fw(\(3.6\))386 1292 y(Th)m(us)40 b(the)i(comp)s(osition)e(is)h(in)m(v)-5 b(arian)m(t,)44 b(i.e.)74 b(for)41 b(t)m(w)m(o)i(p.d.o.-s)f Fu(a)i Fw(=)g Fu(a)p Fw(\()p Fu(x;)15 b(D)s Fw(\))42 b(and)386 1400 y Fu(b)25 b Fw(=)g Fu(b)p Fw(\()p Fu(x;)15 b(D)s Fw(\))31 b(w)m(e)g(ha)m(v)m(e)1433 1556 y Fu(P)1504 1518 y Fm(\003)1544 1556 y Fw(\()p Fu(a)20 b Fp(\016)h Fu(b)p Fw(\))k(=)g Fu(P)1979 1518 y Fm(\003)2019 1556 y Fu(a)20 b Fp(\016)h Fu(P)2224 1518 y Fm(\003)2263 1556 y Fu(b:)-1941 b Fw(\(3.7\))386 1711 y(Moreo)m(v)m(er,)33 b(if)d Fu(a)g Fw(is)g(a)h(p.d.o.)42 b(ha)m(ving)30 b(a)h(sym)m(b)s(ol)f (dep)s(ending)e(in)h Fu(x)i Fw(only)f(on)g(resonan)m(t)386 1819 y(v)-5 b(ariables,)29 b(the)i(same)g(is)e(true)h(for)g Fu(P)1666 1786 y Fm(\003)1706 1819 y Fu(a)p Fw(.)486 1927 y(No)m(w)45 b(w)m(e)h(state)h(the)e(\014rst)f(main)g(result)g(on)h (sim)m(ultaneous)f(normal)g(forms)h(for)386 2035 y Fp(f)p Fu(b)470 2002 y Ft(k)513 2035 y Fw(\()p Fu(x;)15 b(D)s Fw(\))p Fp(g)798 2002 y Ft(d)798 2059 y Fq(1)840 2035 y Fw(.)386 2285 y FE(Theorem)34 b(3.1.)46 b Fk(L)-5 b(et)41 b Fw(1)e Fp(\024)g Fu(\033)j Fp(\024)d(1)h Fk(and)h(supp)-5 b(ose)42 b(that)g Fw(\()p Fu(S)5 b(i)p Fw(\))2612 2299 y Ft(\033)2700 2285 y Fk(holds)42 b(for)f Fu(!)s Fk(.)64 b(Fix)386 2393 y(a)38 b(r)-5 b(esonant)39 b(\(c)-5 b(anonic)g(al\))40 b(b)-5 b(asis)38 b Fp(f)p Fu(\024)1644 2360 y Ft(j)1681 2393 y Fp(g)1726 2360 y Ft(r)1726 2418 y Fq(1)1803 2393 y Fk(if)f Fu(r)g Fw(=)c Fu(dim)2236 2407 y Fl(Z)2285 2393 y Fw(\000)2342 2360 y Ft(!)2342 2421 y Fl(Z)2426 2393 y Fp(\025)h Fw(1)p Fk(.)56 b(If)37 b Fu(\033)g(<)d Fp(1)j Fk(we)g(also)386 2508 y(r)-5 b(e)g(quir)g(e)32 b(that)h Fu(a)915 2475 y Ft(k)915 2534 y Fm(\000)p Ft(j)1007 2508 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))32 b Fk(is)g Fu(C)1414 2475 y Fm(1)1488 2508 y Fw(\()p Fs(R)1583 2475 y Ft(n)1654 2508 y Fp(n)19 b(f)p Fw(0)p Fp(g)p Fw(;)c Fu(G)1965 2475 y Ft(\033)2013 2508 y Fw(\()p Fs(T)2107 2475 y Ft(n)2107 2531 y(x)2156 2508 y Fw(\)\))p Fk(,)32 b Fu(k)d Fw(=)c(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d)p Fk(,)32 b Fu(j)f Fp(2)25 b Fs(Z)3046 2522 y Fq(+)3101 2508 y Fk(.)41 b(Then)386 2631 y(ther)-5 b(e)31 b(exists)g(an)g(el)5 b(liptic)31 b(symb)-5 b(ol)32 b Fu(q)s Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b Fp(\030)1948 2562 y Fr(P)2044 2589 y Fm(1)2044 2658 y Ft(j)t Fq(=0)2186 2631 y Fu(q)2227 2645 y Fm(\000)p Ft(j)2318 2631 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b Fp(2)f Fu(F)13 b(S)2768 2598 y Fq(0)2807 2631 y Fw(\()p Fs(T)2901 2598 y Ft(n)2966 2631 y Fp(\002)i Fs(R)3112 2598 y Ft(n)3165 2631 y Fw(\))31 b Fk(and)386 2758 y(zer)-5 b(o)33 b(or)-5 b(der)34 b(symb)-5 b(ols)34 b Fu(\013)1206 2725 y Ft(k)1249 2758 y Fw(\()p Fu(y)s(;)15 b(\030)t Fw(\))26 b Fp(\030)1573 2690 y Fr(P)1669 2716 y Fm(1)1669 2785 y Ft(j)t Fq(=0)1811 2758 y Fu(\013)1869 2725 y Ft(k)1869 2783 y Fm(\000)p Ft(j)1961 2758 y Fw(\()p Fu(y)s(;)15 b(\030)t Fw(\))26 b Fp(2)f Fu(F)13 b(S)2407 2725 y Fq(0)2446 2758 y Fw(\()p Fs(T)2540 2725 y Ft(r)2600 2758 y Fp(\002)19 b Fs(R)2750 2725 y Ft(n)2803 2758 y Fw(\))32 b Fk(such)h(that)g(for)386 2869 y Fw(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d)432 3134 y Fw(\()p Fp(\000)p Fu(iX)644 3149 y Ft(k)708 3134 y Fw(+)20 b Fu(a)847 3096 y Ft(k)889 3134 y Fw(\()p Fu(x)p Fw(;)15 b Fu(D)s Fw(\)\))22 b Fp(\016)f Fu(q)s Fw(\()p Fu(x)p Fw(;)15 b Fu(D)s Fw(\))26 b(=)e Fu(q)s Fw(\()p Fu(x)p Fw(;)15 b Fu(D)s Fw(\))21 b Fp(\016)g Fw(\()p Fp(\000)p Fu(iX)2240 3149 y Ft(k)2304 3134 y Fw(+)f Fu(\013)2453 3096 y Ft(k)2496 3134 y Fw(\()p Fu(\024x)p Fw(;)15 b Fu(D)s Fw(\))21 b(+)f Fu(R)2969 3149 y Ft(k)3012 3134 y Fw(\()p Fu(x;)15 b(D)s Fw(\)\))p Fu(;)386 3005 y Fw(\(3.8\))386 3302 y Fk(wher)-5 b(e)43 b Fu(R)721 3317 y Ft(k)764 3302 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))44 b Fp(2)f Fu(S)1179 3269 y Fm(\0001)1308 3302 y Fw(\()p Fs(T)1402 3269 y Ft(n)1479 3302 y Fp(\002)27 b Fs(R)1636 3269 y Ft(n)1689 3302 y Fw(\))43 b Fk(and)g Fu(\024x)h Fw(:=)f(\()p Fu(<)g(\024)2441 3269 y Fq(1)2481 3302 y Fu(;)15 b(x)43 b(>;)15 b(:)g(:)g(:)32 b(;)15 b(<)43 b(\024)3070 3269 y Ft(r)3108 3302 y Fu(;)15 b(x)43 b(>)p Fw(\))p Fu(:)386 3410 y Fk(If)i Fw(\000)551 3377 y Ft(!)551 3437 y Fl(Z)649 3410 y Fw(=)j Fp(f)p Fw(0)p Fp(g)p Fk(,)i(the)45 b(symb)-5 b(ols)47 b Fu(q)1530 3424 y Fm(\000)p Ft(j)1621 3410 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))p Fk(,)49 b Fu(j)54 b Fp(2)47 b Fs(Z)2168 3424 y Fq(+)2268 3410 y Fk(ar)-5 b(e)46 b(uniquely)f(determine)-5 b(d)47 b(by)386 3518 y(the)40 b(r)-5 b(e)g(quir)g(ements)41 b Fw([)p Fu(q)1149 3532 y Fq(0)1188 3518 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\)])1392 3532 y Ft(n)1479 3518 y Fw(=)38 b(1)p Fk(,)k Fw([)p Fu(q)1769 3532 y Fm(\000)p Ft(j)1860 3518 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\)])2064 3532 y Ft(n)2151 3518 y Fw(=)38 b(0)p Fk(,)j(for)g Fu(j)j Fp(\025)37 b Fw(1)p Fk(,)42 b(wher)-5 b(e)41 b Fw([)p Fu(a)p Fw(])3193 3532 y Ft(n)3279 3518 y Fw(:=)386 3626 y(\(2)p Fu(\031)s Fw(\))556 3593 y Fm(\000)p Ft(n)674 3553 y Fr(R)717 3658 y Fl(T)767 3639 y FA(n)823 3626 y Fu(a)p Fw(\()p Fu(t)p Fw(\))p Fu(dt)33 b Fk(is)g(the)g(me)-5 b(an)34 b(value)e(of)h Fu(a)p Fk(.)386 3794 y(Pr)-5 b(o)g(of.)43 b Fw(Let)33 b Fu(P)42 b Fp(2)29 b Fu(S)5 b(L)p Fw(\()p Fu(n)p Fw(;)15 b Fs(Z)p Fw(\))29 b(b)s(e)j(as)h(in)e(Theorem)h(2.1,)j (with)c(the)i(\014rst)f Fu(r)j Fw(ro)m(ws)e(b)s(eing)386 3902 y Fu(\024)438 3869 y Fq(1)478 3902 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(\024)746 3869 y Ft(n)794 3902 y Fw(,)29 b(and)g(de\014ne)e Fu(y)i Fw(=)24 b Fu(P)13 b(x)p Fw(.)41 b(In)27 b(view)h(of)h(\(3.7\),)i (w)m(e)e(will)d(consider)i(in)f(the)i(new)386 4010 y(v)-5 b(ariables)35 b(\()p Fu(y)s(;)15 b(\021)s Fw(\),)40 b(with)35 b Fu(\024)1305 3977 y Ft(j)1377 4010 y Fw(=)45 b Fu(~)-54 b(e)1526 4024 y Ft(j)1562 4010 y Fw(.)60 b(Indeed,)37 b(in)e(the)i(original)e(v)-5 b(ariables)35 b Fu(x)g Fw(=)h Fu(P)3233 3977 y Fm(\000)p Fq(1)3327 4010 y Fu(y)386 4122 y Fw(w)m(e)k(observ)m(e)h(that)f Fu(y)1114 4136 y Ft(j)1192 4122 y Fw(=)p Fu(<)g(\024)1426 4089 y Ft(j)1463 4122 y Fu(;)15 b(x)42 b(>)p Fw(,)g(1)g Fp(\024)f Fu(j)46 b Fp(\024)41 b Fu(r)i Fw(if)c Fu(r)k Fp(\025)e Fw(1.)70 b(The)40 b(resonance)g(set)386 4230 y(is)c(the)h(lattice)g Fs(Z)996 4197 y Ft(r)996 4258 y(y)1033 4239 y Fx(0)1092 4230 y Fw(in)f Fs(R)1265 4197 y Ft(r)1265 4258 y(y)1302 4239 y Fx(0)1334 4230 y Fw(,)j(where)d Fu(y)j Fw(=)d(\()p Fu(y)1941 4197 y Fm(0)1965 4230 y Fu(;)15 b(y)2053 4197 y Fm(00)2095 4230 y Fw(\))p Fu(;)g(y)2218 4197 y Fm(0)2278 4230 y Fw(=)36 b(\()p Fu(y)2465 4244 y Fq(1)2504 4230 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(y)2766 4244 y Ft(r)2804 4230 y Fw(\).)61 b(F)-8 b(or)37 b(brevit)m(y)386 4355 y Fu(a)p Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))40 b(is)f(used)f(instead)g (of)i Fu(P)1503 4322 y Fm(\003)1542 4355 y Fu(a)p Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))41 b(=)f Fu(a)p Fw(\()p Fu(P)2102 4322 y Fm(\000)p Fq(1)2197 4355 y Fu(y)s(;)2285 4322 y Ft(tr)2348 4355 y Fu(P)13 b(\021)s Fw(\))40 b(for)f(a)h(sym)m(b)s(ol) d Fu(a)p Fw(\()p Fu(x;)15 b(\030)t Fw(\).)386 4463 y(W)-8 b(e)35 b(write)e(terms)h(of)h(the)f(same)g(order)g(of)g(homogeneit)m(y) h(in)e(b)s(oth)g(sides)g(of)h(\(3.8\))i(in)386 4571 y(the)e(new)f(v)-5 b(ariables)32 b Fu(y)s Fw(.)51 b(The)33 b(\014rst)g(system)h(of)g Fu(d)g Fw(equations)g(in)m(v)m(olv)m(es)f(b)s(oth)g Fu(q)3128 4585 y Fq(0)3167 4571 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))386 4681 y(and)30 b Fu(\013)621 4648 y Ft(k)621 4705 y Fq(0)664 4681 y Fw(\()p Fu(y)747 4648 y Fm(0)770 4681 y Fu(;)15 b(\021)s Fw(\))32 b(\(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d)p Fw(\))813 4845 y Fu(<)g(\022)955 4807 y Ft(k)997 4845 y Fu(;)15 b(D)1112 4860 y Ft(y)1149 4841 y Fx(0)q(0)1220 4845 y Fu(>)25 b(q)1357 4859 y Fq(0)1416 4845 y Fw(+)20 b Fu(a)1555 4807 y Ft(k)1555 4867 y Fq(0)1598 4845 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))p Fu(q)1845 4859 y Fq(0)1910 4845 y Fw(=)25 b Fu(\013)2064 4807 y Ft(k)2064 4867 y Fq(0)2107 4845 y Fw(\()p Fu(y)2190 4807 y Fm(0)2214 4845 y Fu(;)15 b(\021)s Fw(\))p Fu(q)2378 4859 y Fq(0)2418 4845 y Fu(;)76 b(\021)29 b Fp(2)24 b Fs(R)2738 4807 y Ft(n)2811 4845 y Fp(n)d Fw(0)p Fu(:)-2561 b Fw(\(3.9\))386 5005 y(The)37 b(adv)-5 b(an)m(tage)40 b(of)f(the)f(expression)f(\(3.9\))i(is)e(that)i(w)m(e)g(can)f(regard)g (the)g(resonan)m(t)386 5113 y(v)-5 b(ariables)30 b Fu(y)812 5080 y Fm(0)866 5113 y Fw(as)h(parameters)h(if)e Fu(r)f Fp(\025)e Fw(1.)43 b(W)-8 b(e)33 b(can)e(\014nd)f(a)h(nonzero)h (solution)d(of)j(\(3.9\))386 5223 y(if)d(w)m(e)i(c)m(ho)s(ose)g Fu(\013)949 5190 y Ft(k)949 5248 y Fq(0)1023 5223 y Fw(so)g(as)f(to)h (satisfy)495 5417 y Fr(Z)545 5623 y Fl(T)595 5604 y FA(n)p Fx(\000)p FA(r)730 5540 y Fu(a)778 5503 y Ft(k)778 5563 y Fq(0)820 5540 y Fw(\()p Fu(y)903 5503 y Fm(0)927 5540 y Fu(;)15 b(y)1015 5503 y Fm(00)1057 5540 y Fu(;)g(\021)s Fw(\))p Fu(dy)1275 5503 y Fm(0)r(0)1344 5540 y Fw(=)25 b Fu(\013)1498 5503 y Ft(k)1498 5563 y Fq(0)1542 5540 y Fw(\()p Fu(y)1625 5503 y Fm(0)1648 5540 y Fu(;)15 b(\021)s Fw(\))p Fu(;)198 b Fw(1)26 b Fp(\024)f Fu(k)k Fp(\024)24 b Fu(d;)31 b(y)2483 5503 y Fm(0)2532 5540 y Fp(2)25 b Fs(T)2677 5503 y Ft(r)2716 5540 y Fu(;)31 b(\021)d Fp(2)d Fs(R)2991 5503 y Ft(n)3064 5540 y Fp(n)c(f)p Fw(0)p Fp(g)386 5372 y Fw(\(3.10\))p eop %%Page: 9 9 9 8 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f(FIELDS) h(ON)g(TORI)594 b(9)386 434 y Fw(i.e.)39 b(the)26 b(F)-8 b(ourier)25 b(co)s(e\016cien)m(ts)h Fp(F)1523 449 y Ft(y)r Fm(!)p Ft(\020)1671 434 y Fw(\()p Fu(a)1754 401 y Ft(k)1797 434 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))c Fp(\000)g Fu(\013)2154 401 y Ft(k)2154 458 y Fq(0)2197 434 y Fw(\()p Fu(y)2280 401 y Fm(0)2304 434 y Fu(;)k(\021)s Fw(\)\))27 b(v)-5 b(anish)24 b(on)h(the)h(resonan)m(t)386 542 y(set,)31 b(whic)m(h)d(in)h(the)g(canonical)h(v)-5 b(ariables)28 b(is)h(de\014ned)g(b)m(y)g Fu(\020)2417 556 y Ft(r)r Fq(+1)2570 542 y Fw(=)c Fp(\001)15 b(\001)g(\001)i Fu(\020)2828 556 y Ft(n)2900 542 y Fw(=)24 b(0.)41 b(By)30 b(the)386 659 y(comm)m(utativit)m(y)g(w)m(e)f(ha)m(v)m(e)h Fu(<)25 b(\022)1478 626 y Ft(j)1514 659 y Fu(;)15 b(D)1629 674 y Ft(y)1666 655 y Fx(0)q(0)1737 659 y Fu(>)25 b(a)1881 626 y Ft(k)1881 683 y Fq(0)1924 659 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))26 b(=)p Fu(<)f(\022)2369 626 y Ft(k)2411 659 y Fu(;)15 b(D)2526 674 y Ft(y)2563 655 y Fx(0)q(0)2634 659 y Fu(>)25 b(a)2778 615 y Ft(j)2778 685 y Fq(0)2846 659 y Fw(for)k(1)d Fp(\024)f Fu(j;)15 b(k)29 b Fp(\024)386 774 y Fu(d)p Fw(.)45 b(Because)33 b Fu(<)28 b(\022)997 741 y Ft(j)1033 774 y Fu(;)15 b(D)1148 790 y Ft(y)1185 771 y Fx(0)q(0)1258 774 y Fu(>)27 b(\013)1414 741 y Ft(k)1414 799 y Fq(0)1485 774 y Fp(\021)g Fw(0,)33 b(it)f(follo)m(ws)f(from)g (the)h(F)-8 b(rob)s(enius)30 b(theorem)i(and)386 882 y(simple)c(calculations)i(that)h(the)f(general)h(solution)e(of)h (\(3.9\))i(is)e(giv)m(en)g(b)m(y)488 1160 y Fu(q)529 1174 y Fq(0)568 1160 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))27 b(=)e Fu(`)p Fw(\()p Fu(y)1018 1122 y Fm(0)1041 1160 y Fu(;)15 b(\021)s Fw(\))p Fu(e)1206 1122 y Ft(i )r Fq(\()p Ft(y)r(;\021)r Fq(\))1434 1160 y Fu(;)76 b(`)p Fw(\()p Fu(y)1656 1122 y Fm(0)1679 1160 y Fu(;)15 b(\021)s Fw(\))27 b Fp(2)e Fu(G)1987 1122 y Ft(\033)2033 1160 y Fw(\()p Fs(T)2127 1122 y Ft(r)2187 1160 y Fp(\002)20 b Fw(\()p Fs(R)2373 1122 y Ft(n)2446 1160 y Fp(n)h Fw(0\)\))p Fu(;)198 b(or)s(d)2985 1174 y Ft(\021)3027 1160 y Fu(`)25 b Fw(=)g(0)p Fu(;)386 1028 y Fw(\(3.11\))386 1319 y(where)40 b Fu( )s Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))42 b(is)e(the)h(unique)e(solution)g (of)i(the)f(follo)m(wing)f(system)i(of)g(equations)386 1429 y(with)29 b(F)-8 b(ourier)30 b(co)s(e\016cien)m(ts)h(v)-5 b(anishing)28 b(on)i(the)h(resonan)m(t)g(set)g(\000)2619 1396 y Ft(P)10 b(!)2619 1456 y Fl(Z)719 1594 y Fu(<)25 b(\022)861 1557 y Ft(k)903 1594 y Fu(;)15 b(@)991 1609 y Ft(y)1028 1591 y Fx(0)q(0)1099 1594 y Fu(>)25 b( )k Fw(=)c Fu(f)1434 1557 y Ft(k)1424 1617 y Fq(0)1475 1594 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))27 b(:=)e Fp(\000)p Fu(a)1948 1557 y Ft(k)1948 1617 y Fq(0)1991 1594 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))21 b(+)f Fu(\013)2367 1557 y Ft(k)2367 1617 y Fq(0)2410 1594 y Fw(\()p Fu(y)2493 1557 y Fm(0)2517 1594 y Fu(;)15 b(\021)s Fw(\))p Fu(;)198 b Fw(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d:)-2887 b Fw(\(3.12\))386 1748 y(In)30 b(fact,)h Fu( )j Fw(is)29 b(de\014ned)g(b)m(y)1004 1963 y Fp(F)1069 1979 y Ft(y)1106 1960 y Fx(0)q(0)1147 1979 y Fm(!)p Ft(\020)1254 1960 y Fx(0)o(0)1298 1963 y Fw(\()p Fu( )s Fw(\()p Fu(y)1478 1926 y Fm(0)1502 1963 y Fu(;)15 b(y)1590 1926 y Fm(0)q(0)1633 1963 y Fu(;)g(\021)s Fw(\)\))27 b(=)1924 1900 y Fp(F)1989 1915 y Ft(y)2026 1896 y Fx(0)q(0)2067 1915 y Fm(!)p Ft(\020)2174 1896 y Fx(0)o(0)2218 1900 y Fw(\()p Fu(f)2308 1867 y Ft(k)2298 1924 y Fq(0)2350 1900 y Fw(\()p Fu(y)2433 1867 y Fm(0)2457 1900 y Fu(;)15 b(y)2545 1867 y Fm(00)2587 1900 y Fu(;)g(\021)s Fw(\)\))p 1924 1942 824 4 v 2102 2026 a Fu(i)26 b(<)f(\022)2301 1999 y Ft(k)2343 2026 y Fu(;)15 b(\020)2430 1999 y Fm(00)2497 2026 y Fu(>)386 1963 y Fw(\(3.13\))386 2177 y(if)38 b Fu(<)g(\022)633 2144 y Ft(k)675 2177 y Fu(;)15 b(\020)762 2144 y Fm(00)844 2177 y Fu(>)p Fp(6)p Fw(=)38 b(0)h(for)g(some)g(1)g Fp(\024)g Fu(k)j Fp(\024)d Fu(d)p Fw(,)j(and)c(=)g(0,)k(if)c(otherwise.)65 b(In)38 b(view)f(of)386 2285 y(\()p Fu(S)5 b(i)p Fw(\))548 2299 y Ft(\033)633 2285 y Fw(and)38 b(results)e(in)h([20)q(],)j Fu( )s Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))39 b(\(and)f(therefore)g Fu(q)2391 2299 y Fq(0)2430 2285 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\)\))39 b(is)e Fu(G)2881 2252 y Ft(\033)2965 2285 y Fw(in)f Fu(x)p Fw(.)63 b(The)386 2393 y(comp)s(osition)29 b(rule)g(for)h(p.d.o.)41 b(yields)414 2662 y Fu(q)458 2624 y Fm(\000)p Fq(1)455 2688 y(0)552 2662 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))22 b Fp(\016)e Fu(b)884 2624 y Ft(k)927 2662 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))22 b Fp(\016)e Fu(q)1261 2676 y Fq(0)1300 2662 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))27 b(=)p Fu(<)d(\022)1745 2624 y Ft(k)1787 2662 y Fu(;)15 b(\021)1875 2624 y Fm(0)q(0)1944 2662 y Fu(>)25 b Fw(+)p Fu(\013)2169 2624 y Ft(k)2169 2684 y Fq(0)2212 2662 y Fw(\()p Fu(y)2295 2624 y Fm(0)2318 2662 y Fu(;)15 b(\021)s Fw(\))21 b(+)f(\()p Fu(O)s Fw(\()p Fp(j)p Fu(\021)s Fp(j)2793 2624 y Fm(\000)p Fq(1)2889 2662 y Fw(\)\))p Fu(;)31 b Fp(j)p Fu(\021)s Fp(j)c(!)e(1)386 2533 y Fw(\(3.14\))386 2832 y(for)c(all)f(1)25 b Fp(\024)g Fu(k)k Fp(\024)c Fu(d)p Fw(,)e(where)e Fu(b)1359 2799 y Ft(k)1423 2832 y Fw(stands)f(for)h Fu(P)1898 2799 y Fm(\003)1937 2832 y Fw(\()p Fu(iX)2078 2847 y Ft(k)2124 2832 y Fw(+)r Fu(a)2245 2799 y Ft(k)2287 2832 y Fw(\).)38 b(W)-8 b(e)22 b(pro)s(ceed)f(b)m(y)g(induction;)386 2942 y(namely)42 b(w)m(e)h(supp)s(ose)e(that)i(there)g(exist)f Fu(\013)1953 2909 y Ft(k)1953 2968 y Fm(\000)p Ft(j)2045 2942 y Fw(\()p Fu(y)2128 2909 y Fm(0)2151 2942 y Fu(;)15 b(\021)s Fw(\))47 b Fp(2)f Fu(C)2500 2909 y Fm(1)2574 2942 y Fw(\()p Fs(R)2669 2909 y Ft(n)2750 2942 y Fp(n)29 b(f)p Fw(0)p Fp(g)p Fw(;)15 b Fu(G)3071 2909 y Ft(\033)3119 2942 y Fw(\()p Fs(T)3213 2909 y Ft(r)3213 2970 y(y)3250 2951 y Fx(0)3279 2942 y Fw(\)\),)386 3071 y Fu(or)s(d)521 3085 y Ft(\021)563 3071 y Fu(\013)621 3038 y Ft(k)621 3096 y Fm(\000)p Ft(j)712 3071 y Fw(\()p Fu(y)795 3038 y Fm(0)819 3071 y Fu(;)g(\021)s Fw(\))26 b(=)f Fp(\000)p Fu(j)5 b Fw(,)31 b(0)26 b Fp(\024)f Fu(j)31 b Fp(\024)25 b Fu(N)30 b Fp(\000)20 b Fw(1,)31 b(1)25 b Fp(\024)g Fu(k)k Fp(\024)c Fu(d)p Fw(,)31 b(suc)m(h)f(that)884 3332 y Fu(a)932 3294 y Ft(k)975 3332 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))27 b Fp(\030)1304 3218 y Ft(N)7 b Fm(\000)p Fq(1)1315 3245 y Fr(X)1319 3441 y Ft(j)t Fq(=0)1472 3332 y Fu(\013)1530 3294 y Ft(k)1530 3354 y Fm(\000)p Ft(j)1622 3332 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))21 b(+)1980 3218 y Fm(1)1950 3245 y Fr(X)1940 3442 y Ft(j)t Fq(=)p Ft(N)2106 3332 y Fu(a)2154 3294 y Ft(k)2154 3354 y Fm(\000)p Ft(j)2245 3332 y Fw(\()p Fu(y)2328 3294 y Fm(0)2352 3332 y Fu(;)15 b(\021)s Fw(\))p Fu(;)198 b Fw(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d:)-2722 b Fw(\(3.15\))386 3587 y(No)m(w)29 b(w)m(e)g(lo)s(ok)f(for)h(an)f(elliptic)e(sym)m(b)s(ol)i Fu(q)s Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))26 b Fp(\030)f Fw(1)17 b(+)g Fu(q)2349 3601 y Fm(\000)p Ft(N)2470 3587 y Fw(\()p Fu(y)s(;)e(\021)s Fw(\),)31 b Fu(or)s(d)2867 3601 y Ft(\021)2909 3587 y Fu(q)2950 3601 y Fm(\000)p Ft(N)3100 3587 y Fw(=)25 b Fp(\000)p Fu(N)10 b Fw(,)386 3701 y(and)30 b Fu(\013)621 3668 y Ft(k)621 3728 y Fm(\000)p Ft(N)743 3701 y Fw(\()p Fu(y)826 3668 y Fm(0)849 3701 y Fu(;)15 b(\021)s Fw(\))27 b Fp(2)e Fu(C)1157 3668 y Fm(1)1231 3701 y Fw(\()p Fs(R)1326 3668 y Ft(n)1399 3701 y Fp(n)c(f)p Fw(0)p Fp(g)p Fw(;)15 b Fu(G)1712 3668 y Ft(\033)1760 3701 y Fw(\()p Fs(T)1854 3668 y Ft(r)1894 3701 y Fw(\)\))31 b(suc)m(h)f(that)477 4071 y Fu(q)521 4033 y Fm(\000)p Fq(1)615 4071 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))21 b Fp(\016)g Fu(b)947 4033 y Ft(k)990 4071 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))21 b Fp(\016)g Fu(q)s Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))26 b(=)p Fu(<)f(\022)1772 4033 y Ft(k)1814 4071 y Fu(;)15 b(\021)1902 4033 y Fm(0)q(0)1970 4071 y Fu(>)25 b Fw(+)2186 3957 y Ft(N)2152 3984 y Fr(X)2156 4180 y Ft(j)t Fq(=0)2299 4071 y Fu(\013)2357 4033 y Ft(k)2357 4093 y Fm(\000)p Ft(j)2448 4071 y Fw(\()p Fu(y)2531 4033 y Fm(0)2555 4071 y Fu(;)15 b(\021)s Fw(\))21 b(+)f Fu(O)s Fw(\()p Fp(j)p Fu(\021)s Fp(j)2995 4033 y Fm(\000)p Ft(N)7 b Fm(\000)p Fq(1)3208 4071 y Fw(\))p Fu(;)386 3858 y Fw(\(3.16\))386 4324 y(when)42 b Fp(j)p Fu(\021)s Fp(j)k(!)g(1)c Fw(for)h(all)e(1)47 b Fp(\024)e Fu(k)k Fp(\024)c Fu(d)p Fw(.)78 b(W)-8 b(e)44 b(obtain)e(the)h(system)g(of)g(equations,)386 4435 y Fu(k)28 b Fw(=)d(1)p Fu(;)15 b(:)g(:)g(:)33 b(;)15 b(d)31 b Fw(for)f Fu(q)1078 4449 y Fm(\000)p Ft(N)1199 4435 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))32 b(and)e Fu(\013)1672 4402 y Ft(k)1672 4461 y Fm(\000)p Ft(N)1794 4435 y Fw(\()p Fu(y)1877 4402 y Fm(0)1901 4435 y Fu(;)15 b(\021)s Fw(\))727 4607 y Fp(\000)p Fu(i)25 b(<)g(\022)996 4569 y Ft(k)1038 4607 y Fu(;)15 b(@)1126 4622 y Ft(y)1163 4603 y Fx(0)q(0)1234 4607 y Fu(>)25 b(q)1371 4621 y Fm(\000)p Ft(N)1492 4607 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))27 b(=)e Fu(f)1876 4569 y Ft(k)1866 4629 y Fm(\000)p Ft(N)1988 4607 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))26 b(:=)f Fu(\013)2399 4569 y Ft(k)2399 4629 y Fm(\000)p Ft(N)2522 4607 y Fw(\()p Fu(y)2605 4569 y Fm(0)2628 4607 y Fu(;)15 b(\021)s Fw(\))22 b Fp(\000)d Fu(a)2911 4569 y Ft(k)2911 4629 y Fm(\000)p Ft(N)3034 4607 y Fw(\()p Fu(y)s(;)c(\021)s Fw(\))p Fu(:)-2879 b Fw(\(3.17\))386 4777 y(In)30 b(view)f(of)i(\(3.10\))h(and)e (\(3.12\),)j(w)m(e)e(de\014ne)e Fu(\013)1994 4744 y Ft(k)1994 4804 y Fm(\000)p Ft(N)2147 4777 y Fw(b)m(y)886 4949 y Fp(F)951 4964 y Ft(y)988 4945 y Fx(0)1011 4964 y Fm(!)p Ft(\020)1118 4945 y Fx(0)1144 4949 y Fw(\()p Fu(\013)1237 4911 y Ft(k)1237 4971 y Fm(\000)p Ft(N)1359 4949 y Fw(\()p Fu(y)1442 4911 y Fm(0)1466 4949 y Fu(;)15 b(\021)s Fw(\)\))27 b(=)e Fp(F)1812 4964 y Ft(y)r Fm(!)p Ft(\020)1959 4949 y Fw(\()p Fu(a)2042 4911 y Ft(k)2042 4971 y Fm(\000)p Ft(N)2165 4949 y Fw(\()p Fu(y)2248 4911 y Fm(0)2271 4949 y Fu(;)15 b(y)2359 4911 y Fm(0)q(0)2402 4949 y Fu(;)g(\021)s Fw(\)\))p Fp(j)2585 4967 y Ft(\020)5 b Fq(=\()p Ft(\020)2739 4948 y Fx(0)2763 4967 y Ft(;)p Fq(0\))2849 4949 y Fu(:)-2488 b Fw(\(3.18\))386 5108 y(Bt)27 b(the)f(F)-8 b(rob)s(enius)24 b(theorem,)k(\(3.18\))g(and)d(\()p Fu(S)5 b(i)p Fw(\))2046 5122 y Ft(\033)2094 5108 y Fw(,)27 b(\(3.17\))h(admits)e(a)g(unique)e (solution)386 5216 y Fu(q)427 5230 y Fm(\000)p Ft(N)549 5216 y Fw(\()p Fu(y)s(;)15 b(\030)t Fw(\),)41 b(if)c Fp(F)973 5231 y Ft(y)1010 5212 y Fx(0)1033 5231 y Fm(!)p Ft(\020)1140 5212 y Fx(0)1166 5216 y Fw(\()p Fu(q)1242 5230 y Fm(\000)p Ft(N)1364 5216 y Fw(\()p Fu(y)1447 5183 y Fm(0)1471 5216 y Fu(;)15 b(y)1559 5183 y Fm(00)1601 5216 y Fu(;)g(\021)s Fw(\)\))40 b(=)f(0)f(for)g Fu(\020)2186 5183 y Fm(0)2248 5216 y Fp(2)g Fs(Z)2412 5183 y Ft(r)2446 5216 y Fw(.)64 b(Moreo)m(v)m(er,)43 b(in)37 b(view)g(of)386 5324 y(\()p Fu(S)5 b(i)p Fw(\))548 5338 y Ft(\033)596 5324 y Fw(,)30 b Fu(q)692 5338 y Fm(\000)p Ft(N)814 5324 y Fw(\()p Fu(y)s(;)15 b(\030)t Fw(\))31 b(is)f Fu(C)1211 5291 y Fm(1)1285 5324 y Fw(\()p Fs(R)1380 5291 y Ft(n)1453 5324 y Fp(n)21 b(f)p Fw(0)p Fp(g)p Fw(;)15 b Fu(G)1766 5291 y Ft(\033)1814 5324 y Fw(\()p Fs(T)1908 5291 y Ft(n)1957 5324 y Fw(\)\).)486 5432 y(Finally)28 b(w)m(e)j(construct)f(an)h (ellpitic)d(p.d.o.)40 b Fu(q)33 b Fw(b)m(y)793 5586 y Fu(q)s Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))26 b(=)f Fu(:)15 b(:)g(:)21 b Fp(\016)g Fw(\(1)g(+)f Fu(q)1590 5600 y Fm(\000)p Ft(N)1712 5586 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\)\))22 b Fp(\016)e Fu(:)15 b(:)g(:)22 b Fp(\016)e Fw(\(1)h(+)f Fu(q)2465 5600 y Fm(\000)p Fq(1)2559 5586 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\)\))22 b Fp(\016)f Fu(q)2929 5600 y Fq(0)2968 5586 y Fw(\()p Fu(y)s(;)15 b(\021)s Fw(\))p Fu(:)-2813 b Fw(\(3.19\))p eop %%Page: 10 10 10 9 bop 386 239 a Fq(10)693 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g (YOSHINO)386 434 y Fw(The)c(righ)m(t-hand)f(side)g(in)g(\(3.19\))j (de\014nes)e(a)g(formal)g(sym)m(b)s(ol)f(in)g Fu(F)13 b(S)2739 401 y Fq(0)2778 434 y Fw(\()p Fs(T)2872 401 y Ft(n)2933 434 y Fp(\002)f Fw(\()p Fs(R)3111 401 y Ft(n)3176 434 y Fp(n)g Fw(0\)\).)386 542 y(T)-8 b(aking)37 b(an)m(y)g (realization)f(of)h(this)f(formal)g(sym)m(b)s(ol,)i(the)f(ab)s(o)m(v)m (e)h(calculations)e(yield)386 650 y(\(3.8\))c(and)e(the)g(estimates)h (on)g(the)f(reminders.)p 3309 650 4 62 v 3313 592 55 4 v 3313 650 V 3367 650 4 62 v 486 807 a(The)f(next)i(assertion)f(sho)m (ws)g(that)h(\()p Fu(S)5 b(i)p Fw(\))1877 821 y Ft(\033)1955 807 y Fw(is)30 b(optimal.)386 966 y FE(Prop)s(osition)36 b(3.2.)46 b Fk(L)-5 b(et)34 b Fu(!)c Fp(2)c Fs(R)1551 933 y Ft(n)1637 966 y Fk(b)-5 b(e)34 b(nonr)-5 b(esonant)36 b(and)e(assume)g(that)h Fu(!)h Fk(do)-5 b(es)35 b(not)386 1074 y(satisfy)46 b Fw(\()p Fu(S)5 b(i)p Fw(\))844 1088 y Ft(\033)937 1074 y Fk(for)46 b(some)g Fw(1)j Fp(\024)e Fu(\033)52 b Fp(\024)c(1)p Fk(.)79 b(Then)46 b(we)f(c)-5 b(an)46 b(\014nd)g Fu(f)10 b Fw(\()p Fu(x)p Fw(\))48 b Fp(2)g Fu(G)3150 1041 y Ft(\033)3196 1074 y Fw(\()p Fs(T)3290 1041 y Ft(n)3339 1074 y Fw(\))386 1182 y Fk(such)30 b(that)h(the)f(pseudo{di\013er)-5 b(ential)33 b(op)-5 b(er)g(ator)33 b(with)e(symb)-5 b(ol)31 b Fu(<)25 b(!)s(;)15 b(\030)29 b(>)c Fw(+)p Fu(f)10 b Fw(\()p Fu(x)p Fw(\))p Fu(\030)t Fp(j)p Fu(\030)t Fp(j)3280 1149 y Fm(\000)p Fq(1)386 1290 y Fk(c)-5 b(annot)34 b(b)-5 b(e)32 b(tr)-5 b(ansforme)g(d)36 b(via)d(c)-5 b(onjugation)34 b(\(3.8\))g(into)f(a)g (c)-5 b(onstant)34 b(p.d.o.)386 1449 y(Pr)-5 b(o)g(of.)43 b Fw(In)26 b(view)f(of)i(the)g(assumption)e(on)h Fu(!)k Fw(there)d(exists)f(a)h(p)s(ositiv)m(e)e(constan)m(t)j Fu(\016)h(>)c Fw(0)386 1556 y(and)30 b(a)g(sequence)h Fu(\032)1062 1523 y Ft(k)1130 1556 y Fp(2)25 b Fs(Z)1281 1523 y Ft(n)1344 1556 y Fp(n)20 b Fw(0)31 b(\()p Fu(k)e Fw(=)c(1)p Fu(;)15 b Fw(2)p Fu(;)g(:)g(:)g(:)j Fw(\))31 b(suc)m(h)f(that)1208 1724 y(lim)1182 1785 y Ft(k)r Fm(!1)1377 1724 y Fu(e)1419 1686 y Fm(\000)p Fq(2)p Ft(\016)r Fm(j)p Ft(\032)1598 1663 y FA(k)1637 1686 y Fm(j)1657 1663 y Fo(1)p FA(=\033)1764 1724 y Fp(j)c Fu(<)f(!)s(;)15 b(\032)2058 1686 y Ft(k)2126 1724 y Fu(>)25 b Fp(j)2247 1686 y Fm(\000)p Fq(1)2367 1724 y Fw(=)g Fp(1)p Fu(:)-2193 b Fw(\(3.20\))386 1927 y(W)-8 b(e)46 b(c)m(ho)s(ose)h(a)f(function)e Fu(f)10 b Fw(\()p Fu(x)p Fw(\))50 b(=)1671 1859 y Fr(P)1767 1954 y Ft(\021)1843 1903 y Fw(^)1824 1927 y Fu(f)9 b Fw(\()p Fu(\021)s Fw(\))p Fu(e)2038 1894 y Ft(i\021)r(x)2196 1927 y Fp(2)49 b Fu(G)2378 1894 y Ft(\033)2425 1927 y Fw(\()p Fs(T)2519 1894 y Ft(n)2568 1927 y Fw(\))c(suc)m(h)g(that)3100 1903 y(^)3080 1927 y Fu(f)10 b Fw(\()p Fu(\021)s Fw(\))51 b(=)386 2069 y Fu(e)428 2036 y Fm(\000)p Ft(\016)r Fm(j)p Ft(\032)572 2013 y FA(k)611 2036 y Fm(j)631 2013 y Fo(1)p FA(=\033)782 2069 y Fw(if)43 b Fu(\021)50 b Fw(=)d Fu(\032)1139 2036 y Ft(k)1182 2069 y Fw(,)1274 2045 y(^)1254 2069 y Fu(f)10 b Fw(\()p Fu(\021)s Fw(\))48 b(=)f(0)d(for)g Fu(\021)50 b Fp(6)p Fw(=)d Fu(\032)2095 2036 y Ft(k)2138 2069 y Fw(.)81 b(The)43 b(solution)f Fu(q)2842 2083 y Fq(0)2881 2069 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))45 b(of)e(the)386 2194 y(equation)32 b Fu(<)d(!)s(;)15 b(D)1034 2208 y Ft(x)1107 2194 y Fu(>)29 b(q)1248 2208 y Fq(0)1316 2194 y Fw(=)g Fu(f)10 b Fw(\()p Fu(x)p Fw(\))p Fu(\030)t(=)p Fp(j)p Fu(\030)t Fp(j)33 b Fw(satis\014es)f Fp(jF)2236 2208 y Ft(x)p Fm(!)p Ft(\021)2389 2194 y Fw(\()p Fu(q)2465 2208 y Fq(0)2504 2194 y Fw(\()p Fu(x;)15 b(\030)t Fw(\)\))p Fp(j)31 b(\025)e Fu(C)7 b(e)3015 2161 y Ft(\016)r Fm(j)p Ft(\021)r Fm(j)3125 2137 y Fo(1)p FA(=\033)3266 2194 y Fw(for)386 2308 y Fu(\021)29 b Fw(=)c Fu(\032)603 2275 y Ft(k)645 2308 y Fw(,)31 b Fu(k)d Fp(2)d Fs(N)7 b Fw(,)36 b(whic)m(h)29 b(con)m(tradicts)i(the)g(fact)g(that)g Fu(q)2285 2322 y Fq(0)2349 2308 y Fp(2)25 b Fu(G)2507 2275 y Ft(\033)2584 2308 y Fw(in)k Fu(x)p Fw(.)p 3309 2308 V 3313 2250 55 4 v 3313 2308 V 3367 2308 4 62 v 386 2467 a FE(Remark)34 b(3.3.)46 b Fk(In)31 b(The)-5 b(or)g(em)33 b(3.1)f(we)f(c)-5 b(onsider)32 b(p)-5 b(erturb)g(ations)34 b(of)d(c)-5 b(onstant)33 b(ve)-5 b(ctor)386 2574 y(\014elds)37 b Fu(X)697 2589 y Ft(k)740 2574 y Fk(.)53 b(If)36 b Fu(X)995 2589 y Ft(k)1075 2574 y Fk(is)g(a)h(variable)g(c)-5 b(o)g(e\016cient)37 b(ve)-5 b(ctor)37 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b(\014elds,)i Fu(L)e Fw(=)386 3330 y Fu(d)433 3344 y Ft(t)484 3330 y Fw(+)21 b Fu(!)s Fw(\()p Fu(t)p Fw(\))h Fp(^)f Fu(@)891 3344 y Ft(x)936 3330 y Fk(,)34 b Fu(x)28 b Fp(2)g Fs(T)1226 3297 y Fq(1)1268 3330 y Fk(,)34 b Fu(t)28 b Fw(=)g(\()p Fu(t)1558 3344 y Fq(1)1598 3330 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(t)1848 3344 y Ft(n)1895 3330 y Fw(\))29 b Fp(2)f Fs(T)2107 3297 y Ft(n)2155 3330 y Fk(,)35 b Fu(!)s Fw(\()p Fu(t)p Fw(\))29 b(=)2509 3262 y Fr(P)2605 3288 y Ft(n)2605 3357 y(j)t Fq(=1)2747 3330 y Fu(!)2804 3344 y Ft(j)2840 3330 y Fw(\()p Fu(t)p Fw(\))p Fu(dt)3023 3344 y Ft(j)3094 3330 y Fk(b)-5 b(eing)34 b(a)386 3441 y(r)-5 b(e)g(al-value)g(d)37 b(smo)-5 b(oth)38 b(close)-5 b(d)37 b(one)f(form)h(on)f Fu(M)41 b Fp(\021)30 b Fs(T)2238 3408 y Ft(n)2286 3441 y Fk(.)51 b(The)36 b(c)-5 b(orr)g(esp)g(onding)39 b(family)386 3549 y(of)h Fu(n)g Fk(c)-5 b(ommuting)40 b(ve)-5 b(ctor)41 b(\014elds)g(asso)-5 b(ciate)g(d)42 b(with)f Fu(L)f Fk(\(cf.)63 b Fw([6)q(])p Fk(\))40 b(is)g(given)f(by)h Fu(L)3229 3563 y Ft(j)3304 3549 y Fw(=)386 3657 y Fu(@)434 3671 y Ft(t)459 3681 y FA(j)521 3657 y Fw(+)24 b Fu(!)673 3671 y Ft(j)710 3657 y Fw(\()p Fu(t)p Fw(\))p Fu(@)861 3671 y Ft(x)905 3657 y Fk(,)40 b Fw(1)d Fp(\024)g Fu(j)42 b Fp(\024)36 b Fu(n)p Fk(.)60 b(Str)-5 b(aightforwar)g(d)43 b(c)-5 b(alculations)41 b(show)f(the)g(existenc)-5 b(e)386 3767 y(of)35 b(a)h(r)-5 b(e)g(al-value)g(d)37 b(function)e Fu(h)p Fw(\()p Fu(t)p Fw(\))h Fk(on)f Fs(T)1771 3734 y Ft(n)1855 3767 y Fk(such)g(that)h Fu(@)2300 3781 y Ft(t)2325 3791 y FA(j)2363 3767 y Fu(h)p Fw(\()p Fu(t)p Fw(\))30 b(=)f Fu(!)2705 3781 y Ft(j)2742 3767 y Fw(\()p Fu(t)p Fw(\))22 b Fp(\000)g Fu(c)2999 3781 y Ft(j)3036 3767 y Fk(,)35 b(with)h Fu(c)3338 3781 y Ft(j)386 3877 y Fk(b)-5 b(eing)42 b(the)g(me)-5 b(an)43 b(value)f Fw([)p Fu(!)1365 3891 y Ft(j)1402 3877 y Fw(\()p Fp(\001)p Fw(\)])1522 3891 y Ft(n)1612 3877 y Fk(of)g Fu(!)1785 3891 y Ft(j)1821 3877 y Fk(,)j Fw(1)e Fp(\024)f Fu(j)48 b Fp(\024)42 b Fu(n)p Fk(.)70 b(Then)42 b(the)h(family)f Fp(f)p Fu(L)3245 3891 y Ft(j)3282 3877 y Fp(g)3327 3844 y Ft(n)3327 3901 y Fq(1)386 3988 y Fk(is)36 b(tr)-5 b(ansforme)g(d)40 b(into)d(the)g(one)g Fp(f)p Fu(@)1609 4002 y Ft(s)1642 4012 y FA(j)1702 3988 y Fw(+)22 b Fu(c)1834 4002 y Ft(j)1871 3988 y Fu(@)1919 4002 y Ft(y)1961 3988 y Fp(g)2006 3955 y Ft(n)2006 4012 y Fq(1)2090 3988 y Fk(via)36 b(the)h(di\013e)-5 b(omorphism)39 b(of)e Fs(T)3208 3955 y Ft(n)p Fq(+1)3347 3988 y Fk(:)386 4097 y Fu(y)28 b Fw(=)d Fu(x)20 b Fp(\000)g Fu(h)p Fw(\()p Fu(t)p Fw(\))p Fk(,)33 b Fu(s)25 b Fw(=)g Fu(t)p Fk(.)486 4256 y Fw(W)-8 b(e)23 b(no)m(w)g(study)e(the)i(sim)m (ultaneous)e(reduction)g(of)i(a)f(family)f(of)i(comm)m(uting)f(v)m (ector)386 4364 y(\014elds)29 b(with)g(v)-5 b(ariable)29 b(co)s(e\016cien)m(ts)i(on)f Fs(T)1812 4331 y Fq(2)1120 4510 y Fu(X)1195 4524 y Ft(j)1257 4510 y Fw(=)25 b Fu(@)1401 4524 y Ft(t)1451 4510 y Fw(+)20 b Fu(h)1594 4524 y Ft(j)1631 4510 y Fw(\()p Fu(t;)15 b(x)p Fw(\))p Fu(@)1874 4524 y Ft(x)1919 4510 y Fu(;)197 b(j)31 b Fw(=)25 b(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d:)-2253 b Fw(\(3.21\))386 4656 y(The)30 b(comm)m(utativit)m(y)h(relations)e(are)i(giv)m(en)f(b)m(y)863 4801 y Fu(@)911 4815 y Ft(t)941 4801 y Fw(\()p Fu(h)1028 4816 y Ft(k)1091 4801 y Fp(\000)20 b Fu(h)1234 4815 y Ft(j)1271 4801 y Fw(\))26 b(=)f Fu(h)1480 4816 y Ft(k)1523 4801 y Fu(@)1571 4815 y Ft(x)1615 4801 y Fu(h)1667 4815 y Ft(j)1724 4801 y Fp(\000)20 b Fu(h)1867 4815 y Ft(j)1904 4801 y Fu(@)1952 4815 y Ft(x)1996 4801 y Fu(h)2048 4816 y Ft(k)2091 4801 y Fu(;)197 b(j;)15 b(k)30 b Fw(=)25 b(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d:)-2511 b Fw(\(3.22\))386 4961 y(Let)29 b Fu( )609 4928 y Ft(t)606 4986 y(j)668 4961 y Fw(:)c Fs(T)777 4928 y Fq(1)844 4961 y Fp(!)g Fs(T)1019 4928 y Fq(1)1088 4961 y Fw(b)s(e)j(the)h(time)f(map)g(of)h Fu(X)1954 4928 y Ft(j)1991 4961 y Fw(,)g(de\014ned)e(b)m(y)h Fu( )2542 4928 y Ft(t)2539 4986 y(j)2576 4961 y Fw(\()p Fu(\030)t Fw(\))e(=)f Fu(x)2864 4975 y Ft(j)2900 4961 y Fw(\()p Fu(t)p Fw(;)15 b Fu(\030)t Fw(\),)30 b(where)386 5076 y Fu(x)438 5090 y Ft(j)474 5076 y Fw(\()p Fu(t)p Fw(;)15 b Fu(\030)t Fw(\))32 b(is)d(a)i(solution)e(of)1383 5221 y(_)-56 b Fu(x)1404 5235 y Ft(j)1465 5221 y Fw(=)25 b Fu(h)1613 5235 y Ft(j)1650 5221 y Fw(\()p Fu(t;)15 b(x)1810 5235 y Ft(j)1847 5221 y Fw(\))p Fu(;)107 b(x)2066 5235 y Ft(j)2102 5221 y Fw(\(0\))27 b(=)e Fu(\030)t(:)-2023 b Fw(\(3.23\))386 5378 y(Clearly)29 b(\(3.22\))j(is)e(equiv)-5 b(alen)m(t)30 b(to)h Fu( )1661 5345 y Ft(t)1658 5403 y(j)1715 5378 y Fp(\016)20 b Fu( )1842 5345 y Ft(s)1839 5405 y(k)1908 5378 y Fw(=)k Fu( )2065 5345 y Ft(s)2062 5405 y(k)2126 5378 y Fp(\016)c Fu( )2253 5345 y Ft(t)2250 5403 y(j)2287 5378 y Fw(,)31 b Fu(s;)15 b(t)25 b Fp(2)f Fs(R)s Fu(;)37 b(j;)15 b(k)29 b Fw(=)c(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d)p Fw(.)486 5500 y(W)-8 b(e)31 b(recall)f(that)i(the)f(P)m (oincar)m(\023)-43 b(e)32 b(map)e(of)h Fu(X)1994 5514 y Ft(j)2031 5500 y Fw(,)g(giv)m(en)f(b)m(y)h Fu(P)2509 5514 y Ft(j)2572 5500 y Fw(:=)25 b Fu( )2755 5467 y Fq(2)p Ft(\031)2752 5526 y(j)2864 5500 y Fw(:)h Fs(T)2974 5467 y Fq(1)3041 5500 y Fp(!)g Fs(T)3217 5467 y Fq(1)3258 5500 y Fw(,)31 b(is)386 5623 y(a)h(di\013eomorphism)e(of)i Fs(T)1269 5590 y Fq(1)1342 5623 y Fw(and)f(w)m(e)i(assume)f(the)g (orien)m(tation)g(preserving)e(prop)s(ert)m(y)p eop %%Page: 11 11 11 10 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f (FIELDS)h(ON)g(TORI)559 b(11)386 434 y Fw(of)37 b Fu(P)554 448 y Ft(j)591 434 y Fw(.)60 b(The)37 b(rotation)g(n)m(um)m(b)s(er)f (of)h Fu(P)1731 448 y Ft(j)1805 434 y Fw(is)f(denoted)h(b)m(y)g Fu(\032)2433 448 y Ft(j)2505 434 y Fp(2)f Fs(R)46 b Fw(\()p Fu(\032)2787 448 y Ft(j)2861 434 y Fw(is)36 b(also)g(called)386 542 y(the)41 b(rotation)h(n)m(um)m(b)s(er)e(of)h(the)g(v)m(ector)i (\014eld)d Fu(X)2103 556 y Ft(j)2139 542 y Fw(\).)74 b(Moreo)m(v)m(er,)46 b(w)m(e)41 b(supp)s(ose)f(that)386 650 y(there)21 b(exists)f(a)h(smo)s(oth)g(di\013eomorphism)d Fu(u)i Fw(sim)m(ultaneously)f(conjugating)h(the)h(maps)386 758 y Fu(P)444 772 y Fq(1)484 758 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(P)758 773 y Ft(d)843 758 y Fw(to)44 b(the)g(rotations)f Fu(R)1603 772 y Ft(\032)1639 781 y Fo(1)1678 758 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(R)1964 772 y Ft(\032)2000 784 y FA(d)2084 758 y Fw(where)43 b Fu(R)2429 772 y Ft(\032)2465 782 y FA(j)2502 758 y Fw(\()p Fu(z)t Fw(\))48 b(=)e Fu(z)34 b Fw(+)28 b Fu(\032)3005 772 y Ft(j)3042 758 y Fw(,)46 b Fu(z)52 b Fp(2)46 b Fs(T)386 868 y Fw(\()p Fu(j)31 b Fw(=)25 b(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d)p Fw(\),)32 b(so)f(that)1156 1017 y Fu(u)1208 979 y Fm(\000)p Fq(1)1322 1017 y Fp(\016)21 b Fu(P)1446 1031 y Ft(j)1503 1017 y Fp(\016)g Fu(u)k Fw(=)g Fu(R)1811 1031 y Ft(\032)1847 1041 y FA(j)1884 1017 y Fu(;)197 b(j)31 b Fw(=)25 b(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d:)-2218 b Fw(\(3.24\))386 1165 y FE(Theorem)34 b(3.4.)46 b Fk(L)-5 b(et)39 b Fw(1)d Fp(\024)g Fu(\033)j Fp(\024)c Fw(+)p Fp(1)p Fk(.)59 b(If)39 b Fu(d)d Fp(\025)g Fw(2)i Fk(we)h(assume)g(that)h(the)f(Poinc)-5 b(ar)n(\023)-44 b(e)386 1273 y(maps)41 b Fu(P)688 1287 y Fq(1)728 1273 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(P)1002 1288 y Ft(d)1083 1273 y Fk(ar)-5 b(e)40 b(orientation)i(pr)-5 b(eserving)40 b(and)g(that)h(ther)-5 b(e)40 b(exists)g(an)g(index)386 1381 y Fu(j)31 b Fp(2)25 b(f)p Fw(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d)p Fp(g)27 b Fk(such)f(that)h Fw(\(2)p Fu(\031)s Fw(\))1515 1348 y Fm(\000)p Fq(1)1611 1381 y Fu(\032)1658 1395 y Ft(j)1720 1381 y Fk(is)f(irr)-5 b(ational.)42 b(Then)26 b(if)f(we)h(c)-5 b(an)26 b(\014nd)h(a)f Fu(G)3168 1348 y Ft(\033)3240 1381 y Fk(dif-)386 1492 y(fe)-5 b(omorphism)33 b Fu(u)c Fk(on)h Fs(T)1186 1459 y Fq(1)1257 1492 y Fk(satisfying)h (\(3.24\))g(the)f(map)h Fw(\010)p Fk(:)40 b Fu(t)25 b Fw(=)g Fu(s)p Fk(,)30 b Fu(x)25 b Fw(=)g Fu(\036)p Fw(\()p Fu(s;)15 b(y)s Fw(\))p Fk(,)31 b(wher)-5 b(e)386 1600 y Fu(\036)p Fw(\()p Fu(s;)15 b(y)s Fw(\))26 b(=)f Fu(x)815 1614 y Fq(1)854 1600 y Fw(\()p Fu(s;)15 b(u)p Fw(\()p Fu(y)5 b Fp(\000)q Fu(s)p Fw(\(2)p Fu(\031)s Fw(\))1394 1567 y Fm(\000)p Fq(1)1489 1600 y Fu(\032)1536 1614 y Fq(1)1576 1600 y Fw(\)\))24 b Fk(if)g Fu(d)h Fw(=)g(1)g Fk(and)f Fu(\036)p Fw(\()p Fu(s;)15 b(y)s Fw(\))26 b(=)f Fu(x)2584 1614 y Ft(j)2621 1600 y Fw(\()p Fu(s;)15 b(u)p Fw(\()p Fu(y)5 b Fp(\000)q Fu(s)p Fw(\(2)p Fu(\031)s Fw(\))3161 1567 y Fm(\000)p Fq(1)3256 1600 y Fu(\032)3303 1614 y Ft(j)3339 1600 y Fw(\))386 1711 y Fk(if)45 b Fu(d)50 b Fp(\025)e Fw(2)p Fk(,)i(is)45 b(a)i Fu(G)1101 1678 y Ft(\033)1193 1711 y Fk(di\013e)-5 b(omorphism)48 b(of)e Fs(T)2022 1678 y Fq(2)2109 1711 y Fk(tr)-5 b(ansforming)48 b Fu(@)2709 1725 y Ft(t)2769 1711 y Fw(+)29 b Fu(h)2921 1726 y Ft(k)2964 1711 y Fw(\()p Fu(t;)15 b(x)p Fw(\))p Fu(@)3207 1725 y Ft(x)3298 1711 y Fk(to)386 1818 y Fu(@)434 1832 y Ft(s)491 1818 y Fw(+)20 b(\(2)p Fu(\031)s Fw(\))752 1786 y Fm(\000)p Fq(1)848 1818 y Fu(\032)895 1833 y Ft(k)937 1818 y Fu(@)985 1832 y Ft(y)1060 1818 y Fk(for)33 b(al)5 b(l)33 b Fw(1)26 b Fp(\024)e Fu(k)29 b Fp(\024)c Fu(d)p Fk(.)386 1967 y(Pr)-5 b(o)g(of.)43 b Fw(W)-8 b(e)29 b(w)m(an)m(t)g(to)g (reduce)f(the)g(v)m(ector)i(\014elds)d Fu(X)2167 1981 y Ft(j)2232 1967 y Fw(to)i Fu(@)2389 1981 y Ft(t)2434 1967 y Fw(+)16 b Fu(\032)2568 1981 y Ft(j)2604 1967 y Fu(@)2652 1981 y Ft(x)2725 1967 y Fw(b)m(y)28 b(the)g(c)m(hange)h(of) 386 2078 y(v)-5 b(ariables)31 b(on)i Fs(T)953 2045 y Fq(2)994 2078 y Fw(,)g(\()p Fu(t;)15 b(x)p Fw(\))30 b(=)f(\010\()p Fu(s;)15 b(y)s Fw(\))32 b(with)g Fu(t)c Fw(=)h Fu(s)j Fw(and)g Fu(x)d Fw(=)g Fu(\036)p Fw(\()p Fu(s;)15 b(y)s Fw(\))33 b(sim)m(ultaneously)-8 b(.)386 2186 y(By)35 b(simple)d(computations)i(it)g(can)h(b)s(e)e(seen)i(that)g Fu(@)2234 2200 y Ft(s)2294 2186 y Fw(+)22 b(\(2)p Fu(\031)s Fw(\))2557 2153 y Fm(\000)p Fq(1)2653 2186 y Fu(\032)2700 2200 y Ft(j)2736 2186 y Fu(@)2784 2200 y Ft(y)2858 2186 y Fw(=)32 b Fu(@)3009 2200 y Ft(t)3061 2186 y Fw(+)23 b(\()p Fu(\036)3244 2200 y Ft(s)3304 2186 y Fw(+)386 2294 y Fu(\032)433 2308 y Ft(j)469 2294 y Fu(\036)523 2308 y Ft(y)565 2294 y Fw(\))p Fu(@)648 2308 y Ft(x)693 2294 y Fw(.)40 b(Hence)31 b(it)f(is)g(necessary)g(to)i(solv)m(e)e(the)h (equation)1019 2439 y Fu(\036)1073 2453 y Ft(s)1130 2439 y Fw(+)20 b Fu(\032)1268 2453 y Ft(j)1305 2439 y Fu(\036)1359 2453 y Ft(y)1426 2439 y Fw(=)25 b Fu(h)1574 2453 y Ft(j)1611 2439 y Fw(\()p Fu(s;)15 b(\036)p Fw(\()p Fu(s;)g(y)s Fw(\)\))p Fu(;)198 b(j)31 b Fw(=)25 b(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d:)-2354 b Fw(\(3.25\))486 2598 y(By)27 b(the)h(de\014nition)e (of)h Fu(x)1333 2612 y Ft(j)1370 2598 y Fw(\()p Fu(t)p Fw(;)15 b Fu(\030)t Fw(\))28 b(in)f(\(3.23\),)j(w)m(e)e(obtain)f(for)h Fu(\030)h Fw(=)c Fu(u)p Fw(\()p Fu(y)18 b Fp(\000)d Fu(s)p Fw(\(2)p Fu(\031)s Fw(\))3136 2565 y Fm(\000)p Fq(1)3231 2598 y Fu(\032)3278 2612 y Ft(j)3314 2598 y Fw(\),)386 2706 y Fu(t)25 b Fw(=)g Fu(s)487 2851 y(@)535 2865 y Ft(s)572 2851 y Fu(\036)c Fw(+)f(\(2)p Fu(\031)s Fw(\))908 2814 y Fm(\000)p Fq(1)1003 2851 y Fu(\032)1050 2865 y Ft(j)1087 2851 y Fu(@)1135 2865 y Ft(y)1177 2851 y Fu(\036)83 b Fw(=)114 b(_)-57 b Fu(x)1519 2865 y Ft(j)1556 2851 y Fw(\()p Fu(t;)15 b(\030)t Fw(\))21 b Fp(\000)f Fw(\(2)p Fu(\031)s Fw(\))2025 2814 y Fm(\000)p Fq(1)2120 2851 y Fu(\032)2167 2865 y Ft(j)2204 2851 y Fu(@)2252 2866 y Ft(\030)2290 2851 y Fu(x)2342 2865 y Ft(j)2379 2851 y Fw(\()p Fu(t)p Fw(;)15 b Fu(\030)t Fw(\))p Fu(u)2618 2814 y Fm(0)2642 2851 y Fw(\()p Fu(y)23 b Fp(\000)d Fu(s)p Fw(\(2)p Fu(\031)s Fw(\))3049 2814 y Fm(\000)p Fq(1)3144 2851 y Fu(\032)3191 2865 y Ft(j)3228 2851 y Fw(\))1314 2995 y(+)82 b(\(2)p Fu(\031)s Fw(\))1637 2957 y Fm(\000)p Fq(1)1733 2995 y Fu(\032)1780 3009 y Ft(j)1817 2995 y Fu(@)1865 3010 y Ft(\030)1903 2995 y Fu(x)1955 3009 y Ft(j)1991 2995 y Fw(\()p Fu(t)p Fw(;)15 b Fu(\030)t Fw(\))p Fu(u)2230 2957 y Fm(0)2254 2995 y Fw(\()p Fu(y)24 b Fp(\000)c Fu(s)p Fw(\(2)p Fu(\031)s Fw(\))2662 2957 y Fm(\000)p Fq(1)2757 2995 y Fu(\032)2804 3009 y Ft(j)2840 2995 y Fw(\))26 b(=)56 b(_)-56 b Fu(x)3049 3009 y Ft(j)3085 2995 y Fw(\()p Fu(t)p Fw(;)15 b Fu(\030)t Fw(\))1314 3130 y(=)82 b Fu(h)1519 3144 y Ft(j)1556 3130 y Fw(\()p Fu(t;)15 b(x)1716 3144 y Ft(j)1753 3130 y Fw(\()p Fu(t)p Fw(;)g Fu(\030)t Fw(\)\))27 b(=)e Fu(h)2150 3144 y Ft(j)2187 3130 y Fw(\()p Fu(s;)15 b(\036)p Fw(\()p Fu(s;)g(y)s Fw(\)\))p Fu(:)-2234 b Fw(\(3.26\))386 3278 y(Hence)31 b Fu(\036)g Fw(transforms)f Fu(@)1243 3292 y Ft(t)1293 3278 y Fw(+)20 b Fu(h)1436 3292 y Ft(j)1473 3278 y Fw(\()p Fu(t;)15 b(x)p Fw(\))p Fu(@)1716 3292 y Ft(x)1792 3278 y Fw(in)m(to)30 b Fu(@)2024 3292 y Ft(s)2082 3278 y Fw(+)20 b(\(2)p Fu(\031)s Fw(\))2343 3245 y Fm(\000)p Fq(1)2438 3278 y Fu(\032)2485 3292 y Ft(j)2522 3278 y Fu(@)2570 3292 y Ft(y)2611 3278 y Fw(.)42 b(If)30 b Fu(d)c Fp(\025)f Fw(2)31 b(for)g Fu(i)25 b Fp(6)p Fw(=)h Fu(j)5 b Fw(,)386 3389 y Fu(@)434 3403 y Ft(t)475 3389 y Fw(+)12 b Fu(h)610 3403 y Ft(i)638 3389 y Fw(\()p Fu(t;)j(x)p Fw(\))p Fu(@)881 3403 y Ft(x)952 3389 y Fw(is)25 b(transformed)h(to)g Fu(@)1698 3403 y Ft(s)1747 3389 y Fw(+)12 b Fu(g)1873 3403 y Ft(i)1901 3389 y Fw(\()p Fu(s;)j(y)s Fw(\))p Fu(@)2150 3403 y Ft(y)2218 3389 y Fw(for)26 b(some)g Fu(g)2619 3403 y Ft(i)2673 3389 y Fp(2)f Fu(G)2831 3356 y Ft(\033)2877 3389 y Fw(\()p Fs(T)2971 3356 y Fq(2)3013 3389 y Fw(;)15 b Fs(R)s Fw(\).)45 b(The)386 3497 y(comm)m(utativit)m(y)31 b(implies)d(that)946 3642 y(\()p Fu(@)1029 3656 y Ft(s)1087 3642 y Fw(+)19 b(\(2)p Fu(\031)s Fw(\))1347 3605 y Fm(\000)p Fq(1)1443 3642 y Fu(\032)1490 3656 y Ft(j)1527 3642 y Fu(@)1575 3656 y Ft(y)1616 3642 y Fw(\))p Fu(g)1694 3656 y Ft(i)1723 3642 y Fw(\()p Fu(s;)c(y)s Fw(\))26 b(=)f(0)p Fu(;)198 b(i)25 b Fw(=)g(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d;)32 b(i)25 b Fp(6)p Fw(=)g Fu(j:)-2660 b Fw(\(3.27\))386 3801 y(Since)29 b(\(2)p Fu(\031)s Fw(\))793 3768 y Fm(\000)p Fq(1)889 3801 y Fu(\032)936 3815 y Ft(j)1003 3801 y Fw(is)g(irrational) g(it)h(is)g(readily)f(v)m(eri\014ed)g(from)h(the)h(F)-8 b(ourier)30 b(expansion)386 3909 y(that)24 b(the)g Fu(g)769 3923 y Ft(i)821 3909 y Fw(are)g(constan)m(ts.)40 b(Because)25 b(the)f(rotation)g(n)m(um)m(b)s(er)e(is)h(in)m(v)-5 b(arian)m(tly)22 b(de\014ned)386 4017 y(for)30 b(v)m(ector)i(\014elds)d(on)h Fs(T)1216 3984 y Fq(2)1257 4017 y Fw(,)h(w)m(e)g(ha)m(v)m(e)g Fu(g)1699 4031 y Ft(i)1728 4017 y Fw(\()p Fu(s;)15 b(y)s Fw(\))26 b Fp(\021)e Fw(\(2)p Fu(\031)s Fw(\))2220 3984 y Fm(\000)p Fq(1)2316 4017 y Fu(\032)2363 4031 y Ft(i)2391 4017 y Fw(.)486 4125 y(In)30 b(order)h(to)h(pro)m(v)m(e)g(that)f(the)h (map)f(\010)f(is)h(a)g Fu(G)2092 4092 y Ft(\033)2170 4125 y Fw(di\013eomorphism)d(on)j Fs(T)2996 4092 y Fq(2)3068 4125 y Fw(w)m(e)h(note)386 4233 y(that)41 b(the)g(Jacobian)g(of)g(\010) g(do)s(es)f(not)h(v)-5 b(anish,)43 b(b)s(ecause)d(it)h(is)f(equal)g(to) i Fu(@)3049 4247 y Ft(y)3090 4233 y Fu(\036)p Fw(.)73 b(F)-8 b(or)386 4343 y Fu(\030)29 b Fw(=)c Fu(u)p Fw(\()p Fu(y)20 b Fp(\000)c Fu(s)p Fw(\(2)p Fu(\031)s Fw(\))1003 4310 y Fm(\000)p Fq(1)1098 4343 y Fu(\032)1145 4357 y Ft(j)1181 4343 y Fw(\))29 b(w)m(e)g(ha)m(v)m(e)h Fu(@)1633 4357 y Ft(y)1674 4343 y Fu(\036)p Fw(\()p Fu(s;)15 b(y)s Fw(\))26 b(=)f Fu(@)2099 4358 y Ft(\030)2137 4343 y Fu(x)2189 4357 y Ft(j)2226 4343 y Fw(\()p Fu(t)p Fw(;)15 b Fu(\030)t Fw(\))p Fu(u)2465 4310 y Fm(0)2489 4343 y Fw(\()p Fu(y)20 b Fp(\000)c Fu(s)p Fw(\(2)p Fu(\031)s Fw(\))2889 4310 y Fm(\000)p Fq(1)2984 4343 y Fu(\032)3031 4357 y Ft(j)3067 4343 y Fw(\).)41 b(Since)386 4451 y Fu(u)32 b Fw(is)f(a)h Fu(G)712 4418 y Ft(\033)790 4451 y Fw(di\013eomorphism)d(w)m(e)j(ha)m (v)m(e)h Fu(u)1829 4418 y Fm(0)1853 4451 y Fw(\()p Fu(y)s Fw(\))28 b Fp(6)p Fw(=)f(0)33 b(for)e(ev)m(ery)i Fu(y)d Fp(2)e Fs(R)r Fw(.)51 b(On)31 b(the)h(other)386 4559 y(hand,)d Fu(\022)s Fw(\()p Fu(t)p Fw(;)15 b Fu(z)t Fw(\))26 b(:=)f Fu(@)1068 4573 y Ft(z)1108 4559 y Fu(x)1160 4573 y Ft(j)1197 4559 y Fw(\()p Fu(t)p Fw(;)15 b Fu(z)t Fw(\))31 b(satis\014es)1201 4692 y(_)1183 4716 y Fu(\022)d Fw(=)d Fu(@)1398 4730 y Ft(x)1442 4716 y Fu(h)1494 4730 y Ft(j)1531 4716 y Fw(\()p Fu(t;)15 b(x)1691 4730 y Ft(j)1728 4716 y Fw(\()p Fu(t)p Fw(;)g Fu(z)t Fw(\)\))p Fu(\022)s(;)198 b(\022)s Fp(j)2292 4730 y Ft(t)p Fq(=0)2436 4716 y Fw(=)25 b(1)-2191 b(\(3.28\))386 4867 y(whic)m(h)41 b(implies)f(that)k Fu(\022)s Fw(\()p Fu(t)p Fw(;)15 b Fu(\030)t Fw(\))46 b(=)f Fu(@)1632 4882 y Ft(\030)1670 4867 y Fu(x)1722 4881 y Ft(j)1759 4867 y Fw(\()p Fu(t)p Fw(;)15 b Fu(\030)t Fw(\))47 b Fp(6)p Fw(=)e(0)f(for)e(all)g Fu(t)k Fp(2)f Fs(R)s Fw(,)52 b Fu(\030)e Fp(2)45 b Fs(R)s Fw(.)84 b(Th)m(us)386 4975 y Fu(@)434 4989 y Ft(y)476 4975 y Fu(\036)p Fw(\()p Fu(s;)15 b(y)s Fw(\))31 b Fp(6)p Fw(=)f(0)k(for)f(all)f Fu(s)f Fp(2)f Fs(R)r Fw(,)41 b Fu(y)33 b Fp(2)d Fs(R)42 b Fw(and)33 b(the)h(map)f(\010)g(is)f(a)i Fu(G)2660 4942 y Ft(\033)2740 4975 y Fw(di\013eomorphism.)386 5083 y(In)40 b(order)g(to)h(complete)g(the)f(pro)s(of,)j(w)m(e)e(will)d(sho)m(w)i (that)h Fu(\036)p Fw(\()p Fu(s)27 b Fw(+)g(2)p Fu(\031)s(;)15 b(y)s Fw(\))43 b(=)f Fu(\036)p Fw(\()p Fu(s;)15 b(y)s Fw(\))386 5191 y(and)36 b Fu(\036)p Fw(\()p Fu(s;)15 b(y)28 b Fw(+)c(2)p Fu(\031)s Fw(\))36 b(=)g Fu(\036)p Fw(\()p Fu(s;)15 b(y)s Fw(\).)60 b(The)36 b(latter)h(relation)f(is)f (easy)-8 b(.)61 b(T)-8 b(o)37 b(see)g(the)g(former)386 5299 y(one,)43 b(w)m(e)e(recall)f(that)h Fu(x)1246 5313 y Ft(j)1283 5299 y Fw(\(2)p Fu(\031)s(;)15 b(u)p Fw(\()p Fu(y)s Fw(\)\))44 b(=)e Fu(u)p Fw(\()p Fu(y)30 b Fw(+)d Fu(\032)2127 5313 y Ft(j)2163 5299 y Fw(\))41 b(b)m(y)g(the)f (de\014nition)e(of)j Fu(u)p Fw(.)71 b(The)386 5407 y(function)37 b Fu(x)802 5421 y Ft(j)838 5407 y Fw(\()p Fu(t)26 b Fw(+)f(2)p Fu(\031)s(;)15 b(u)p Fw(\()p Fu(y)29 b Fp(\000)d Fu(\032)1473 5421 y Ft(j)1534 5407 y Fp(\000)g Fu(s\032)1721 5421 y Ft(j)1757 5407 y Fu(=)p Fw(2)p Fu(\031)s Fw(\)\))39 b(is)f(a)g(solution)f(of)h(\(3.23\))j(with)36 b(initial)386 5515 y(v)-5 b(alue)32 b Fu(\030)i Fw(=)29 b Fu(x)849 5529 y Ft(j)886 5515 y Fw(\(2)p Fu(\031)s(;)15 b(u)p Fw(\()p Fu(y)26 b Fp(\000)c Fu(\032)1359 5529 y Ft(j)1418 5515 y Fp(\000)f Fu(s\032)1600 5529 y Ft(j)1636 5515 y Fu(=)p Fw(2)p Fu(\031)s Fw(\)\))32 b(=)d Fu(u)p Fw(\()p Fu(y)c Fp(\000)d Fu(s\032)2323 5529 y Ft(j)2359 5515 y Fu(=)p Fw(2)p Fu(\031)s Fw(\).)50 b(By)33 b(uniqueness)e(it)i(is)386 5623 y(therefore)e(equal)f(to)h Fu(x)1171 5637 y Ft(j)1207 5623 y Fw(\()p Fu(t;)15 b(u)p Fw(\()p Fu(y)24 b Fp(\000)c Fu(s\032)1652 5637 y Ft(j)1688 5623 y Fu(=)p Fw(2)p Fu(\031)s Fw(\)\).)p 3309 5623 4 62 v 3313 5565 55 4 v 3313 5623 V 3367 5623 4 62 v eop %%Page: 12 12 12 11 bop 386 239 a Fq(12)693 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g (YOSHINO)386 434 y FE(Remark)k(3.5.)46 b Fk(a\))40 b(L)-5 b(et)40 b Fu(d)d Fw(=)g(1)j Fk(and)g(supp)-5 b(ose)41 b(either)e(that)i Fu(\033)f Fw(=)d(1)j Fk(and)g Fu(\032=)p Fw(\(2)p Fu(\031)s Fw(\))f(=)386 542 y Fu(\032)433 556 y Fq(1)472 542 y Fu(=)p Fw(\(2)p Fu(\031)s Fw(\))32 b Fk(satis\014es)e(the)h(Bruno)f(c)-5 b(ondition)31 b Fw([9)q(])f Fk(or)h(that)g Fu(\033)d Fw(=)d Fp(1)30 b Fk(and)g Fu(\032=)p Fw(\(2)p Fu(\031)s Fw(\))i Fk(satis\014es)386 650 y(the)g(usual)g(Sie) -5 b(gel)32 b(c)-5 b(ondition.)43 b(Then)32 b(by)f(the)h(glob)-5 b(al)33 b(r)-5 b(e)g(duction)34 b(the)-5 b(or)g(ems)33 b(due)f(to)g(M.)386 758 y(Herman)37 b(and)h(A.)d(Bruno,)j(the)f(cir)-5 b(cle)37 b(map)h Fu(P)45 b Fw(=)32 b Fu(P)2238 772 y Fq(1)2315 758 y Fk(is)k(c)-5 b(onjugate)g(d)38 b(to)f(a)g(r)-5 b(otation)386 866 y(by)32 b(a)h Fu(G)655 833 y Fq(1)727 866 y Fk(or)g Fu(C)916 833 y Fm(1)1023 866 y Fk(di\013e)-5 b(omorphism.)386 973 y(b\))36 b(L)-5 b(et)37 b Fu(d)c Fw(=)g(1)k Fk(and)g Fw(1)c Fu(<)g(\033)i(<)e Fw(+)p Fp(1)p Fk(.)53 b(We)36 b(ar)-5 b(e)38 b(not)f(awar)-5 b(e)39 b(of)d(any)i(r)-5 b(esult)37 b(on)g(glob)-5 b(al)386 1081 y(r)g(e)g(duction)34 b(in)f Fu(G)966 1048 y Ft(\033)1044 1081 y Fk(Gevr)-5 b(ey)34 b(classes.)386 1189 y(c\))h(L)-5 b(et)35 b Fu(d)30 b Fp(\025)f Fw(2)p Fk(.)49 b(If)35 b Fu(\033)e Fw(=)c(+)p Fp(1)34 b Fk(or)i Fu(\033)d Fw(=)c(1)p Fk(,)36 b(ther)-5 b(e)35 b(is)g(a)h(lo)-5 b(c)g(al)36 b(simultane)-5 b(ous)37 b(r)-5 b(e)g(duction)386 1297 y(the)g(or)g(em)32 b(due)d(to)h(J.)f(Moser)h Fw([32)q(])p Fk(,)g(under)g(a)g(simultane)-5 b(ous)32 b(Sie)-5 b(gel)29 b(c)-5 b(ondition.)43 b(A)29 b(glo-)386 1405 y(b)-5 b(al)34 b Fu(C)598 1372 y Fm(1)705 1405 y Fk(simultane)-5 b(ous)34 b(r)-5 b(e)g(duction)34 b(the)-5 b(or)g(em)35 b(is)e(shown)i(by)d(Y.)h (Katznelson,)h(B.)f(Kr)-5 b(a)386 1513 y(and)33 b(D.)g(Ornstein)f Fw([28)r(])p Fk(.)386 1621 y(d\))44 b(If)f Fu(d)h Fp(\025)h Fw(1)e Fk(and)h Fw(1)h Fp(\024)f Fu(\033)k(<)c Fw(+)p Fp(1)p Fk(,)h(T.)e(Gr)-5 b(amchev)45 b(and)f(M.)f(Y)-7 b(oshino)44 b Fw([22)q(])g Fk(pr)-5 b(o-)386 1729 y(ve)g(d)33 b(lo)-5 b(c)g(al)35 b(simultane)-5 b(ous)35 b(r)-5 b(e)g(ductions)35 b(of)e(c)-5 b(ommuting)34 b Fu(G)2369 1696 y Ft(\033)2449 1729 y Fk(cir)-5 b(cle)33 b(maps)h(under)g(we)-5 b(ak)386 1837 y(exp)g(onential)35 b(arithmetic)e(c)-5 b(onditions.)840 2099 y Fw(4.)51 b Fv(Nekhor)n(oshev)31 b(type)j(estima)-6 b(tes)32 b(f)n(or)i Fu(G)2588 2066 y Ft(\033)2668 2099 y Fv(p.d.o.)486 2261 y Fw(The)39 b(next)g(result)g(sho)m(ws)g(that)i(p) s(erturbations)c(with)h Fu(G)2501 2228 y Ft(\033)2587 2261 y Fw(p.d.o.)69 b(are)40 b(conjuga-)386 2369 y(ted)h(with)f Fu(G)843 2336 y Ft(\022)922 2369 y Fw(elliptic)f(p.d.o.)72 b(for)41 b(some)h Fu(\022)j(>)e(\033)h Fw(if)c(sim)m(ultaneous)f(small) h(divisor)386 2477 y(conditions)30 b(in)g(the)h(nonresonan)m(t)g(case)i (are)e(satis\014ed.)43 b(This)29 b(sho)m(ws)i(sharp)f(con)m(trast)386 2585 y(to)39 b(the)f(case)h(of)g(comm)m(uting)f(di\013eomorphisms,)f (where)h(ev)m(en)h(w)m(eak)m(er)g(arithmetic)386 2693 y(conditions)29 b(are)i(su\016cien)m(t.)386 2870 y FE(Theorem)j(4.1.)46 b Fk(L)-5 b(et)25 b Fw(\000)1244 2837 y Ft(!)1244 2898 y Fl(Z)1320 2870 y Fw(=)g Fp(f)p Fw(0)p Fp(g)h Fk(and)g(let)g Fw(\()p Fu(S)5 b(i)p Fw(\))2029 2884 y Fm(1)2129 2870 y Fk(in)26 b Fp(x)p Fk(2)f(hold)i(for)f(some)g Fu(\034)35 b Fp(\024)25 b(\000)p Fw(1)p Fk(.)39 b(L)-5 b(et)386 2985 y Fu(\022)27 b Fw(=)e(max)q Fp(f)p Fw(1)p Fu(;)15 b Fw(1)k(+)f Fu(\034)10 b Fp(g)p Fk(.)42 b(Asssume)32 b(that)g Fu(a)1778 2952 y Ft(k)1821 2985 y Fw(\()p Fu(x;)15 b(D)s Fw(\))33 b(\(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d)p Fw(\))32 b Fk(is)g(a)g Fu(G)2828 2952 y Ft(\033)2906 2985 y Fk(p.d.o.)42 b(Then)386 3093 y(ther)-5 b(e)45 b(exist)f(symb)-5 b(ols)46 b Fu(q)1232 3107 y Fm(\000)p Ft(j)1323 3093 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))45 b Fk(of)g(the)f(c)-5 b(onjugating)45 b(p.d.o.)78 b(with)45 b(the)g(fol)5 b(lowing)386 3211 y(pr)-5 b(op)g(erties:)63 b(ther)-5 b(e)43 b(exists)f(a)g(neighb)-5 b(ourho)g(o)g(d)2056 3188 y Fw(~)2041 3211 y Fu(S)47 b Fp(\032)42 b Fs(C)2316 3178 y Ft(n)2411 3211 y Fk(of)g(the)g(unit)g(spher)-5 b(e)44 b Fu(S)3238 3178 y Ft(n)p Fm(\000)p Fq(1)386 3319 y Fk(such)33 b(that)h(for)f(every)f Fu(s)25 b(>)g(n=)p Fw(2)p Fk(,)33 b Fu(")26 b(>)f Fw(0)32 b Fk(we)h(c)-5 b(an)34 b(\014nd)f Fu(M)i(>)25 b Fw(0)p Fk(,)33 b Fu(N)i(>)25 b Fw(0)33 b Fk(satisfying)628 3503 y Fw(sup)633 3597 y Ft(\030)s Fm(2)725 3580 y Fq(~)714 3597 y Ft(S)780 3503 y Fp(k)p Fu(@)878 3465 y Ft(\015)873 3526 y(x)923 3503 y Fu(q)964 3517 y Fm(\000)p Ft(j)1055 3503 y Fw(\()p Fu(:;)15 b(\030)t Fw(\))p Fp(k)1279 3517 y Ft(H)1342 3498 y FA(s)1464 3503 y Fp(\024)83 b Fu("M)1758 3465 y Ft(j)1795 3503 y Fu(N)1878 3465 y Fm(j)p Ft(\015)t Fm(j)1962 3503 y Fw(\(\()p Fu(\022)s(j)25 b Fw(+)20 b Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!\))2428 3465 y Ft(\033)2477 3503 y Fu(;)108 b(\015)30 b Fp(2)25 b Fs(Z)2838 3465 y Ft(n)2838 3526 y Fq(+)2893 3503 y Fu(;)48 b(j)31 b Fp(2)25 b Fs(N)6 b Fu(;)-2818 b Fw(\(4.1\))915 3738 y(sup)919 3832 y Ft(\030)s Fm(2)1012 3816 y Fq(~)1000 3832 y Ft(S)1067 3738 y Fp(j)p Fu(\013)1150 3701 y Ft(k)1150 3761 y Fm(\000)p Ft(j)1241 3738 y Fw(\()p Fu(\030)t Fw(\))p Fp(j)84 b(\024)f Fu("M)1758 3701 y Ft(j)1795 3738 y Fw(\(\()p Fu(\022)s(j)5 b Fw(\)!\))2048 3701 y Ft(\033)2096 3738 y Fu(;)202 b(k)28 b Fw(=)d(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d;)32 b(j)f Fp(2)24 b Fs(N)7 b Fu(;)-2712 b Fw(\(4.2\))386 3998 y Fk(wher)-5 b(e)25 b Fp(k)q(\001)q(k)751 4012 y Ft(H)814 3994 y FA(s)876 3998 y Fk(denotes)f(the)g(usual)g(Sob)-5 b(olev)25 b(norm.)40 b(It)24 b(fol)5 b(lows)25 b(that)2712 3930 y Fr(P)2808 3956 y Fm(1)2808 4025 y Ft(j)t Fq(=0)2950 3998 y Fu(q)2991 4012 y Fm(\000)p Ft(j)3082 3998 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b Fp(2)386 4122 y Fu(F)13 b(S)518 4089 y Fq(0)513 4150 y Ft(\022)r(\033)627 4122 y Fk(and)804 4054 y Fr(P)900 4080 y Fm(1)900 4149 y Ft(j)t Fq(=0)1042 4122 y Fu(\013)1100 4136 y Fm(\000)p Ft(j)1191 4122 y Fw(\()p Fu(\030)t Fw(\))27 b Fp(2)e Fu(F)13 b(S)1550 4089 y Fq(0)1590 4122 y Fw(\()p Fs(R)1685 4089 y Ft(n)1758 4122 y Fp(n)21 b(f)p Fw(0)p Fp(g)p Fw(\))p Fk(.)44 b(Set)33 b Fu(\022)2268 4089 y Fm(0)2316 4122 y Fw(=)26 b Fu(\022)35 b Fk(if)d Fu(\033)d(>)c Fw(1)34 b Fk(or)f Fu(\034)j(>)25 b Fw(0)p Fk(,)34 b(and)386 4249 y Fu(\022)432 4216 y Fm(0)480 4249 y Fw(=)25 b(1)17 b(+)h Fu(\032)p Fk(,)31 b Fw(0)26 b Fu(<)f(\032)g Fp(\034)g Fw(1)32 b Fk(if)f Fu(\022)c Fw(=)e(1)32 b Fk(\(i.e.)41 b Fu(\033)28 b Fw(=)d(1)32 b Fk(and)g Fu(\034)j Fp(\024)25 b Fw(0)p Fk(\).)42 b(Then)32 b(ther)-5 b(e)32 b(exist)f Fu(G)3293 4216 y Ft(\022)r(\033)386 4360 y Fk(r)-5 b(e)g(alizations)35 b Fu(q)s Fw(\()p Fu(x;)15 b(\030)t Fw(\))p Fk(,)34 b Fu(\013)1233 4327 y Ft(k)1276 4360 y Fw(\()p Fu(\030)t Fw(\))p Fk(,)f Fw(1)26 b Fp(\024)e Fu(k)29 b Fp(\024)c Fu(d)33 b Fk(and)g(p)-5 b(ositive)34 b(c)-5 b(onstants)34 b Fu(C)7 b Fk(,)32 b Fu(\016)37 b Fk(such)32 b(that)725 4554 y Fw(sup)706 4632 y Ft(x)p Fm(2)p Fl(T)843 4613 y FA(n)895 4554 y Fp(j)p Fu(@)973 4516 y Ft(\015)968 4576 y(x)1018 4554 y Fu(@)1071 4509 y Ft(\014)1066 4583 y(\030)1118 4554 y Fu(R)1187 4569 y Ft(k)1230 4554 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))p Fp(j)27 b(\024)e Fu(C)1656 4506 y Fm(j)p Ft(\014)s Fm(j)p Fq(+)p Fm(j)p Ft(\015)t Fm(j)1649 4580 y Fq(2)1876 4554 y Fw(\()p Fu(\015)5 b Fw(!\))2023 4516 y Ft(\033)2071 4554 y Fw(\()p Fu(\014)g Fw(!\))2222 4516 y Ft(\033)r(\022)2299 4493 y Fx(0)2327 4554 y Fp(j)p Fu(\030)t Fp(j)2421 4516 y Fm(\000j)p Ft(\015)t Fm(j)2575 4554 y Fw(exp\()p Fp(\000)p Fu(\016)s Fp(j)p Fu(\030)t Fp(j)2957 4516 y Fq(1)p Ft(=)p Fq(\()p Ft(\033)r(\022)r Fq(\))3166 4554 y Fw(\))p Fu(;)-2840 b Fw(\(4.3\))386 4798 y Fk(for)40 b(al)5 b(l)39 b Fu(\014)5 b(;)15 b(\015)43 b Fp(2)37 b Fs(Z)1023 4765 y Ft(n)1023 4820 y Fq(+)1078 4798 y Fk(,)j Fu(\030)h Fp(2)c Fs(R)1385 4765 y Ft(n)1477 4798 y Fk(and)j Fw(1)d Fp(\024)g Fu(k)k Fp(\024)36 b Fu(d)p Fk(.)62 b(Thus)40 b Fu(R)2488 4813 y Ft(k)2530 4798 y Fw(\()p Fu(x;)15 b(D)s Fw(\))38 b Fp(2)f Fu(O)s(S)3039 4759 y Fm(\0001)3034 4827 y Ft(\033)r(\022)3169 4798 y Fw(\()p Fs(T)3263 4765 y Ft(n)3311 4798 y Fw(\))p Fk(,)386 4905 y Fw(1)k Fp(\024)f Fu(k)j Fp(\024)d Fu(d)p Fk(.)68 b(Final)5 b(ly,)43 b(if)e(the)g(symb)-5 b(ols)43 b Fu(a)1922 4919 y Ft(j)1959 4905 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))41 b Fk(dep)-5 b(end)43 b(on)e(one)h(b)-5 b(ase)41 b(variable,)386 5013 y(the)29 b(system)g(c)-5 b(an)29 b(b)-5 b(e)29 b(tr)-5 b(ansforme)g(d)31 b(into)e(a)g(simultane) -5 b(ous)30 b(normal)h(form)e(without)h(any)386 5121 y(r)-5 b(estrictions)37 b(on)f Fu(\034)45 b Fk(in)35 b(the)h(Diophantine)h(c)-5 b(ondition)37 b(for)f Fu(!)2479 5088 y Fq(1)2518 5121 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(!)2795 5088 y Ft(d)2836 5121 y Fk(.)50 b(In)35 b(this)h(c)-5 b(ase)386 5229 y(ther)g(e)37 b(is)f(no)g(loss)h(of)g(r)-5 b(e)g(gularity)37 b(in)f(the)g(r)-5 b(emainders;)40 b(mor)-5 b(e)37 b(pr)-5 b(e)g(cisely)38 b(if)d Fu(a)3063 5244 y Ft(k)3106 5229 y Fw(\()p Fu(x;)15 b(D)s Fw(\))p Fk(,)386 5337 y Fw(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d)33 b Fk(ar)-5 b(e)33 b Fu(G)1030 5304 y Ft(\033)1109 5337 y Fk(p.d.o.)43 b(then)33 b Fu(R)1645 5352 y Ft(k)1688 5337 y Fw(\()p Fu(x;)15 b(D)s Fw(\))33 b Fk(ar)-5 b(e)34 b Fu(G)2188 5304 y Ft(\033)2267 5337 y Fk(smo)-5 b(othing)34 b(p.d.o.)486 5515 y Fw(First)42 b(w)m(e)h(need)g(a)g(tec)m(hnical)g (assertion,)j(where)c Fu(C)2341 5482 y Fm(1)2334 5539 y Fq(0)2416 5515 y Fw(\()p Fs(T)2510 5482 y Ft(n)2558 5515 y Fw(\))i(denotes)f(the)g(set)g(of)386 5623 y(smo)s(oth)30 b(functions)f(with)g(mean)i(v)-5 b(alue)29 b(zero.)p eop %%Page: 13 13 13 12 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f (FIELDS)h(ON)g(TORI)559 b(13)386 434 y FE(Lemma)33 b(4.2.)46 b Fk(Supp)-5 b(ose)39 b(that)h Fw(\()p Fu(S)5 b(i)p Fw(\))1670 448 y Fm(1)1784 434 y Fk(holds.)60 b(Then)38 b(for)h(every)f Fu(f)2754 449 y Ft(k)2796 434 y Fw(\()p Fu(x)p Fw(\))e Fp(2)f Fu(C)3122 401 y Fm(1)3115 458 y Fq(0)3196 434 y Fw(\()p Fs(T)3290 401 y Ft(n)3339 434 y Fw(\))386 542 y Fk(such)h(that)h Fu(X)860 557 y Ft(k)903 542 y Fu(f)948 556 y Ft(r)1017 542 y Fw(=)31 b Fu(X)1194 556 y Ft(r)1232 542 y Fu(f)1277 557 y Ft(k)1319 542 y Fk(,)37 b Fw(1)32 b Fp(\024)f Fu(r)m(;)15 b(k)35 b Fp(\024)c Fu(d)37 b Fk(ther)-5 b(e)36 b(exists)h(a)f(unique)f(solution)i Fu(q)s Fw(\()p Fu(x)p Fw(\))32 b Fp(2)386 652 y Fu(C)458 619 y Fm(1)451 676 y Fq(0)532 652 y Fw(\()p Fs(T)626 619 y Ft(n)675 652 y Fw(\))h Fk(of)g(the)g(system)g Fu(<)25 b(!)1454 619 y Ft(k)1497 652 y Fu(;)15 b(@)1585 666 y Ft(x)1654 652 y Fu(>)25 b(q)s Fw(\()p Fu(x)p Fw(\))h(=)f Fu(f)2083 667 y Ft(k)2125 652 y Fw(\()p Fu(x)p Fw(\))p Fk(,)33 b Fw(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d)p Fk(,)33 b(satisfying)1463 818 y Fw(^)-51 b Fu(u)p Fw(\()p Fu(\030)t Fw(\))26 b(=)f(0)p Fu(;)201 b(\030)30 b Fp(2)24 b Fw(\000)2228 781 y Ft(!)2228 841 y Fl(Z)2279 818 y Fu(:)-1918 b Fw(\(4.4\))386 985 y Fk(F)-7 b(uthermor)i(e,)35 b(for)e(some)g Fu(C)1343 999 y Ft(\034)1411 985 y Fu(>)25 b Fw(0)33 b Fk(this)h(unique)d(solution)j(satis\014es)g(the)f(estimate) 1113 1152 y Fp(k)p Fu(q)s Fp(k)1247 1166 y Ft(H)1310 1147 y FA(s)1373 1152 y Fp(\024)25 b Fu(C)1534 1166 y Ft(\034)1618 1152 y Fw(max)1592 1213 y Fq(1)p Fm(\024)p Ft(k)r Fm(\024)p Ft(d)1827 1152 y Fp(k)p Fu(f)1917 1167 y Ft(k)1960 1152 y Fp(k)2005 1171 y Ft(H)2068 1152 y FA(s)p Fo(+)p FA(\034)14 b Fu(;)202 b(s)24 b Fp(\025)h Fw(0)p Fu(:)-2261 b Fw(\(4.5\))386 1397 y Fk(Pr)-5 b(o)g(of.)43 b Fw(By)35 b(F)-8 b(ourier)35 b(transform)g(w)m(e)g(obtain)g Fu(<)e(!)2147 1364 y Ft(k)2190 1397 y Fu(;)15 b(\021)37 b(>)j Fw(^)-52 b Fu(q)s Fw(\()p Fu(\021)s Fw(\))34 b(=)2735 1373 y Fr(b)2717 1397 y Fu(f)2762 1412 y Ft(k)2804 1397 y Fw(\()p Fu(\021)s Fw(\).)56 b(It)36 b(follo)m(ws)386 1518 y(that)608 1494 y Fr(b)589 1518 y Fu(f)634 1533 y Ft(k)676 1518 y Fw(\()p Fu(\021)s Fw(\))h(=)f(0)h(when)f Fu(<)f(!)1430 1485 y Ft(k)1473 1518 y Fu(;)15 b(\021)39 b(>)p Fw(=)d(0.)60 b(The)36 b(compatibilit)m(y)f(condition)g(and)h(the) 386 1626 y(de\014nition)28 b(of)i(\000)950 1593 y Ft(!)950 1653 y Fl(Z)1031 1626 y Fw(implies)e(that)j(w)m(e)f(can)h(decomp)s(ose) 749 1889 y Fs(Z)814 1851 y Ft(n)877 1889 y Fp(n)21 b Fw(\000)1000 1851 y Ft(!)1000 1912 y Fl(Z)1075 1889 y Fw(=)1229 1775 y Ft(d)1197 1802 y Fr([)1171 1997 y Ft(m)p Fq(=1)1339 1889 y Fu(Z)1408 1851 y Fm(0)1431 1889 y Fw(\()p Fu(m)p Fw(\))p Fu(;)197 b(Z)1872 1851 y Fm(0)1895 1889 y Fw(\()p Fu(m)p Fw(\))2060 1802 y Fr(\\)2177 1889 y Fu(Z)2246 1851 y Fm(0)2269 1889 y Fw(\()p Fu(j)5 b Fw(\))26 b(=)f Fp(;)p Fu(;)47 b Fw(1)25 b Fp(\024)g Fu(j)31 b(<)25 b(m)g Fp(\024)g Fu(d)-2812 b Fw(\(4.6\))386 2152 y(with)41 b(the)h(prop)s(ert)m(y)g Fu(<)j(!)1333 2119 y Ft(k)1375 2152 y Fu(;)15 b(\021)49 b(>)p Fp(6)p Fw(=)c(0)e(for)f Fu(\021)48 b Fp(2)d Fu(Z)2203 2119 y Fm(0)2226 2152 y Fw(\()p Fu(k)s Fw(\),)h(1)g Fp(\024)f Fu(k)j Fp(\024)d Fu(d)p Fw(.)77 b(Hence)43 b(for)386 2260 y(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d)31 b Fw(the)f(in)m(v)m(erse)g(F)-8 b(ourier)30 b(transform)g(giv)m(es,)728 2554 y Fp(k)p Fu(q)s Fp(k)862 2568 y Ft(H)925 2549 y FA(s)988 2554 y Fp(\024)25 b Fw(\(2)p Fu(\031)s Fw(\))1254 2517 y Fm(\000)p Ft(n=)p Fq(2)1490 2441 y Ft(d)1443 2468 y Fr(X)1444 2666 y Ft(k)r Fq(=1)1589 2372 y Fr(0)1589 2535 y(@)1730 2468 y(X)1669 2669 y Ft(\021)r Fm(2)p Ft(Z)1806 2650 y Fx(0)1829 2669 y Fq(\()p Ft(k)r Fq(\))2124 2493 y Fp(j)p Fu(\021)s Fp(j)2222 2460 y Fq(2)p Ft(s)p 1947 2533 524 4 v 1947 2617 a Fp(j)h Fu(<)f(!)2154 2590 y Ft(k)2196 2617 y Fu(;)15 b(\021)29 b(>)c Fp(j)2431 2590 y Fq(2)2481 2554 y Fp(j)2524 2530 y Fr(b)2506 2554 y Fu(f)2551 2569 y Ft(k)2593 2554 y Fw(\()p Fu(\021)s Fw(\))p Fp(j)2736 2517 y Fq(2)2777 2372 y Fr(1)2777 2535 y(A)2857 2394 y Fq(1)p Ft(=)p Fq(2)574 2940 y Fp(\024)83 b Fu(C)793 2954 y Ft(\034)876 2940 y Fw(max)851 3002 y Fq(1)p Fm(\024)p Ft(k)r Fm(\024)p Ft(d)1086 2758 y Fr(0)1086 2921 y(@)1221 2854 y(X)1165 3055 y Ft(\021)r Fm(2)p Fl(Z)1301 3036 y FA(n)1337 3055 y Fm(n)p Fq(0)1433 2940 y Fu(<)25 b(\021)k(>)1674 2903 y Fq(2\()p Ft(s)p Fq(+)p Ft(\034)8 b Fq(\))1919 2940 y Fp(j)1963 2916 y Fr(b)1944 2940 y Fu(f)1989 2955 y Ft(k)2032 2940 y Fw(\()p Fu(\021)s Fw(\))p Fp(j)2175 2903 y Fq(2)2216 2758 y Fr(1)2216 2921 y(A)2295 2780 y Fq(1)p Ft(=)p Fq(2)2430 2940 y Fw(=)25 b Fu(C)2591 2954 y Ft(\034)2675 2940 y Fw(max)2650 3002 y Fq(1)p Fm(\024)p Ft(k)r Fm(\024)p Ft(d)2885 2940 y Fp(k)p Fu(f)2975 2955 y Ft(k)3017 2940 y Fp(k)3062 2960 y Ft(H)3125 2941 y FA(s)p Fo(+)p FA(\034)3245 2940 y Fu(;)386 3231 y Fw(where)30 b Fu(<)25 b(\021)j(>)p Fw(=)d(\(1)c(+)f Fp(j)p Fu(\021)s Fp(j)1275 3198 y Fq(2)1315 3231 y Fw(\))1350 3198 y Fq(1)p Ft(=)p Fq(2)1461 3231 y Fw(.)p 3309 3231 4 62 v 3313 3173 55 4 v 3313 3231 V 3367 3231 4 62 v 386 3416 a Fk(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)34 b(4.1.)43 b Fw(W)-8 b(e)27 b(will)d(supp)s(ose)h(that)i Fu(\034)37 b Fw(is)25 b(an)h(in)m(teger)h(\(the)g(pro)s(of)f(is)f(the)386 3524 y(same)32 b(for)g Fu(\034)38 b Fp(62)27 b Fs(Z)p Fw(,)h(but)j(there)i (are)f(some)g(tec)m(hnical)g(complications\).)45 b(Clearly)-8 b(,)31 b(after)386 3638 y(conjugation)f(with)f(the)h(sym)m(b)s(ol)f Fu(e)1594 3605 y Ft(i )r Fq(\()p Ft(x;\030)s Fq(\))1819 3638 y Fw(,)i(w)m(e)f(ma)m(y)h(supp)s(ose)e(that)i Fu(a)2792 3605 y Ft(k)2792 3662 y Fq(0)2834 3638 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b(=)f Fu(\013)3220 3652 y Fq(0)3260 3638 y Fw(\()p Fu(\030)t Fw(\))386 3746 y(for)k(1)d Fp(\024)f Fu(k)k Fp(\024)24 b Fu(d)30 b Fw(and)g(hence)g(that)g Fu( )f Fw(=)c(0,)30 b Fu(q)1889 3760 y Fq(0)1953 3746 y Fw(=)25 b(1)30 b(and)g(the)f(conjugating)h(p.d.o.)41 b(is)28 b(of)386 3854 y(the)34 b(form)f Fu(q)s Fw(\()p Fu(x;)15 b(\030)t Fw(\))32 b Fp(\030)f Fw(1)23 b(+)1309 3786 y Fr(P)1405 3812 y Fm(1)1405 3881 y Ft(j)t Fq(=1)1547 3854 y Fu(q)1588 3868 y Fm(\000)p Ft(j)1679 3854 y Fw(\()p Fu(x;)15 b(\030)t Fw(\).)53 b(Then,)34 b(b)m(y)f(the)h(comp)s(osition)f (rule,)h Fu(q)3284 3868 y Fm(\000)p Ft(j)386 3965 y Fw(satis\014es)c (the)g(follo)m(wing)f(equations)400 4246 y Fu(<)c(!)556 4209 y Ft(k)599 4246 y Fu(;)15 b(D)714 4260 y Ft(x)783 4246 y Fu(>)25 b(q)920 4260 y Fm(\000)p Ft(j)1037 4246 y Fw(=)g Fu(F)1204 4209 y Ft(k)1191 4269 y Fm(\000)p Ft(j)1282 4246 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))22 b(+)e Fu(G)1673 4209 y Ft(k)1673 4269 y Fm(\000)p Ft(j)1763 4246 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))22 b Fp(\000)e Fu(a)2130 4209 y Ft(k)2130 4269 y Fm(\000)p Ft(j)2221 4246 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))21 b(+)f Fu(\013)2597 4209 y Ft(k)2597 4269 y Fm(\000)p Ft(j)2689 4246 y Fw(\()p Fu(\030)t Fw(\))p Fu(;)107 b Fw(1)25 b Fp(\024)g Fu(k)k Fp(\024)c Fu(d;)386 4118 y Fw(\(4.7\))386 4413 y(where)835 4651 y Fu(F)906 4614 y Ft(k)893 4674 y Fm(\000)p Ft(j)985 4651 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b(=)f Fp(\000)1403 4533 y Ft(j)t Fm(\000)p Fq(1)1399 4565 y Fr(X)1402 4759 y Ft(r)r Fq(=1)1609 4565 y Fr(X)1545 4766 y Fm(j)p Ft(\014)s Fm(j\024)p Ft(j)t Fm(\000)p Ft(r)1846 4590 y Fw(1)p 1828 4631 82 4 v 1828 4714 a Fu(\014)5 b Fw(!)1920 4651 y Fu(D)1998 4607 y Ft(\014)1995 4681 y(\030)2045 4651 y Fu(a)2093 4614 y Ft(k)2093 4678 y Fm(\000)p Fq(\()p Ft(j)t Fm(\000)p Ft(r)r Fm(\000j)p Ft(\014)s Fm(j)p Fq(\))2465 4651 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))p Fu(@)2724 4614 y Ft(\014)2719 4674 y(x)2772 4651 y Fu(q)2813 4665 y Fm(\000)p Ft(r)2906 4651 y Fw(\()p Fu(x;)g(\030)t Fw(\))-2726 b(\(4.8\))386 4922 y(and)1175 5161 y Fu(G)1247 5123 y Ft(k)1247 5183 y Fm(\000)p Ft(j)1338 5161 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b(=)1670 5042 y Ft(j)t Fm(\000)p Fq(1)1666 5074 y Fr(X)1670 5269 y Ft(r)r Fq(=1)1812 5161 y Fu(q)1853 5175 y Fm(\000)p Ft(r)1946 5161 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))p Fu(\013)2210 5123 y Ft(k)2210 5187 y Fm(\000)p Fq(\()p Ft(j)t Fm(\000)p Ft(r)r Fq(\))2446 5161 y Fw(\()p Fu(\030)t Fw(\))p Fu(:)-2199 b Fw(\(4.9\))386 5413 y(Clearly)29 b(in)g(order)h(to)h(solv)m(e)g(\(4.7\))h(w)m(e)f(m)m(ust)f(c)m(ho)s (ose)698 5586 y Fu(\013)756 5549 y Ft(k)756 5609 y Fm(\000)p Ft(j)848 5586 y Fw(\()p Fu(\030)t Fw(\))c(=)f Fp(\000)p Fw([)p Fu(F)1251 5549 y Ft(k)1238 5609 y Fm(\000)p Ft(j)1329 5586 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\)])1533 5600 y Ft(n)1602 5586 y Fp(\000)20 b Fw([)p Fu(G)1790 5549 y Ft(k)1790 5609 y Fm(\000)p Ft(j)1881 5586 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\)])2085 5600 y Ft(n)2154 5586 y Fw(+)20 b([)p Fu(a)2318 5549 y Ft(k)2318 5609 y Fm(\000)p Ft(j)2409 5586 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\)])2613 5600 y Ft(n)2662 5586 y Fu(;)197 b Fw(1)26 b Fp(\024)f Fu(k)j Fp(\024)d Fu(d:)-2908 b Fw(\(4.10\))p eop %%Page: 14 14 14 13 bop 386 239 a Fq(14)693 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g (YOSHINO)486 434 y Fw(Because)k Fu(a)884 401 y Ft(k)926 434 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))31 b Fp(2)d Fu(S)1313 401 y Fq(0)1308 456 y Ft(\033)1355 434 y Fw(\()p Fs(T)1449 401 y Ft(n)1498 434 y Fw(\),)34 b(1)29 b Fp(\024)g Fu(k)k Fp(\024)28 b Fu(d)33 b Fw(there)g(exists)g(a)g(complex)f(neigh)m(b)s (our-)386 549 y(ho)s(o)s(d)627 526 y(~)612 549 y Fu(S)h Fw(in)28 b Fs(C)866 516 y Ft(n)947 549 y Fw(of)h(the)h(unit)d(sphere)h Fu(S)1738 516 y Ft(n)p Fm(\000)p Fq(1)1904 549 y Fw(in)g Fs(R)2068 516 y Ft(n)2150 549 y Fw(so)h(that)g(for)g(\014xed)f Fu(s)d(>)g(n=)p Fw(2)k(and)g Fu(\034)386 656 y Fw(there)h(exists)h(a)f (p)s(ositiv)m(e)g(constan)m(t)h Fu(A)g Fw(suc)m(h)f(that)775 808 y(sup)779 903 y Ft(\030)s Fm(2)872 886 y Fq(~)860 903 y Ft(S)927 808 y Fp(k)p Fu(@)1025 771 y Ft(\014)1020 831 y(x)1073 808 y Fu(a)1121 771 y Ft(k)1121 831 y Fm(\000)p Ft(j)1212 808 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)1436 828 y Ft(H)1499 809 y FA(s)p Fo(+)p FA(\034)1645 808 y Fp(\024)25 b Fu(A)1809 771 y Ft(j)t Fq(+)p Fm(j)p Ft(\014)s Fm(j)p Fq(+1)2073 808 y Fw(\()p Fu(j)5 b Fw(!)p Fu(\014)g Fw(!\))2291 771 y Ft(\033)2340 808 y Fu(;)197 b(j)31 b Fp(2)25 b Fs(Z)2781 822 y Fq(+)2835 808 y Fu(;)46 b(\014)31 b Fp(2)24 b Fs(Z)3138 771 y Ft(n)3138 831 y Fq(+)3193 808 y Fu(:)-2832 b Fw(\(4.11\))386 1040 y(By)29 b(the)g(Sc)m(hauder)g(lemma)f(for)h(m)m(ultiplication)d(in)h(Sob)s (olev)h(spaces,)i(if)e Fu(s)17 b Fw(+)g Fu(\034)35 b(>)25 b(n=)p Fw(2)386 1148 y(there)30 b(exists)h Fu(!)d(>)d Fw(0)30 b(suc)m(h)g(that)953 1294 y Fp(k)p Fu(f)10 b(g)s Fp(k)1144 1314 y Ft(H)1207 1295 y FA(s)p Fo(+)p FA(\034)1351 1294 y Fp(\024)25 b Fu(!)s Fp(k)p Fu(f)10 b Fp(k)1652 1314 y Ft(H)1715 1295 y FA(s)p Fo(+)p FA(\034)1834 1294 y Fp(k)p Fu(g)s Fp(k)1970 1314 y Ft(H)2033 1295 y FA(s)p Fo(+)p FA(\034)2154 1294 y Fu(;)197 b(f)5 b(;)15 b(g)28 b Fp(2)d Fu(H)2706 1256 y Ft(s)p Fq(+)p Ft(\034)2699 1320 y Fq(0)2837 1294 y Fw(\()p Fs(T)2931 1257 y Ft(n)2980 1294 y Fw(\))p Fu(:)-2654 b Fw(\(4.12\))386 1446 y(Estimates)32 b(\(4.1\))j(and)c(\(4.2\))j(will)c(b)s(e)i(pro)m(v)m(ed)g(b)m(y)h (induction.)44 b(The)32 b(estimates)h(\(4.1\))386 1554 y(and)e(\(4.2\))i(hold)d(for)h Fu(j)i Fw(=)27 b(0.)44 b(Supp)s(ose)30 b(that)i(\(4.1\))h(and)e(\(4.2\))j(are)d(true)h(for)f (0)c Fp(\024)g Fu(j)33 b Fp(\024)386 1669 y Fu(i)20 b Fp(\000)g Fw(1.)41 b(By)31 b(\(4.12\))i(w)m(e)e(obtain)e(for)i Fu(\030)e Fp(2)1771 1646 y Fw(~)1756 1669 y Fu(S)35 b Fw(that)769 1905 y Fp(k)p Fu(@)867 1867 y Ft(\015)862 1927 y(x)912 1905 y Fu(F)983 1867 y Ft(k)970 1927 y Fm(\000)p Ft(i)1054 1905 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)1278 1924 y Ft(H)1341 1905 y FA(s)p Fo(+)p FA(\034)40 b Fp(\024)25 b Fu(!)1666 1791 y Ft(i)p Fm(\000)p Fq(1)1657 1818 y Fr(X)1661 2013 y Ft(r)r Fq(=1)1863 1818 y Fr(X)1804 2020 y Fm(j)p Ft(\014)s Fm(j\024)p Ft(i)p Fm(\000)p Ft(r)2082 1818 y Fr(X)2069 2017 y Ft(\015)2109 1998 y Fx(0)2131 2017 y Fm(\024)p Ft(\015)2242 1777 y Fr(\022)2362 1850 y Fu(\015)2350 1958 y(\015)2402 1925 y Fm(0)2467 1777 y Fr(\023)2577 1843 y Fw(1)p 2559 1884 82 4 v 2559 1967 a Fu(\014)5 b Fw(!)386 1905 y(\(4.13\))524 2171 y Fp(\002)83 b(k)p Fu(@)776 2134 y Ft(\015)t Fm(\000)p Ft(\015)911 2110 y Fx(0)771 2194 y Ft(x)939 2171 y Fu(D)1017 2127 y Ft(\014)1014 2200 y(\030)1064 2171 y Fu(a)1112 2134 y Ft(k)1112 2198 y Fm(\000)p Fq(\()p Ft(i)p Fm(\000)p Ft(r)r Fm(\000j)p Ft(\014)s Fm(j)p Fq(\))1475 2171 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)1699 2191 y Ft(H)1762 2172 y FA(s)p Fo(+)p FA(\034)1883 2171 y Fp(k)p Fu(@)1981 2134 y Ft(\014)s Fq(+)p Ft(\015)2119 2110 y Fx(0)1976 2194 y Ft(x)2146 2171 y Fu(q)2187 2185 y Fm(\000)p Ft(r)2280 2171 y Fw(\()p Fp(\001)p Fu(;)g(\030)t Fw(\))p Fp(k)2504 2191 y Ft(H)2567 2172 y FA(s)p Fo(+)p FA(\034)524 2416 y Fp(\024)83 b Fu("!)s(A)872 2303 y Ft(i)p Fm(\000)p Fq(1)864 2330 y Fr(X)867 2524 y Ft(r)r Fq(=1)1069 2330 y Fr(X)1010 2531 y Fm(j)p Ft(\014)s Fm(j\024)p Ft(i)p Fm(\000)p Ft(r)1288 2330 y Fr(X)1275 2528 y Ft(\015)1315 2509 y Fx(0)1338 2528 y Fm(\024)p Ft(\015)1448 2288 y Fr(\022)1568 2361 y Fu(\015)1556 2469 y(\015)1608 2436 y Fm(0)1673 2288 y Fr(\023)1755 2416 y Fu(A)1823 2379 y Ft(i)p Fm(\000)p Ft(r)r Fq(+)p Fm(j)p Ft(\015)t Fm(j\000j)p Ft(\015)2186 2356 y Fx(0)2208 2379 y Fm(j)2231 2416 y Fw(\(\()p Fu(i)22 b Fp(\000)e Fu(r)i Fp(\000)e(j)p Fu(\014)5 b Fp(j)p Fw(\)!\()p Fu(\015)27 b Fp(\000)20 b Fu(\015)3017 2379 y Fm(0)3041 2416 y Fw(\)!\))3136 2379 y Ft(\033)3183 2416 y Fu(M)3281 2379 y Ft(r)524 2681 y Fp(\002)83 b Fu(N)761 2644 y Fm(j)p Ft(\014)s Fm(j)p Fq(+)p Ft(\034)8 b Fq(+)p Fm(j)p Ft(\015)1053 2620 y Fx(0)1074 2644 y Fm(j)1098 2681 y Fw(\(\()p Fu(\022)s(r)23 b Fw(+)c Fp(j)p Fu(\014)5 b Fp(j)21 b Fw(+)f Fp(j)p Fu(\015)1663 2644 y Fm(0)1687 2681 y Fp(j)g Fw(+)g Fu(\034)10 b Fw(\)!\))1968 2644 y Ft(\033)2041 2681 y Fp(\024)25 b Fu("M)2277 2644 y Ft(i)2306 2681 y Fu(N)2389 2644 y Fm(j)p Ft(\015)t Fm(j)2473 2681 y Fw(\(\()p Fu(\022)s(i)20 b Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!\))2928 2644 y Ft(\033)2976 2681 y Fu(!)s(AN)3187 2644 y Ft(\034)3231 2681 y Fu(K)386 2828 y Fw(with)672 3041 y Fu(K)32 b Fw(=)886 2928 y Ft(i)p Fm(\000)p Fq(1)877 2955 y Fr(X)881 3149 y Ft(r)r Fq(=1)1008 3041 y Fw(\()p Fu(AN)r(=)-5 b(M)10 b Fw(\))1359 3004 y Ft(i)p Fm(\000)p Ft(r)1552 2955 y Fr(X)1493 3156 y Fm(j)p Ft(\014)s Fm(j\024)p Ft(i)p Fm(\000)p Ft(r)1771 2955 y Fr(X)1758 3153 y Ft(\015)1798 3134 y Fx(0)1821 3153 y Fm(\024)p Ft(\015)1931 2913 y Fr(\022)2051 2986 y Fu(\015)2039 3094 y(\015)2091 3061 y Fm(0)2156 2913 y Fr(\023)386 3041 y Fw(\(4.14\))518 3351 y Fp(\002)83 b Fw(\()p Fu(A=)-5 b(N)10 b Fw(\))933 3314 y Fm(j)p Ft(\015)t Fm(j\000j)p Ft(\015)1128 3290 y Fx(0)1151 3314 y Fm(j)1190 3223 y Fr(\022)1267 3290 y Fw(\()p Fu(i)21 b Fp(\000)f Fu(r)i Fp(\000)e(j)p Fu(\014)5 b Fp(j)p Fw(\)!\()p Fu(\015)27 b Fp(\000)20 b Fu(\015)2017 3257 y Fm(0)2040 3290 y Fw(\)!\()p Fu(\022)s(r)j Fw(+)d Fp(j)p Fu(\014)5 b Fp(j)21 b Fw(+)f Fu(\034)30 b Fw(+)20 b Fp(j)p Fu(\015)2792 3257 y Fm(0)2816 3290 y Fp(j)p Fw(\)!)p 1267 3330 1636 4 v 1891 3414 a(\()p Fu(\022)s(i)g Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!)2912 3223 y Fr(\023)2979 3246 y Ft(\033)518 3640 y Fp(\024)681 3526 y Ft(i)p Fm(\000)p Fq(1)672 3553 y Fr(X)676 3748 y Ft(r)r Fq(=1)803 3640 y Fw(\()p Fu(AN)r(=)-5 b(M)10 b Fw(\))1154 3602 y Ft(i)p Fm(\000)p Ft(r)1347 3553 y Fr(X)1288 3755 y Fm(j)p Ft(\014)s Fm(j\024)p Ft(i)p Fm(\000)p Ft(r)1566 3553 y Fr(X)1553 3751 y Ft(\015)1593 3732 y Fx(0)1616 3751 y Fm(\024)p Ft(\015)1736 3578 y Fw(\()p Fp(j)p Fu(\015)1848 3545 y Fm(0)1872 3578 y Fp(j)21 b Fw(+)e(1\))c Fu(:)g(:)g(:)j Fw(\()p Fp(j)p Fu(\015)2338 3545 y Fm(0)2362 3578 y Fp(j)i Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)21 b(\000)f(j)p Fu(\015)2789 3545 y Fm(0)2812 3578 y Fp(j)p Fw(\))p 1736 3619 1137 4 v 2137 3702 a(\()p Fu(\015)26 b Fp(\000)20 b Fu(\015)2388 3676 y Fm(0)2411 3702 y Fw(\)!)518 3950 y Fp(\002)83 b Fw(\()p Fu(A=)-5 b(N)10 b Fw(\))933 3912 y Fm(j)p Ft(\015)t Fm(j\000j)p Ft(\015)1128 3889 y Fx(0)1151 3912 y Fm(j)1190 3821 y Fr(\022)1668 3888 y Fw(\()p Fu(\015)26 b Fp(\000)19 b Fu(\015)1918 3855 y Fm(0)1942 3888 y Fw(\)!)p 1267 3929 V 1267 4012 a(\()p Fp(j)p Fu(\015)1379 3986 y Fm(0)1403 4012 y Fp(j)h Fw(+)g(1\))15 b Fu(:)g(:)g(:)i Fw(\()p Fp(j)p Fu(\015)1868 3986 y Fm(0)1892 4012 y Fp(j)k Fw(+)f Fp(j)p Fu(\015)5 b Fp(j)20 b(\000)g(j)p Fu(\015)2319 3986 y Fm(0)2343 4012 y Fp(j)p Fw(\))2414 3821 y Fr(\023)2481 3844 y Ft(\033)518 4205 y Fp(\002)672 4077 y Fr(\022)749 4144 y Fw(\()p Fu(i)h Fp(\000)f Fu(r)j Fp(\000)c(j)p Fu(\014)5 b Fp(j)p Fw(\)!\()p Fp(j)p Fu(\015)1359 4111 y Fm(0)1385 4144 y Fp(j)20 b Fw(+)g(1\))15 b Fu(:)g(:)g(:)i Fw(\()p Fp(j)p Fu(\015)1850 4111 y Fm(0)1874 4144 y Fp(j)k Fw(+)f Fp(j)p Fu(\015)5 b Fp(j)21 b(\000)e(j)p Fu(\015)2301 4111 y Fm(0)2325 4144 y Fp(j)p Fw(\)\()p Fu(\022)s(r)k Fw(+)d Fp(j)p Fu(\014)5 b Fp(j)21 b Fw(+)f Fp(j)p Fu(\015)2916 4111 y Fm(0)2939 4144 y Fp(j)h Fw(+)f Fu(\034)10 b Fw(\)!)p 749 4185 2438 4 v 1775 4268 a(\()p Fu(\022)s(i)20 b Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!)3196 4077 y Fr(\023)3263 4100 y Ft(\033)518 4494 y Fp(\024)681 4380 y Ft(i)p Fm(\000)p Fq(1)672 4407 y Fr(X)676 4602 y Ft(r)r Fq(=1)803 4494 y Fw(\()p Fu(AN)r(=)-5 b(M)10 b Fw(\))1154 4456 y Ft(i)p Fm(\000)p Ft(r)1347 4407 y Fr(X)1288 4609 y Fm(j)p Ft(\014)s Fm(j\024)p Ft(i)p Fm(\000)p Ft(r)1566 4407 y Fr(X)1553 4605 y Ft(\015)1593 4587 y Fx(0)1616 4605 y Fm(\024)p Ft(\015)1711 4494 y Fw(\()p Fu(A=)-5 b(N)10 b Fw(\))1972 4456 y Fm(j)p Ft(\015)t Fm(j\000j)p Ft(\015)2167 4433 y Fx(0)2190 4456 y Fm(j)2214 4494 y Fu(C)2286 4455 y Ft(\033)r Fm(\000)p Fq(1)2279 4520 y(1)2422 4494 y Fu(C)2494 4456 y Ft(\033)2487 4516 y Fq(2)386 4744 y Fw(where)1106 4923 y Fu(C)1171 4937 y Fq(1)1235 4923 y Fw(=)1742 4861 y(\()p Fu(\015)26 b Fp(\000)20 b Fu(\015)1993 4828 y Fm(0)2016 4861 y Fw(\)!)p 1341 4902 1137 4 v 1341 4985 a(\()p Fp(j)p Fu(\015)1453 4959 y Fm(0)1477 4985 y Fp(j)h Fw(+)f(1\))15 b Fu(:)g(:)g(:)i Fw(\()p Fp(j)p Fu(\015)1943 4959 y Fm(0)1967 4985 y Fp(j)j Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)21 b(\000)f(j)p Fu(\015)2394 4959 y Fm(0)2418 4985 y Fp(j)p Fw(\))2513 4923 y Fp(\024)25 b Fw(1)-2268 b(\(4.15\))539 5288 y Fu(C)604 5302 y Fq(2)669 5288 y Fw(=)774 5226 y(\()p Fu(i)21 b Fp(\000)f Fu(r)j Fp(\000)d(j)p Fu(\014)5 b Fp(j)p Fw(\)!\()p Fp(j)p Fu(\015)1385 5193 y Fm(0)1410 5226 y Fp(j)21 b Fw(+)f(1\))15 b Fu(:)g(:)g(:)i Fw(\()p Fp(j)p Fu(\015)1876 5193 y Fm(0)1900 5226 y Fp(j)j Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)21 b(\000)f(j)p Fu(\015)2327 5193 y Fm(0)2351 5226 y Fp(j)p Fw(\)\()p Fu(\022)s(r)i Fw(+)e Fp(j)p Fu(\014)5 b Fp(j)21 b Fw(+)f Fp(j)p Fu(\015)2941 5193 y Fm(0)2965 5226 y Fp(j)g Fw(+)g Fu(\034)10 b Fw(\)!)p 774 5267 2438 4 v 1800 5350 a(\()p Fu(\022)s(i)20 b Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!)386 5111 y(\(4.16\))436 5538 y Fp(\024)542 5476 y Fw(\()p Fu(i)21 b Fp(\000)f Fu(r)j Fp(\000)c(j)p Fu(\014)5 b Fp(j)p Fw(\)!\()p Fu(\022)s(r)24 b Fw(+)c Fp(j)p Fu(\014)5 b Fp(j)21 b Fw(+)f Fp(j)p Fu(\015)5 b Fp(j)21 b Fw(+)f Fu(\034)10 b Fw(\)!)p 542 5517 1278 4 v 987 5600 a(\()p Fu(\022)s(i)21 b Fw(+)e Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!)1854 5538 y Fp(\024)1960 5476 y Fw(\()p Fu(\022)s(i)20 b Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)21 b(\000)f Fw(\()p Fu(\022)j Fp(\000)d Fw(1\)\()p Fu(i)h Fp(\000)f Fu(r)s Fw(\))g(+)g Fu(\034)10 b Fw(\)!)p 1960 5517 1188 4 v 2361 5600 a(\()p Fu(\022)s(i)20 b Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!)3183 5538 y Fp(\024)25 b Fw(1)p eop %%Page: 15 15 15 14 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f (FIELDS)h(ON)g(TORI)559 b(15)386 434 y Fw(for)37 b(all)f Fu(\015)5 b(;)15 b(\015)809 401 y Fm(0)833 434 y Fu(;)g(\014)43 b Fp(2)36 b Fs(Z)1129 401 y Ft(n)1129 456 y Fq(+)1184 434 y Fw(,)j Fu(\015)1300 401 y Fm(0)1360 434 y Fp(\024)e Fu(\015)5 b Fw(,)39 b Fu(r)m(;)15 b(i)38 b Fp(2)e Fs(N)7 b Fw(,)45 b Fu(i)25 b Fp(\000)g Fu(r)39 b Fp(\025)e Fw(1.)62 b(In)36 b(\(4.16\))k(the)d(inequalit)m(y)386 542 y(\()p Fu(\022)24 b Fp(\000)d Fw(1\)\()p Fu(i)i Fp(\000)e Fu(r)s Fw(\))28 b Fp(\025)g Fu(\022)23 b Fp(\000)f Fw(1)28 b Fp(\025)g Fu(\034)42 b Fw(if)31 b Fu(i)22 b Fp(\000)f Fu(r)31 b Fp(\025)d Fw(1)k(w)m(as)h(used.)45 b(Putting)31 b(\(4.15\))k(and)c(\(4.16\))386 650 y(in)m(to)h(\(4.14\),)j(and)d(c)m (ho)s(osing)g(\014rst)g Fu(N)38 b(>)28 b(A)k Fw(and)g(then)g Fu(M)38 b(>)28 b Fw(2)p Fu(AN)43 b Fw(large)32 b(enough)g(so)386 762 y(that)p 582 688 44 4 v 30 w Fu(\016)d Fw(:=)c Fu(AN)r(=)-5 b(M)41 b Fw(satis\014es)29 b(the)g(inequalit)m(y)p 1994 688 V 28 w Fu(\016)t Fw(\(1)19 b Fp(\000)f Fu(A=)-5 b(N)10 b Fw(\))2452 729 y Fm(\000)p Ft(n)2570 694 y Fr(P)2666 720 y Fm(1)2666 789 y Ft(\027)t Fq(=0)2814 762 y Fw(2)2859 729 y Fm(\000)p Ft(\027)2958 762 y Fw(\()p Fu(\027)24 b Fw(+)18 b(2\))3231 729 y Ft(n)3304 762 y Fp(\024)386 873 y Fw(\(6)p Fu(C)531 887 y Ft(\034)575 873 y Fu(!)s(AN)786 840 y Ft(\034)829 873 y Fw(\))864 840 y Fm(\000)p Fq(1)959 873 y Fw(,)30 b(w)m(e)h(get)763 1135 y Fu(K)90 b Fp(\024)1092 1022 y Ft(i)p Fm(\000)p Fq(1)1084 1049 y Fr(X)1087 1243 y Ft(r)r Fq(=1)1215 1135 y Fw(\()p Fu(AN)r(=)-5 b(M)10 b Fw(\))1566 1098 y Ft(i)p Fm(\000)p Ft(r)1759 1049 y Fr(X)1700 1250 y Fm(j)p Ft(\014)s Fm(j\024)p Ft(i)p Fm(\000)p Ft(r)1978 1049 y Fr(X)1965 1247 y Ft(\015)2005 1228 y Fx(0)2027 1247 y Fm(\024)p Ft(\015)2123 1135 y Fw(\()p Fu(A=)-5 b(N)10 b Fw(\))2384 1098 y Fm(j)p Ft(\015)t Fm(j\000j)p Ft(\015)2579 1074 y Fx(0)2601 1098 y Fm(j)386 1135 y Fw(\(4.17\))930 1478 y Fp(\024)1092 1364 y Ft(i)p Fm(\000)p Fq(1)1084 1391 y Fr(X)1087 1586 y Ft(r)r Fq(=1)1215 1478 y Fw(\()p Fu(AN)r(=)-5 b(M)10 b Fw(\))1566 1440 y Ft(i)p Fm(\000)p Ft(r)1759 1391 y Fr(X)1700 1593 y Fm(j)p Ft(\014)s Fm(j\024)p Ft(i)p Fm(\000)p Ft(r)1950 1478 y Fw(\(1)21 b Fp(\000)f Fu(A=)-5 b(N)10 b Fw(\))2368 1440 y Fm(\000)p Ft(n)930 1820 y Fp(\024)1094 1759 y Fu(AN)p 1094 1799 152 4 v 1120 1883 a(M)1255 1820 y Fw(\(1)21 b Fp(\000)f Fu(A=)-5 b(N)10 b Fw(\))1673 1783 y Fm(\000)p Ft(n)1799 1707 y(i)p Fm(\000)p Fq(1)1791 1734 y Fr(X)1794 1928 y Ft(r)r Fq(=1)1922 1820 y Fw(\()p Fu(AN)r(=)-5 b(M)10 b Fw(\))2273 1783 y Ft(i)p Fm(\000)p Ft(r)r Fm(\000)p Fq(1)2482 1820 y Fw(\()p Fu(i)21 b Fp(\000)e Fu(r)k Fw(+)d(1\))2894 1783 y Ft(n)930 2124 y Fp(\024)p 1084 2050 44 4 v 83 w Fu(\016)t Fw(\(1)h Fp(\000)e Fu(A=)-5 b(N)10 b Fw(\))1545 2086 y Fm(\000)p Ft(n)1694 2010 y Fm(1)1663 2038 y Fr(X)1664 2232 y Ft(\027)t Fq(=0)1810 2124 y Fw(2)1855 2086 y Fm(\000)p Ft(\027)1953 2124 y Fw(\()p Fu(\027)27 b Fw(+)19 b(2\))2230 2086 y Ft(n)2304 2124 y Fp(\024)24 b Fw(\(6)p Fu(C)2544 2138 y Ft(\034)2588 2124 y Fu(!)s(AN)2799 2086 y Ft(\034)2843 2124 y Fw(\))2878 2086 y Fm(\000)p Fq(1)2972 2124 y Fu(:)386 2377 y Fw(Hence,)31 b(b)m(y)g(the)f(c)m(hoice)h(of)g Fu(M)40 b Fw(and)30 b Fu(N)10 b Fw(,)31 b(w)m(e)f(obtain)715 2555 y(sup)720 2649 y Ft(\030)s Fm(2)812 2632 y Fq(~)801 2649 y Ft(S)867 2555 y Fp(k)p Fu(@)965 2517 y Ft(\015)960 2577 y(x)1010 2555 y Fu(F)1081 2517 y Ft(k)1068 2577 y Fm(\000)p Ft(i)1152 2555 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)1376 2574 y Ft(H)1439 2555 y FA(s)p Fo(+)p FA(\034)41 b Fp(\024)25 b Fw(\(3)p Fu(C)1826 2569 y Ft(\034)1870 2555 y Fw(\))1905 2517 y Fm(\000)p Fq(1)1999 2555 y Fu("M)2139 2517 y Ft(i)2168 2555 y Fu(N)2251 2517 y Fm(j)p Ft(\015)t Fm(j)2335 2555 y Fw(\(\()p Fu(\022)s(i)20 b Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!\))2790 2517 y Ft(\033)2838 2555 y Fu(;)107 b(\015)30 b Fp(2)25 b Fs(Z)3198 2517 y Ft(n)3198 2577 y Fq(+)3253 2555 y Fu(:)-2892 b Fw(\(4.18\))386 2827 y(Next,)31 b(for)g(an)m(y)f Fu(\030)f Fp(2)1115 2804 y Fw(~)1099 2827 y Fu(S)769 3084 y Fp(k)p Fu(@)867 3047 y Ft(\015)862 3107 y(x)912 3084 y Fu(G)984 3047 y Ft(k)984 3107 y Fm(\000)p Ft(i)1067 3084 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)1291 3104 y Ft(H)1354 3085 y FA(s)p Fo(+)p FA(\034)1557 3084 y Fp(\024)1720 2971 y Ft(i)p Fm(\000)p Fq(1)1711 2998 y Fr(X)1715 3192 y Ft(r)r Fq(=1)1857 3084 y Fp(k)p Fu(@)1955 3047 y Ft(\015)2001 3084 y Fu(q)2042 3098 y Fm(\000)p Ft(r)2134 3084 y Fw(\()p Fp(\001)p Fu(;)g(\030)t Fw(\))p Fp(k)2358 3104 y Ft(H)2421 3085 y FA(s)p Fo(+)p FA(\034)2542 3084 y Fp(j)p Fu(\013)2625 3047 y Ft(k)2625 3111 y Fm(\000)p Fq(\()p Ft(i)p Fm(\000)p Ft(r)r Fq(\))2852 3084 y Fw(\()p Fu(\030)t Fw(\))p Fp(j)-2605 b Fw(\(4.19\))1557 3324 y Fp(\024)83 b Fu(")1753 3287 y Fq(2)1793 3324 y Fu(M)1891 3287 y Ft(i)1919 3324 y Fu(N)2002 3287 y Fm(j)p Ft(\015)t Fm(j)p Fq(+)p Ft(\034)2180 3324 y Fw(\(\()p Fu(\022)s(i)20 b Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!\))2635 3287 y Ft(\033)2683 3324 y Fu(L)-2359 b Fw(\(4.20\))386 3497 y(where)742 3736 y Fu(L)25 b Fw(=)933 3622 y Ft(i)p Fm(\000)p Fq(1)925 3650 y Fr(X)928 3844 y Ft(r)r Fq(=1)1071 3608 y Fr(\022)1148 3674 y Fw(\()p Fu(\022)s(r)d Fw(+)e Fp(j)p Fu(\015)5 b Fp(j)21 b Fw(+)f Fu(\034)10 b Fw(\)!\()p Fu(\022)s Fw(\()p Fu(i)21 b Fp(\000)f Fu(r)s Fw(\)\)!)p 1148 3715 958 4 v 1434 3798 a(\()p Fu(\022)s(i)g Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!)2115 3608 y Fr(\023)2182 3630 y Ft(\033)386 3736 y Fw(\(4.21\))588 4055 y(=)742 3927 y Fr(\022)819 3993 y Fw(\()p Fu(\034)30 b Fw(+)20 b(1\)!)p 819 4034 303 4 v 825 4117 a Fu(\022)s(i)f Fw(+)h Fp(j)p Fu(\015)5 b Fp(j)1131 3927 y Fr(\023)1198 3949 y Ft(\033)1269 3941 y(i)p Fm(\000)p Fq(1)1260 3969 y Fr(X)1264 4163 y Ft(r)r Fq(=1)1406 3927 y Fr(\022)1657 3993 y Fw(\()p Fu(\034)30 b Fw(+)20 b(2\))15 b Fu(:)g(:)g(:)j Fw(\()p Fu(\034)30 b Fw(+)20 b(1)h(+)f Fu(\022)s(r)i Fw(+)d Fp(j)p Fu(\015)5 b Fp(j)21 b(\000)f Fw(1\))p 1483 4034 1609 4 v 1483 4117 a(\()p Fu(\022)s Fw(\()p Fu(i)h Fp(\000)f Fu(r)s Fw(\))g(+)g(1\))15 b Fu(:)g(:)g(:)i Fw(\()p Fu(\022)s Fw(\()p Fu(i)j Fp(\000)g Fu(r)s Fw(\))g(+)g Fu(\022)s(r)i Fw(+)e Fp(j)p Fu(\015)5 b Fp(j)21 b(\000)f Fw(1\))3102 3927 y Fr(\023)3168 3949 y Ft(\033)3230 4055 y Fu(:)386 4313 y Fw(Since)29 b Fu(i)21 b Fp(\000)f Fu(r)28 b Fp(\025)c Fw(1,)31 b Fu(r)d Fp(\025)d Fw(1)31 b(and)f Fu(\022)d Fp(\025)e Fw(\(1)c(+)f Fu(\034)10 b Fw(\))31 b(w)m(e)g(ha)m(v)m(e)822 4538 y(~)-47 b Fu(c)83 b Fw(:=)1304 4477 y(\()p Fu(\034)31 b Fw(+)20 b(2\))15 b Fu(:)g(:)g(:)i Fw(\()p Fu(\034)31 b Fw(+)20 b(1)g(+)g Fu(\022)s(r)i Fw(+)e Fp(j)p Fu(\015)5 b Fp(j)21 b(\000)f Fw(1\))p 1131 4517 V 1131 4601 a(\()p Fu(\022)s Fw(\()p Fu(i)g Fp(\000)g Fu(r)s Fw(\))g(+)g(1\))15 b Fu(:)g(:)g(:)i Fw(\()p Fu(\022)s Fw(\()p Fu(i)k Fp(\000)f Fu(r)s Fw(\))g(+)g Fu(\022)s(r)i Fw(+)e Fp(j)p Fu(\015)5 b Fp(j)20 b(\000)g Fw(1\))955 4788 y Fp(\024)1501 4727 y Fu(\022)e(:)d(:)g(:)h Fw(\()p Fu(\022)23 b Fw(+)c Fu(\022)s(r)j Fw(+)e Fp(j)p Fu(\015)5 b Fp(j)21 b(\000)f Fw(1\))p 1131 4767 V 1131 4851 a(\()p Fu(\022)s Fw(\()p Fu(i)g Fp(\000)g Fu(r)s Fw(\))g(+)g(1\))15 b Fu(:)g(:)g(:)i Fw(\()p Fu(\022)s Fw(\()p Fu(i)k Fp(\000)f Fu(r)s Fw(\))g(+)g Fu(\022)s(r)i Fw(+)e Fp(j)p Fu(\015)5 b Fp(j)20 b(\000)g Fw(1\))2774 4788 y Fp(\024)25 b Fw(1)p Fu(:)-2554 b Fw(\(4.22\))386 5014 y(It)30 b(is)g(readily)f(v)m (eri\014ed)g(from)h(\(4.21\))j(that,)e(for)f(all)f Fu(\015)i Fp(2)24 b Fs(Z)2388 4981 y Ft(n)2388 5037 y Fq(+)740 5283 y Fu(L)83 b Fp(\024)1038 5155 y Fr(\022)1115 5222 y Fw(\()p Fu(\034)31 b Fw(+)20 b(1\)!)p 1115 5262 303 4 v 1121 5345 a Fu(\022)s(i)g Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)1428 5155 y Fr(\023)1495 5178 y Ft(\033)1565 5169 y(i)p Fm(\000)p Fq(1)1557 5197 y Fr(X)1560 5391 y Ft(r)r Fq(=1)1733 5283 y Fw(1)-1392 b(\(4.23\))885 5570 y(=)1038 5441 y Fr(\022)1115 5508 y Fw(\()p Fu(\034)31 b Fw(+)20 b(1\)!)p 1115 5549 V 1121 5632 a Fu(\022)s(i)g Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)1428 5441 y Fr(\023)1495 5464 y Ft(\033)1557 5570 y Fw(\()p Fu(i)21 b Fp(\000)e Fw(1\))26 b Fp(\024)1985 5508 y Fw(\(\()p Fu(\034)31 b Fw(+)20 b(1\)!\))2357 5475 y Ft(\033)p 1946 5549 498 4 v 1946 5632 a Fw(\()p Fu(\022)s(i)g Fw(+)g Fp(j)p Fu(\015)5 b Fp(j)p Fw(\))2306 5606 y Ft(\033)r Fm(\000)p Fq(1)2480 5570 y Fp(\024)25 b Fw(\(\()p Fu(\034)31 b Fw(+)19 b(1\)!\))2947 5532 y Ft(\033)2996 5570 y Fu(:)p eop %%Page: 16 16 16 15 bop 386 239 a Fq(16)693 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g (YOSHINO)386 434 y Fw(Hence,)c(b)m(y)d(c)m(ho)s(osing)h Fu(")h(>)g Fw(0)f(and)f Fu(N)35 b(>)25 b Fw(0)f(so)g(that)g Fu("N)2253 401 y Ft(\034)2296 434 y Fw(\(\()p Fu(\034)18 b Fw(+)6 b(1\)!\))2641 401 y Ft(\033)2714 434 y Fp(\024)25 b Fw(1)p Fu(=)p Fw(3)g(w)m(e)f(obtain,)386 542 y(from)30 b(\(4.1\))i(and)e(\(4.2\))i(with)d(1)d Fp(\024)f Fu(j)31 b Fp(\024)25 b Fu(i)20 b Fp(\000)g Fw(1)753 700 y(sup)757 794 y Ft(\030)s Fm(2)850 777 y Fq(~)838 794 y Ft(S)905 700 y Fp(k)p Fu(@)1003 663 y Ft(\015)998 723 y(x)1048 700 y Fu(G)1120 663 y Ft(k)1120 723 y Fm(\000)p Ft(i)1202 700 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)1426 720 y Ft(H)1489 701 y FA(s)p Fo(+)p FA(\034)1635 700 y Fp(\024)25 b Fw(3)1776 663 y Fm(\000)p Fq(1)1871 700 y Fu("M)2011 663 y Ft(i)2040 700 y Fu(N)2123 663 y Fm(j)p Ft(\015)t Fm(j)2206 700 y Fw(\(\()p Fu(\022)s(i)c Fw(+)f Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!\))2662 663 y Ft(\033)2710 700 y Fu(;)197 b(\015)31 b Fp(2)24 b Fs(Z)3160 663 y Ft(n)3160 723 y Fq(+)3215 700 y Fu(:)-2854 b Fw(\(4.24\))486 933 y(Finally)-8 b(,)29 b(observ)m(e)i(that)g(\(4.11\))h(leads)e(to)489 1204 y(sup)494 1298 y Ft(\030)s Fm(2)586 1281 y Fq(~)575 1298 y Ft(S)641 1204 y Fp(k)p Fu(@)739 1167 y Ft(\015)734 1227 y(x)784 1204 y Fu(a)832 1167 y Ft(k)832 1227 y Fm(\000)p Ft(j)924 1204 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)1148 1224 y Ft(H)1211 1205 y FA(s)p Fo(+)p FA(\034)1357 1204 y Fp(\024)25 b Fw(3)1498 1167 y Fm(\000)p Fq(1)1592 1204 y Fu("M)1732 1167 y Ft(j)1769 1204 y Fu(N)1852 1167 y Fm(j)p Ft(\015)t Fm(j)1936 1204 y Fw(\(\()p Fu(\022)s(j)h Fw(+)20 b Fp(j)p Fu(\015)5 b Fp(j)p Fw(\)!\))2403 1167 y Ft(\033)2451 1204 y Fu(;)197 b(j)31 b Fp(2)25 b Fs(N)7 b Fu(;)51 b(\015)31 b Fp(2)25 b Fs(Z)3192 1167 y Ft(n)3192 1227 y Fq(+)3246 1204 y Fu(:)386 1072 y Fw(\(4.25\))386 1437 y(pro)m(vided)35 b(that)i Fu(M)45 b Fp(\025)36 b Fw(3)p Fu(A=")h Fw(and)f Fu(N)45 b Fp(\025)36 b Fu(A)p Fw(.)59 b(By)36 b(Lemma)h(4.2,)i(\(4.10\),)i(\(4.24\))d(and)386 1551 y(\(4.25\))33 b(w)m(e)d(ha)m(v)m(e,)i(for)e(an)m(y)h Fu(\030)e Fp(2)1498 1528 y Fw(~)1483 1551 y Fu(S)604 1706 y Fp(j)p Fu(\013)687 1669 y Ft(k)687 1729 y Fm(\000)p Ft(i)771 1706 y Fw(\()p Fu(\030)t Fw(\))p Fp(j)84 b(\024)f(j)p Fw([)p Fu(F)1269 1669 y Ft(k)1256 1729 y Fm(\000)p Ft(i)1340 1706 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\)])1544 1720 y Ft(n)1592 1706 y Fp(j)21 b Fw(+)f Fp(j)p Fw([)p Fu(G)1851 1669 y Ft(k)1851 1729 y Fm(\000)p Ft(i)1934 1706 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\)])2138 1720 y Ft(n)2186 1706 y Fp(j)21 b Fw(+)e Fp(j)p Fw([)p Fu(a)2420 1669 y Ft(k)2420 1729 y Fm(\000)p Ft(i)2504 1706 y Fw(\()p Fp(\001)p Fu(;)c(\030)t Fw(\)])2708 1720 y Ft(n)2757 1706 y Fp(j)994 1854 y(\024)83 b(k)p Fu(F)1264 1817 y Ft(k)1251 1877 y Fm(\000)p Ft(i)1335 1854 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)1559 1874 y Ft(L)1607 1855 y Fo(2)1667 1854 y Fw(+)20 b Fp(k)p Fu(G)1875 1817 y Ft(k)1875 1877 y Fm(\000)p Ft(i)1958 1854 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)2182 1874 y Ft(L)2230 1855 y Fo(2)2291 1854 y Fw(+)20 b Fp(k)p Fu(a)2475 1817 y Ft(k)2475 1877 y Fm(\000)p Ft(i)2558 1854 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)2782 1874 y Ft(L)2830 1855 y Fo(2)994 2003 y Fp(\024)83 b(k)p Fu(F)1264 1965 y Ft(k)1251 2025 y Fm(\000)p Ft(i)1335 2003 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)1559 2022 y Ft(H)1622 2003 y FA(s)p Fo(+)p FA(\034)35 b Fw(+)20 b Fp(k)p Fu(G)1970 1965 y Ft(k)1970 2025 y Fm(\000)p Ft(i)2053 2003 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)2277 2022 y Ft(H)2340 2003 y FA(s)p Fo(+)p FA(\034)36 b Fw(+)20 b Fp(k)p Fu(a)2665 1965 y Ft(k)2665 2025 y Fm(\000)p Ft(i)2749 2003 y Fw(\()p Fp(\001)p Fu(;)15 b(\030)t Fw(\))p Fp(k)2973 2022 y Ft(H)3036 2003 y FA(s)p Fo(+)p FA(\034)-2755 b Fw(\(4.26\))994 2149 y Fp(\024)83 b Fu(")p Fw(\()p Fu(C)1290 2163 y Ft(\034)1333 2149 y Fw(\))1368 2111 y Fm(\000)p Fq(1)1463 2149 y Fu(M)1561 2111 y Ft(i)1589 2149 y Fw(\(\()p Fu(\022)s(i)p Fw(\)!\))1831 2111 y Ft(\033)1905 2149 y Fp(\024)25 b Fu("M)2141 2111 y Ft(i)2169 2149 y Fw(\(\()p Fu(\022)s(i)p Fw(\)!\))2411 2111 y Ft(\033)2459 2149 y Fu(:)386 2297 y Fw(Th)m(us)31 b(\(4.1\))i(and)e(\(4.2\))j(for)e(1)c Fp(\024)f Fu(j)33 b Fp(\024)28 b Fu(i)21 b Fp(\000)g Fw(1)32 b(imply)e(\(4.2\))k(for)d Fu(j)i Fw(=)28 b Fu(i)p Fw(,)k(while)e(\(4.1\))k(for)386 2405 y Fu(j)d Fw(=)25 b Fu(i)30 b Fw(follo)m(ws)g(from)g(Lemma)g(4.2)i (applied)c(to)750 2554 y Fu(f)795 2569 y Ft(k)862 2554 y Fw(=)d Fu(F)1029 2517 y Ft(k)1016 2577 y Fm(\000)p Ft(i)1099 2554 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))22 b(+)e Fu(G)1490 2517 y Ft(k)1490 2577 y Fm(\000)p Ft(i)1572 2554 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))21 b Fp(\000)f Fu(a)1938 2517 y Ft(k)1938 2577 y Fm(\000)p Ft(i)2021 2554 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))22 b(+)e Fu(\013)2398 2517 y Ft(k)2398 2577 y Fm(\000)p Ft(i)2481 2554 y Fw(\()p Fu(\030)t Fw(\))p Fu(;)198 b Fw(1)25 b Fp(\024)g Fu(k)k Fp(\024)c Fu(d;)-2842 b Fw(\(4.27\))386 2712 y(and)25 b(the)h(fact)h(that)g(the)f Fu(H)1313 2679 y Ft(s)p Fq(+)p Ft(\034)1443 2712 y Fw(\()p Fs(T)1537 2679 y Ft(n)1586 2712 y Fw(\))g(norm)g(of)g Fu(f)2025 2727 y Ft(k)2093 2712 y Fw(giv)m(en)g(b)m(y)f(\(4.27\),)30 b(do)s(es)25 b(not)h(dep)s(end)386 2822 y(on)k Fu(\013)570 2789 y Ft(k)570 2848 y Fm(\000)p Ft(i)653 2822 y Fw(.)p 3309 2822 4 62 v 3313 2764 55 4 v 3313 2822 V 3367 2822 4 62 v 497 3012 a(5.)51 b Fv(Global)33 b(pr)n(oper)-6 b(ties)32 b(of)i(per)-6 b(turba)g(tions)31 b(of)j(resonant)e(vector)1805 3120 y(fields)486 3282 y Fw(Let)41 b Fp(L)722 3296 y Ft(j)800 3282 y Fw(b)s(e)g(de\014ned)f(b)m(y)h(\(1.3\))i(with)d Fu(a)1890 3296 y Ft(j)1970 3282 y Fp(2)k Fu(G)2147 3249 y Ft(\033)2193 3282 y Fw(\()p Fs(T)2287 3249 y Ft(n)2336 3282 y Fw(\).)74 b(W)-8 b(e)42 b(sa)m(y)g(that)g Fp(L)3076 3296 y Ft(j)3154 3282 y Fw(is)e Fu(G)3328 3249 y Ft(\033)386 3390 y Fw(globally)21 b(h)m(yp)s(o)s(elliptic)f(\(1)26 b Fp(\024)f Fu(\033)k Fp(\024)24 b(1)p Fw(\))g(if)e(ev)m(ery)h (distribution)d Fu(u)j Fw(suc)m(h)f(that)i Fp(L)3057 3404 y Ft(j)3093 3390 y Fu(u)i Fp(2)e Fu(G)3328 3357 y Ft(\033)386 3498 y Fw(\()p Fu(j)41 b Fw(=)35 b(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d)p Fw(\))37 b(is)e(in)g Fu(G)1267 3465 y Ft(\033)1313 3498 y Fw(.)59 b(Set)36 b Fu(r)h Fw(=)e(dim)1895 3512 y Fl(Z)1958 3498 y Fw(\000)2015 3465 y Ft(!)2015 3525 y Fl(Z)2102 3498 y Fw(and)g(for)h Fu(r)i Fp(\025)c Fw(1,)39 b(tak)m(e)e(a)g(canonical)386 3609 y(basis)24 b Fu(\021)652 3576 y Fq(1)692 3609 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(\021)957 3576 y Ft(r)1021 3609 y Fw(of)25 b(\000)1176 3576 y Ft(!)1176 3637 y Fl(Z)1226 3609 y Fw(.)39 b(W)-8 b(e)27 b(supp)s(ose)d(that)i(it)f(is)f(p)s (ossible)f(to)j(\014nd)e Fu(\013)2821 3623 y Ft(j)2857 3609 y Fw(\()p Fu(y)s Fw(\))i Fp(2)f Fu(G)3159 3576 y Ft(\033)3205 3609 y Fw(\()p Fs(T)3299 3576 y Ft(r)3339 3609 y Fw(\))386 3717 y(\()p Fu(j)31 b Fw(=)25 b(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d)p Fw(\))32 b(for)e(whic)m(h)f(the)i (system)637 3870 y Fu(<)25 b(!)793 3832 y Ft(j)829 3870 y Fu(;)15 b(@)917 3884 y Ft(x)987 3870 y Fu(>)25 b( )k Fw(=)c Fu(a)1315 3884 y Ft(j)1351 3870 y Fw(\()p Fu(x)p Fw(\))c Fp(\000)f Fu(\013)1643 3884 y Ft(j)1680 3870 y Fw(\()p Fu(<)25 b(\021)1859 3832 y Fq(1)1899 3870 y Fu(;)15 b(x)26 b(>;)15 b(:)g(:)g(:)31 b(;)15 b(<)25 b(\021)2448 3832 y Ft(r)2487 3870 y Fu(;)15 b(x)25 b(>)p Fw(\))p Fu(;)198 b Fw(1)25 b Fp(\024)g Fu(j)31 b Fp(\024)25 b Fu(d)-2924 b Fw(\(5.1\))386 4042 y(admits)34 b(a)i(unique)e(solution)g Fu( )k Fw(with)1748 4018 y(^)1730 4042 y Fu( )s Fw(\()p Fu(\030)t Fw(\))c(=)g(0)h(for)h Fu(\030)h Fp(2)69 b Fw(\000)2534 4009 y Ft(!)2534 4069 y Fl(Z)2584 4042 y Fw(.)56 b(Note)36 b(that)g(\(5.1\))i(is)386 4150 y(w)m(eak)m(er)32 b(than)e(the)g(Siegel) g(t)m(yp)s(e)g(conditions)f(in)g Fs(Z)2123 4117 y Ft(n)2186 4150 y Fp(n)21 b Fw(\000)2309 4117 y Ft(!)2309 4177 y Fl(Z)2359 4150 y Fw(.)486 4268 y(If)j(the)i(m)m(ultiplication)c(op)s (erator)j Fu(q)j Fw(is)c(de\014ned)g(b)m(y)h Fu(q)s(v)s Fw(\()p Fu(x)p Fw(\))h(=)f Fu(e)2623 4235 y Ft(i )r Fq(\()p Ft(x)p Fq(\))2795 4268 y Fu(v)s Fw(\()p Fu(x)p Fw(\),)i(the)e(pro)s(of) 386 4376 y(of)30 b(Theorem)h(3.1)g(and)f(\(5.1\))i(imply)c(that)402 4641 y Fu(q)446 4604 y Fm(\000)p Fq(1)560 4641 y Fp(\016)21 b(L)689 4655 y Ft(j)745 4641 y Fp(\016)g Fu(q)28 b Fw(=)d Fu(L)1038 4655 y Ft(j)1100 4641 y Fw(:=)p Fu(<)g(!)1352 4604 y Ft(j)1388 4641 y Fu(;)15 b(D)1503 4655 y Ft(x)1573 4641 y Fu(>)25 b Fw(+)p Fu(\013)1798 4655 y Ft(j)1834 4641 y Fw(\()p Fu(<)g(\021)2013 4604 y Fq(1)2053 4641 y Fu(;)15 b(x)26 b(>;)15 b(:)g(:)g(:)31 b(;)15 b(<)26 b(\021)2603 4604 y Ft(r)2641 4641 y Fu(;)15 b(x)26 b(>)p Fw(\))91 b(1)25 b Fp(\024)g Fu(j)31 b Fp(\024)25 b Fu(d:)386 4515 y Fw(\(5.2\))386 4794 y(Therefore)30 b(the)h(system)f Fp(L)1319 4808 y Ft(j)1355 4794 y Fu(u)c Fw(=)f Fu(f)1574 4808 y Ft(j)1610 4794 y Fw(,)30 b(\(1)c Fp(\024)f Fu(j)31 b Fp(\024)25 b Fu(d)p Fw(\))31 b(is)e(equiv)-5 b(alen)m(t)30 b(to)h(the)g(system)1407 4942 y Fu(L)1469 4956 y Ft(j)1505 4942 y Fu(v)e Fw(=)c Fu(g)1717 4956 y Ft(j)1754 4942 y Fu(;)197 b Fw(1)26 b Fp(\024)f Fu(j)30 b Fp(\024)25 b Fu(d)-1967 b Fw(\(5.3\))386 5110 y(where)30 b Fu(g)692 5124 y Ft(j)754 5110 y Fw(=)25 b Fu(e)892 5077 y Ft(i )r Fq(\()p Ft(x)p Fq(\))1064 5110 y Fu(f)1109 5124 y Ft(j)1145 5110 y Fw(\()p Fu(x)p Fw(\),)31 b(1)25 b Fp(\024)g Fu(j)31 b Fp(\024)25 b Fu(d)p Fw(.)486 5218 y(W)-8 b(e)25 b(de\014ne)f(the)g(c) m(hange)i(of)e(v)-5 b(ariables)23 b Fu(x)i Fw(=)g Fu(T)19 b Fw(~)-51 b Fu(x)p Fw(,)31 b(~)-50 b Fu(x)25 b Fw(=)g(\()p Fu(y)s(;)15 b(z)t Fw(\))25 b(with)e Fu(y)28 b Fw(=)d(\()p Fu(y)3000 5232 y Fq(1)3040 5218 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(y)3301 5232 y Ft(r)3339 5218 y Fw(\))386 5326 y(and)33 b Fu(z)i Fw(=)c(\()p Fu(z)822 5340 y Fq(1)862 5326 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(z)1120 5340 y Ft(n)p Fm(\000)p Ft(r)1257 5326 y Fw(\))34 b(b)m(y)f Fu(y)1500 5340 y Ft(j)1567 5326 y Fw(=)p Fu(<)e(\021)1788 5293 y Ft(j)1824 5326 y Fu(;)15 b(x)32 b(>)p Fw(,)i(\(1)e Fp(\024)e Fu(j)37 b Fp(\024)30 b Fu(r)s Fw(\),)35 b(and)e Fu(z)2827 5340 y Ft(j)2894 5326 y Fw(=)p Fu(<)e(\024)3119 5293 y Ft(j)3156 5326 y Fu(;)15 b(x)31 b(>)p Fw(,)386 5434 y(\(1)26 b Fp(\024)f Fu(j)31 b Fp(\024)25 b Fu(n)20 b Fp(\000)f Fu(r)s Fw(\).)41 b(Then)29 b(the)i(system)f(\(5.3\))j(b)s (ecomes)848 5586 y Fu(<)25 b(\022)990 5549 y Ft(j)1025 5586 y Fu(;)15 b(D)1140 5600 y Ft(z)1206 5586 y Fu(>)25 b(w)e Fw(+)d Fu(\013)1539 5600 y Ft(j)1575 5586 y Fw(\()p Fu(y)s Fw(\))p Fu(w)29 b Fw(=)c Fu(h)1935 5600 y Ft(j)1971 5586 y Fw(\()p Fu(y)s(;)15 b(z)t Fw(\))p Fu(;)199 b(j)31 b Fw(=)25 b(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d;)-2511 b Fw(\(5.4\))p eop %%Page: 17 17 17 16 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f (FIELDS)h(ON)g(TORI)559 b(17)386 434 y Fw(where)40 b Fu(\022)705 401 y Ft(j)784 434 y Fp(2)j Fs(R)948 401 y Ft(n)p Fm(\000)p Ft(r)1130 434 y Fw(are)f(giv)m(en)f(b)m(y)g(\(0)p Fu(;)15 b(\022)1844 401 y Ft(j)1881 434 y Fw(\))43 b(=)2073 401 y Ft(tr)2136 434 y Fu(T)13 b(!)2262 401 y Ft(j)2298 434 y Fw(,)44 b(1)g Fp(\024)f Fu(j)48 b Fp(\024)43 b Fu(d)p Fw(.)73 b(The)41 b(partial)386 542 y(F)-8 b(ourier)30 b(transform)f Fu(z)h Fp(!)25 b Fu(\020)37 b Fw(reduces)30 b(\(5.4\))i(to)694 705 y(\()p Fu(<)25 b(\022)871 668 y Ft(j)907 705 y Fu(;)15 b(\020)32 b(>)25 b Fw(+)p Fu(\013)1244 719 y Ft(j)1280 705 y Fw(\()p Fu(y)s Fw(\)\))20 b(^)-65 b Fu(w)t Fw(\()p Fu(y)s(;)15 b(\020)7 b Fw(\))26 b(=)1850 681 y(^)1829 705 y Fu(h)1881 719 y Ft(j)1917 705 y Fw(\()p Fu(y)s(;)15 b(\020)7 b Fw(\))p Fu(;)198 b(\020)32 b Fp(2)25 b Fs(Z)2568 668 y Ft(n)p Fm(\000)p Ft(r)2699 705 y Fu(;)30 b(j)h Fw(=)25 b(1)p Fu(;)15 b(:)g(:)g(:)33 b(;)15 b(d:)-2867 b Fw(\(5.5\))386 884 y(De\014ne)29 b(\010\()p Fu(y)s(;)15 b(\020)7 b Fw(\))25 b(=)1058 815 y Fr(P)1154 842 y Ft(d)1154 910 y(j)t Fq(=1)1296 884 y Fp(j)h Fu(<)f(\022)1489 851 y Ft(j)1524 884 y Fu(;)15 b(\020)32 b(>)25 b Fw(+)p Fu(\013)1861 898 y Ft(j)1898 884 y Fw(\()p Fu(y)s Fw(\))p Fp(j)k Fw(and)g(let)g Fu(d)2423 898 y Fq(0)2488 884 y Fp(\024)c Fu(n)16 b Fp(\000)h Fu(r)32 b Fw(b)s(e)c(the)h(rank)f(of)386 1011 y Fu(\022)432 978 y Fq(1)471 1011 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(\022)733 978 y Ft(d)773 1011 y Fw(.)41 b(Then)29 b(w)m(e)i(ha)m(v)m(e)386 1177 y FE(Theorem)j(5.1.)46 b Fk(Supp)-5 b(ose)32 b Fw(\000)1439 1144 y Ft(!)1439 1204 y Fl(Z)1515 1177 y Fp(6)p Fw(=)25 b Fp(f)p Fw(0)p Fp(g)31 b Fk(and)h(let)f Fu(\021)2126 1144 y Fq(1)2166 1177 y Fu(;)15 b(:)g(:)g(:)31 b(;)15 b(\021)2430 1144 y Ft(r)2500 1177 y Fk(b)-5 b(e)30 b(a)h(c)-5 b(anonic)g(al)33 b(b)-5 b(asis)31 b(of)386 1292 y Fw(\000)443 1259 y Ft(!)443 1319 y Fl(Z)493 1292 y Fk(.)42 b(Assume)32 b(that)i(\(5.1\))g(has)g(a)f(unique)f(solution)i(and)f(that)h Fu(a)2605 1259 y Ft(k)2605 1317 y Fm(\000)p Ft(j)2697 1292 y Fw(\()p Fu(x;)15 b(\030)t Fw(\))26 b Fp(2)f Fu(C)3087 1259 y Fm(1)3161 1292 y Fw(\()p Fs(R)3256 1259 y Ft(n)3256 1319 y(\030)3329 1292 y Fp(n)386 1409 y(f)p Fw(0)p Fp(g)p Fw(;)15 b Fu(G)633 1376 y Ft(\033)681 1409 y Fw(\()p Fs(T)775 1376 y Ft(n)775 1431 y(x)824 1409 y Fw(\)\))33 b Fk(if)f Fu(\033)d(<)c Fp(1)p Fk(.)41 b(Then)386 1517 y(\(i\))f(The)h(system)g(\(5.4\))h(is)e Fu(G)1440 1484 y Ft(\033)1527 1517 y Fk(glob)-5 b(al)5 b(ly)41 b(hyp)-5 b(o)g(el)5 b(liptic)43 b(if)d(ther)-5 b(e)41 b(exists)g Fu(c)f(>)f Fw(0)h Fk(such)386 1625 y(that)d Fw(\010\()p Fu(y)s(;)15 b(\020)7 b Fw(\))30 b Fp(\025)g Fu(c)36 b Fk(for)g(al)5 b(l)36 b Fu(\020)h Fp(2)30 b Fs(Z)1566 1592 y Ft(n)p Fm(\000)p Ft(r)1733 1625 y Fk(and)37 b Fu(y)c Fp(2)d Fs(T)2141 1592 y Ft(r)2181 1625 y Fk(.)50 b(If)35 b Fu(d)2404 1639 y Fq(0)2475 1625 y Fw(=)30 b Fu(n)22 b Fp(\000)g Fu(r)38 b Fk(the)e(pr)-5 b(e)g(c)g(e)g(e)g(ding)386 1733 y(c)g(ondition)29 b(is)f(r)-5 b(eplac)g(e)g(d)30 b(by)d(a)h(discr)-5 b(ete)29 b(c)-5 b(ondition)29 b Fw(\010\()p Fu(y)s(;)15 b(\020)7 b Fw(\))25 b Fp(6)p Fw(=)g(0)j Fk(for)g(al)5 b(l)28 b Fu(\020)k Fp(2)25 b Fs(Z)3072 1700 y Ft(n)p Fm(\000)p Ft(r)3231 1733 y Fk(and)386 1841 y(al)5 b(l)38 b Fu(y)e Fp(2)d Fs(T)755 1808 y Ft(n)803 1841 y Fk(.)56 b(We)36 b(note)i(that)h(the)e(c)-5 b(ompletely)39 b(r)-5 b(esonant)39 b(c)-5 b(ase)38 b Fu(r)e Fw(=)d Fu(n)23 b Fp(\000)g Fw(1)38 b Fk(satis\014es)386 1949 y Fu(d)433 1963 y Fq(0)498 1949 y Fw(=)25 b Fu(n)20 b Fp(\000)g Fu(r)s Fk(.)386 2056 y(\(ii\))36 b(Supp)-5 b(ose)39 b Fu(d)950 2070 y Fq(0)1022 2056 y Fw(=)32 b Fu(n)22 b Fp(\000)h Fu(r)s Fk(.)53 b(Then)37 b(the)g(system)g(\(5.4\))h(is)f Fu(G)2524 2023 y Ft(\033)2606 2056 y Fk(glob)-5 b(al)5 b(ly)38 b(hyp)-5 b(o)g(el)5 b(liptic)386 2164 y(for)33 b Fw(1)26 b Fu(<)f(\033)j Fp(\024)d(1)33 b Fk(if)f(and)h(only)h(if)e (the)h(discr)-5 b(ete)34 b(c)-5 b(ondition)34 b(is)e(satis\014e)-5 b(d.)386 2331 y(Pr)g(o)g(of.)43 b Fw(By)30 b(the)h(represen)m(taion)f (\(5.5\),)j(ev)m(ery)e(solution)e Fu(w)k Fw(satis\014es)48 b(^)-64 b Fu(w)s Fw(\()p Fu(y)s Fw(;)15 b Fu(\020)7 b Fw(\))386 2449 y(=)519 2425 y(^)497 2449 y Fu(h)549 2463 y Ft(j)586 2449 y Fw(\()p Fu(y)s Fw(;)15 b Fu(\020)7 b Fw(\)\()p Fu(<)41 b(\022)984 2416 y Ft(j)1020 2449 y Fu(;)15 b(\020)47 b(>)40 b Fw(+)p Fu(\013)1387 2463 y Ft(j)1424 2449 y Fw(\()p Fu(y)s Fw(\)\))1577 2416 y Fm(\000)p Fq(1)1712 2449 y Fw(for)f(some)h(1)h Fp(\024)f Fu(j)46 b Fp(\024)40 b Fu(d)p Fw(.)69 b(Hence)40 b Fu(w)i Fw(is)d(in)f Fu(G)3303 2416 y Ft(\033)3349 2449 y Fw(.)386 2572 y(If)e Fu(d)530 2586 y Fq(0)606 2572 y Fw(=)g Fu(n)24 b Fp(\000)g Fu(r)s Fw(,)39 b(w)m(e)e(note)g(that)1547 2503 y Fr(P)1643 2530 y Ft(d)1643 2599 y(j)t Fq(=1)1785 2572 y Fp(j)f Fu(<)g(\022)1999 2539 y Ft(j)2035 2572 y Fu(;)15 b(\020)43 b(>)35 b Fp(j)i(!)f(1)g Fw(as)i Fp(j)p Fu(\020)7 b Fp(j)36 b(!)g(1)p Fw(.)60 b(Hence)386 2688 y(the)34 b(condition)e(\010)f Fp(\025)f Fu(c)k Fw(can)g(b)s(e)g (replaced)f(b)m(y)g(the)h(discrete)g(condition.)49 b(Th)m(us)33 b(\(i\))h(is)386 2796 y(pro)m(v)m(ed.)486 2904 y(As)c(to)i(\(ii\))d(w)m (e)i(de\014ne)f(the)h(distribution)c Fu(w)r Fw(\()p Fu(y)s(;)15 b(z)t Fw(\))33 b(b)m(y)49 b(^)-64 b Fu(w)s Fw(\()p Fu(y)s(;)15 b(\020)7 b Fw(\))26 b Fp(\021)f Fw(0)31 b(if)f Fu(\020)i Fp(6)p Fw(=)25 b Fu(\020)3133 2871 y Fq(0)3172 2904 y Fw(,)31 b(and)405 3012 y(^)-64 b Fu(w)r Fw(\()p Fu(y)s(;)15 b(\020)623 2979 y Fq(0)663 3012 y Fw(\))26 b(=)f Fu(\016)s Fw(\()p Fu(y)20 b Fp(\000)d Fu(y)1099 2979 y Fq(0)1138 3012 y Fw(\),)30 b(with)d Fu(y)1481 2979 y Fq(0)1546 3012 y Fp(2)d Fs(T)1690 2979 y Ft(r)1759 3012 y Fw(and)k Fu(\020)1981 2979 y Fq(0)2045 3012 y Fp(2)d Fs(Z)2196 2979 y Ft(n)p Fm(\000)p Ft(r)2356 3012 y Fw(satisfying)i Fu(\020)2806 2979 y Fq(0)2862 3012 y Fw(+)16 b Fu(\013)3007 3027 y Ft(k)3050 3012 y Fw(\()p Fu(y)3133 2979 y Fq(0)3173 3012 y Fw(\))25 b(=)g(0)386 3120 y(for)e(1)j Fp(\024)f Fu(k)k Fp(\024)c Fu(d)p Fw(.)38 b(Here)25 b Fu(\016)s Fw(\()p Fu(y)10 b Fp(\000)d Fu(y)1435 3087 y Fq(0)1475 3120 y Fw(\))24 b(stands)f(for)h(the)g(Dirac)f(measure)h(massed)g(at)g Fu(y)k Fw(=)d Fu(y)3310 3087 y Fq(0)3349 3120 y Fw(.)386 3228 y(The)30 b(distribution)c Fu(w)r Fw(\()p Fu(y)s(;)15 b(z)t Fw(\))33 b(giv)m(es)d(a)h(nonsmo)s(oth)f(zero)h(solution)e(of)h (\(5.4\).)p 3309 3228 4 62 v 3313 3170 55 4 v 3313 3228 V 3367 3228 4 62 v 386 3394 a FE(Remark)k(5.2.)46 b Fk(\(a\))36 b(If)e Fu(d)1291 3408 y Fq(0)1360 3394 y Fu(<)28 b(n)21 b Fp(\000)h Fu(r)37 b Fk(the)e(c)-5 b(ondition)36 b Fw(\010\()p Fu(y)s(;)15 b(\020)7 b Fw(\))29 b Fp(\025)f Fu(c)35 b Fk(is)g(not)g(a)g(discr)-5 b(ete)386 3504 y(c)g(ondition)41 b(b)-5 b(e)g(c)g(ause)41 b(the)f(ve)-5 b(ctors)41 b Fw(\()p Fu(<)e(\022)1778 3471 y Fq(1)1816 3504 y Fu(;)15 b(\020)46 b(>;)15 b(:)g(:)g(:)31 b(;)15 b(<)39 b(\022)2385 3471 y Ft(d)2424 3504 y Fu(;)15 b(\020)46 b(>)p Fw(\))39 b(\()p Fu(\020)46 b Fp(2)38 b Fs(Z)2980 3471 y Ft(n)p Fm(\000)p Ft(r)3111 3504 y Fw(\))i Fk(\014l)5 b(l)40 b(a)386 3614 y(dense)e(subset)h(of)f(a)h Fu(d)1162 3628 y Fq(0)1201 3614 y Fk({dimensional)i(subsp)-5 b(ac)g(e)39 b(of)g Fs(R)2302 3581 y Ft(d)2349 3614 y Fk(.)58 b(We)38 b(note)h(that)g(if)f Fu(d)3146 3628 y Fq(0)3221 3614 y Fu(<)d(d)386 3722 y Fk(this)e(actual)5 b(ly)34 b(o)-5 b(c)g(curs.)486 3830 y(\(b\))33 b(If)h(the)g(simultane)-5 b(ous)36 b Fu(G)1499 3797 y Ft(\033)1579 3830 y Fk(Sie)-5 b(gel)33 b(c)-5 b(ondition)36 b(holds,)f(the)g(solvability)g(of)f(\(5.1\))386 3938 y(is)f(sup)-5 b(er\015uous;)35 b(that)g(is)e(inste)-5 b(ad)35 b(of)e(the)h(Diophantine)h(c)-5 b(onditions)35 b(on)f Fu(!)2986 3905 y Ft(j)3022 3938 y Fk(,)g(the)f(sol-)386 4046 y(vability)i(of)h(\(5.1\))g(for)g(sp)-5 b(e)g(cial)37 b Fu(a)1531 4060 y Ft(j)1567 4046 y Fw(\()p Fu(x)p Fw(\))f Fk(is)f(enough.)50 b(This)36 b(al)5 b(lows)37 b(us)e(to)h(char)-5 b(acterize)386 4156 y(lar)g(ger)36 b(classes)g(of)g(\014rst)f(or)-5 b(der)37 b(op)-5 b(er)g(ators)39 b(on)c Fs(T)2078 4123 y Fq(2)2154 4156 y Fk(with)h(nonr)-5 b(esonant)38 b Fu(!)33 b Fw(=)c Fu(!)3126 4123 y Fq(1)3200 4156 y Fk(than)386 4264 y(those)j(in)g(Pr)-5 b(op)g(osition)34 b(3.2)e(of)g Fw([4)q(])p Fk(.)41 b(Note)32 b(that)h(the)e(Gevr)-5 b(ey)32 b(sp)-5 b(ac)g(es)34 b Fu(G)2838 4231 y Ft(\033)2884 4264 y Fw(\()p Fs(T)2978 4231 y Ft(n)3027 4264 y Fw(\))e Fk(ar)-5 b(e)32 b(not)386 4372 y(metrisable)i(if)e Fw(1)26 b Fp(\024)e Fu(\033)29 b(<)c Fp(1)32 b Fk(and)i(ther)-5 b(efor)g(e)34 b(Bair)-5 b(e's)34 b(the)-5 b(or)g(em)34 b(is)f(not)g(applic)-5 b(able.)486 4539 y Fw(No)m(w)27 b(w)m(e)g(study)f(the)h(global)f(prop)s(erties)f(of)i Fp(f)p Fu(b)2091 4506 y Ft(k)2134 4539 y Fw(\()p Fu(x;)15 b(D)s Fw(\))p Fp(g)28 b Fw(in)e(the)h(nonresonan)m(t)g(case)386 4649 y(\000)443 4616 y Ft(!)443 4676 y Fl(Z)538 4649 y Fw(=)45 b Fp(f)p Fw(0)p Fp(g)p Fw(.)77 b(Let)43 b(a)f(corresp)s (onding)e(normal)h(form)h Fp(f)p Fu(!)2401 4616 y Ft(k)2472 4649 y Fp(\001)28 b Fu(\030)k Fw(+)c Fu(\013)2754 4616 y Ft(k)2797 4649 y Fw(\()p Fu(\030)t Fw(\))p Fp(g)2956 4616 y Ft(d)2956 4673 y Fq(1)3040 4649 y Fw(b)s(e)42 b(as)g(in)386 4763 y(Theorem)e(3.1)h(and)e(1)j Fp(\024)g Fu(\022)h Fp(\024)e(1)p Fw(.)70 b(W)-8 b(e)42 b(sa)m(y)e(that)h Fp(f)p Fu(!)2351 4730 y Ft(k)2394 4763 y Fu(\030)31 b Fw(+)26 b Fu(\013)2620 4730 y Ft(k)2663 4763 y Fw(\()p Fu(\030)t Fw(\))p Fp(g)2822 4730 y Ft(d)2822 4788 y Fq(1)2904 4763 y Fw(satis\014es)39 b(the)386 4871 y(sim)m(ultaneous)29 b(inhomogeneous)h Fu(\022)s Fw(-Siegel)f(condition)g(if)545 5030 y(lim)15 b(inf)431 5095 y Fm(j)p Ft(\030)s Fm(j!1)p Ft(;\030)s Fm(2)p Fl(Z)796 5072 y FA(n)796 5116 y Fo(+)841 5095 y Fm(n)p Fq(0)926 5030 y Fw(exp\()p Fu(")p Fp(j)p Fu(\030)t Fp(j)1236 4993 y Fq(1)p Ft(=\022)1347 5030 y Fw(\))41 b(max)1397 5091 y Fq(1)p Fm(\024)p Ft(k)r Fm(\024)p Ft(d)1633 5030 y Fp(j)p Fu(!)1718 4993 y Ft(k)1763 5030 y Fp(\001)s Fu(\030)7 b Fw(+)s Fu(\013)1970 4993 y Ft(k)2014 5030 y Fw(\()p Fu(\030)t Fw(\))p Fp(j)26 b Fu(>)f Fw(0)p Fu(;)137 b Fw(for)30 b(ev)m(ery)22 b Fu(")k(>)f Fw(0)46 b(\()p Fu(T)13 b(S)5 b(i)p Fw(\))3335 5045 y Ft(\022)386 5249 y Fw(for)30 b(1)c Fp(\024)f Fu(\022)i(<)e Fp(1)p Fw(,)31 b(and)1150 5404 y(lim)15 b(inf)1036 5469 y Fm(j)p Ft(\030)s Fm(j!1)p Ft(;\030)s Fm(2)p Fl(Z)1401 5446 y FA(n)1401 5490 y Fo(+)1445 5469 y Fm(n)p Fq(0)1531 5404 y Fp(j)p Fu(\030)t Fp(j)1625 5366 y Ft(N)1733 5404 y Fw(max)1708 5465 y Fq(1)p Fm(\024)p Ft(k)r Fm(\024)p Ft(d)1943 5404 y Fp(j)p Fu(!)2028 5366 y Ft(k)2091 5404 y Fp(\001)20 b Fu(\030)25 b Fw(+)19 b Fu(\013)2349 5366 y Ft(k)2393 5404 y Fw(\()p Fu(\030)t Fw(\))p Fp(j)26 b Fu(>)f Fw(0)p Fu(;)347 b Fw(\()p Fu(T)13 b(S)5 b(i)p Fw(\))3299 5418 y Fm(1)386 5623 y Fw(for)30 b(some)h Fu(N)k Fp(2)25 b Fs(R)39 b Fw(when)29 b Fu(\022)f Fw(=)d Fp(1)p Fw(.)p eop %%Page: 18 18 18 17 bop 386 239 a Fq(18)693 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g (YOSHINO)486 434 y Fw(If)g Fu(\022)g Fw(=)c Fp(1)p Fw(,)32 b(\(TSi\))1108 448 y Fm(1)1213 434 y Fw(is)e(w)m(ell)h(de\014ned)f(in)g (the)h(sense)h(that)g(it)e(do)s(es)i(not)f(dep)s(end)f(on)386 542 y(realizations)j(of)977 474 y Fr(P)1073 500 y Fm(1)1073 569 y Ft(r)r Fq(=0)1216 542 y Fu(\013)1274 556 y Fm(\000)p Ft(r)1367 542 y Fw(\()p Fu(\030)t Fw(\).)51 b(Ho)m(w)m(ev)m(er,)37 b(if)32 b(1)f Fp(\024)g Fu(\022)i(<)d Fp(1)p Fw(,)35 b(the)f(condition)e(\()p Fu(T)13 b(S)5 b(i)p Fw(\))3335 557 y Ft(\022)386 656 y Fw(is)28 b(indep)s(enden)m(t)f(of)j (realizations)e(if)g Fu(q)33 b Fw(is)28 b(a)i Fu(G)1965 623 y Ft(\022)2032 656 y Fw(p.d.o.)41 b(In)28 b(this)g(case,)j Fu(\013)2858 623 y Ft(j)2895 656 y Fw(\()p Fu(D)s Fw(\))f(and)e(the)386 766 y(remainders)h Fu(R)922 733 y Ft(k)964 766 y Fw(\()p Fu(x;)15 b(D)s Fw(\))31 b(are)g Fu(G)1459 733 y Ft(\022)1528 766 y Fw(p.d.o.)40 b(as)31 b(w)m(ell.)386 942 y FE(Theorem)j(5.3.)46 b Fk(Assume)34 b(that)i Fu(!)1627 909 y Ft(j)1663 942 y Fk(,)e Fu(j)g Fw(=)28 b(1)p Fu(;)15 b(:)g(:)g(:)33 b(;)15 b(d)p Fk(,)35 b(ar)-5 b(e)35 b(simultane)-5 b(ously)36 b(nonr)-5 b(eso-)386 1050 y(nant)32 b(and)h(satisfy)f Fw(\()p Fu(S)5 b(i)p Fw(\))1216 1064 y Ft(\033)1295 1050 y Fk(for)32 b(some)g Fw(1)26 b Fp(\024)f Fu(\033)j Fp(\024)d(1)p Fk(.)42 b(L)-5 b(et)31 b(the)h(hyp)-5 b(otheses)34 b(of)e(The)-5 b(or)g(em)386 1158 y(3.1)44 b(b)-5 b(e)43 b(true.)74 b(Then)43 b Fw(\()p Fu(T)13 b(S)5 b(i)p Fw(\))1408 1172 y Fm(1)1527 1158 y Fk(holds)45 b(for)f(one)f(simultane)-5 b(ous)45 b(c)-5 b(anonic)g(al)45 b(form)f(of)386 1268 y Fp(f)p Fu(b)470 1282 y Ft(j)507 1268 y Fp(g)552 1235 y Ft(d)552 1293 y Fq(1)624 1268 y Fk(if)32 b(and)g(only)h(if)e(it)h (holds)h(for)g(every)e(simultane)-5 b(ous)34 b(c)-5 b(anonic)g(al)33 b(form)g(of)f Fp(f)p Fu(b)3224 1235 y Ft(j)3261 1268 y Fp(g)3306 1235 y Ft(d)3306 1293 y Fq(1)3347 1268 y Fk(.)386 1382 y(Mor)-5 b(e)g(over,)36 b Fp(f)p Fu(b)900 1349 y Ft(j)937 1382 y Fp(g)982 1349 y Ft(d)982 1407 y Fq(1)1058 1382 y Fk(is)e(glob)-5 b(al)5 b(ly)36 b(hyp)-5 b(o)g(el)5 b(liptic)38 b(on)d Fs(T)2150 1349 y Ft(n)2233 1382 y Fk(if)f(and)i(only)g(if)e(a)h(normal)i(form)386 1494 y(of)29 b Fp(f)p Fu(b)573 1461 y Ft(j)610 1494 y Fp(g)655 1461 y Ft(d)655 1518 y Fq(1)724 1494 y Fk(satis\014es)h Fw(\()p Fu(T)13 b(S)5 b(i)p Fw(\))1290 1508 y Fm(1)1365 1494 y Fk(.)41 b(If)28 b Fw(\()p Fu(S)5 b(i)p Fw(\))1687 1508 y Fm(1)1791 1494 y Fk(holds)31 b(and)e Fu(a)2240 1461 y Ft(j)2277 1494 y Fw(\()p Fu(x;)15 b(D)s Fw(\))p Fk(,)30 b Fu(j)h Fw(=)25 b(1)p Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(d)p Fk(,)30 b(ar)-5 b(e)29 b Fu(G)3328 1461 y Ft(\033)386 1606 y Fk(p.d.o.)48 b(for)36 b(some)f Fw(1)29 b Fp(\024)f Fu(\033)k(<)d Fp(1)34 b Fk(then)h(the)g(op)-5 b(er)g(ator)38 b Fp(f)p Fu(b)2321 1573 y Ft(j)2358 1606 y Fp(g)2403 1573 y Ft(d)2403 1630 y Fq(1)2478 1606 y Fk(is)c Fu(G)2649 1573 y Ft(\022)2722 1606 y Fk(glob)-5 b(al)5 b(ly)36 b(hyp)-5 b(o)g(el)5 b(li-)386 1718 y(ptic)30 b(if)g(and)g(only)h(if)f(a)g(normal)h(form)g(of)f Fp(f)p Fu(b)1892 1685 y Ft(j)1929 1718 y Fp(g)1974 1685 y Ft(d)1974 1742 y Fq(1)2045 1718 y Fk(satis\014es)h Fw(\()p Fu(T)13 b(S)5 b(i)p Fw(\))2612 1733 y Ft(\022)2681 1718 y Fk(for)31 b Fu(\022)c Fp(\025)e Fu(\032\033)s Fk(,)31 b(wher)-5 b(e)386 1826 y Fu(\032)25 b Fw(=)g(max)p Fp(f)p Fw(1)p Fu(;)15 b Fw(1)22 b(+)e Fu(\034)10 b Fp(g)p Fk(.)386 2002 y(Pr)-5 b(o)g(of.)43 b Fw(The)33 b(pro)s(of)g(follo)m(ws)f(from)h (Theorem)g(3.1)i(\(resp)s(ectiv)m(ely)-8 b(,)35 b(the)e(Nekhoroshev)386 2110 y(t)m(yp)s(e)22 b(estimates\))h(if)f Fu(\022)27 b Fw(=)e Fp(1)d Fw(\(resp)s(ectiv)m(ely)-8 b(,)24 b Fu(\022)k Fp(\025)d Fu(\032\033)s Fw(\),)f(and)e(the)g(follo)m(wing)f(prop)s (erties:)386 2217 y(Since)32 b Fu(q)s Fw(\()p Fu(x;)15 b(D)s Fw(\))33 b(is)f(elliptic)e(in)h Fu(G)1517 2184 y Ft(\032\033)1633 2217 y 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v 1032 2798 a(6.)51 b Fv(Logarithmic)34 b(Ha)n(usdorff)e(dimension)486 2960 y Fw(F)-8 b(or)31 b Fi(\014)d Fw(=)d(\()p Fu(\014)919 2974 y Fq(1)959 2960 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(\014)1227 2975 y Ft(d)1268 2960 y Fw(\))26 b Fp(2)55 b Fu(R)1515 2927 y Ft(d)1585 2960 y Fw(and)30 b(1)c Fp(\024)f Fu(\033)j Fp(\024)d(1)31 b Fw(w)m(e)f(de\014ne)682 3137 y Fu(N)10 b(G)837 3100 y Ft(\033)883 3137 y Fw(\()p Fi(\014)s Fw(;)15 b Fu(d)p Fw(\))85 b(=)d Fp(f)p Fu(!)29 b Fw(=)c(\()p Fu(!)1663 3100 y Fq(1)1702 3137 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(!)1979 3100 y Ft(d)2020 3137 y Fw(\))25 b Fp(2)g Fs(R)2226 3100 y Ft(nd)2341 3137 y Fw(:)g Fp(f)p Fu(<)g(!)2592 3100 y Ft(j)2629 3137 y Fu(;)15 b(\030)29 b(>)c Fw(+)p Fu(\014)2956 3151 y Ft(j)2993 3137 y Fp(g)3038 3100 y Ft(d)3038 3160 y Fq(1)386 3137 y Fw(\(6.1\))1917 3272 y(do)s(es)30 b(not)h(satisfy)e(\()p Fu(T)13 b(S)5 b(i)p Fw(\))2795 3286 y Ft(\033)2843 3272 y Fp(g)p Fu(;)-2527 b Fw(\(6.2\))785 3407 y Fu(N)10 b(G)940 3370 y Ft(\033)986 3407 y Fw(\()p Fu(d)p Fw(\))85 b(=)d Fu(N)10 b(G)1496 3370 y Ft(\033)1543 3407 y Fw(\(0;)15 b Fu(d)p Fw(\))p Fu(:)-1384 b Fw(\(6.3\))386 3578 y(Clearly)21 b Fu(N)10 b(G)849 3545 y Ft(\033)895 3578 y Fw(\()p Fu(d)p Fw(\))t Fp(n)t Fw(\000)1122 3545 y Ft(!)1122 3605 y Fl(Z)1196 3578 y Fw(is)21 b(the)i(exceptional)f(set)h(for)f(whic)m(h)f(\()p Fu(S)5 b(i)p Fw(\))2576 3592 y Ft(\033)2646 3578 y Fw(fails,)23 b(and)e(therefore)386 3686 y(no)32 b(reduction)f(to)i(normal)f(forms)f (is)h(p)s(ossible.)44 b(Similarly)-8 b(,)29 b(if)j Fi(\014)f Fp(6)p Fw(=)d(0,)34 b Fu(N)10 b(G)3014 3653 y Ft(\033)3060 3686 y Fw(\()p Fi(\014)s Fw(;)15 b Fu(d)p Fw(\))34 b(is)386 3796 y(related)d(to)h(the)f(exceptional)h(sets)f(of)h Fp(f)p Fu(!)1826 3763 y Ft(j)1862 3796 y Fp(g)1907 3763 y Ft(d)1907 3820 y Fq(1)1979 3796 y Fw(for)f(whic)m(h)f(the)i(system)f Fp(f)p Fu(L)2947 3811 y Ft(k)3011 3796 y Fw(+)20 b Fu(\014)3153 3811 y Ft(k)3197 3796 y Fp(g)3242 3763 y Ft(d)3242 3820 y Fq(1)3314 3796 y Fw(is)386 3904 y(not)36 b(globally)f Fu(G)972 3871 y Ft(\033)1054 3904 y Fw(h)m(yp)s(o)s(elliptic.)54 b(The)36 b(sizes)f(of)h Fu(N)10 b(G)2271 3871 y Ft(\033)2318 3904 y Fw(\()p Fu(d)p Fw(\))37 b(and)e Fu(N)10 b(G)2809 3871 y Ft(\033)2855 3904 y Fw(\()p Fu(\014)5 b Fw(;)15 b(1\))38 b(will)33 b(b)s(e)386 4012 y(estimated)38 b(in)f(terms)h(of)g (the)g(Hausdor\013)g(dimension)d(and)i(logarithmic)g(Hausdor\013)386 4119 y(dimension)28 b(asso)s(ciated)j(with)e(the)h(Gevrey)h(index)e Fu(\033)s Fw(.)486 4227 y(De\014ne)h(the)h(follo)m(wing)e(set)h(for)h Fu(n)24 b(>)h(d)810 4398 y(E)877 4413 y Fh(\014)930 4398 y Fw(\()p Fu(\033)s Fw(\))h(=)f Fu(E)1244 4413 y Fh(\014)1297 4398 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g(d)p Fw(\))-1218 b(\(6.4\))656 4559 y(=)83 b Fp(f)p Fu(X)33 b Fp(2)25 b Fs(R)1109 4522 y Ft(nd)1223 4559 y Fw(:)h Fp(j)p FE(q)p Fu(X)i Fp(\000)20 b Fi(\014)s Fp(j)25 b Fu(<)g(e)1799 4522 y Fm(\000j)p Fg(q)p Fm(j)1937 4498 y Fo(1)p FA(=\033)2075 4559 y Fw(for)30 b(in\014nitely)d(man)m(y)k FE(q)25 b Fp(2)g Fs(Z)3074 4522 y Ft(n)3117 4559 y Fp(g)p Fu(:)386 4732 y Fw(Note)32 b(that)f(if)e Fu(X)38 b Fw(is)29 b(iden)m(ti\014ed)g (with)g Fu(!)f Fw(=)d(\()p Fu(!)1970 4699 y Fq(1)2009 4732 y Fu(;)15 b(:)g(:)g(:)32 b(;)15 b(!)2286 4699 y Ft(d)2327 4732 y Fw(\),)31 b(w)m(e)g(ha)m(v)m(e)1428 4903 y Fu(E)1495 4917 y Fq(0)1535 4903 y Fw(\()p Fu(\033)s Fw(\))84 b Fp(\032)e Fu(N)10 b(G)2052 4865 y Ft(\033)2099 4903 y Fw(\()p Fu(d)p Fw(\))-1830 b(\(6.5\))1415 5038 y Fu(E)1482 5053 y Fh(\014)1535 5038 y Fw(\()p Fu(\033)s Fw(\))84 b Fp(\032)e Fu(N)10 b(G)2052 5000 y Ft(\033)2099 5038 y Fw(\()p Fi(\014)s Fw(;)15 b Fu(d)p Fw(\))p Fu(:)-1958 b Fw(\(6.6\))386 5221 y FE(De\014nition)35 b(6.1.)46 b Fk(L)-5 b(et)33 b Fu(J)42 b Fk(b)-5 b(e)33 b(a)g(set)f(in)h Fs(R)1835 5188 y Ft(nd)1956 5221 y Fk(and)h(de\014ne)f(the)g(function)g Fu(f)42 b Fk(as)33 b Fu(f)10 b Fw(\()p Fu(r)s Fw(\))25 b(=)386 5335 y Fu(r)430 5302 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)658 5335 y Fw(\(log)17 b(1)p Fu(=r)s Fw(\))995 5302 y Fm(\000)p Ft(s)1087 5335 y Fk(.)42 b(The)33 b(lo)-5 b(garithmic)35 b(Hausdor\013)e(me)-5 b(asur)g(e)34 b(of)f Fu(J)42 b Fk(is)1169 5524 y Ff(L)1230 5486 y Ft(s)1266 5524 y Fw(\()p Fu(J)9 b Fw(\))26 b(:=)32 b(lim)1542 5585 y Ft(\016)r Fm(!)p Fq(0)1697 5524 y Fw(inf)1714 5585 y Fe(C)1756 5597 y FA(\016)1957 5438 y Fr(X)1823 5635 y Ft(C)5 b Fm(2)p Fe(C)1967 5647 y FA(\016)2001 5635 y Fq(:)p Fm([)p Ft(C)g Fm(\033)p Ft(J)2238 5524 y Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(C)d Fw(\)\))p Fu(;)p eop %%Page: 19 19 19 18 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f (FIELDS)h(ON)g(TORI)559 b(19)386 434 y Fk(wher)-5 b(e)32 b(the)f(in\014mum)g(is)g(over)g(al)5 b(l)32 b Fu(\016)i Fk(c)-5 b(overs,)32 b Ff(C)2004 449 y Ft(\016)2073 434 y Fk(of)f Fu(J)40 b Fk(and)31 b Fu(L)p Fw(\()p Fu(C)7 b Fw(\))31 b Fk(is)g(the)g(diameter)h(of)386 542 y Fu(C)7 b Fk(.)41 b(The)33 b(lo)-5 b(garithmic)35 b(Hausdor\013)e(dimension)h (is)803 687 y Fw(Ldim)23 b(\()p Fu(J)9 b Fw(\))26 b(:=)f(sup)o Fp(f)p Fu(s)h Fw(:)f Ff(L)1674 650 y Ft(s)1710 687 y Fw(\()p Fu(J)9 b Fw(\))27 b(=)e Fp(1g)g Fw(=)g(inf)6 b Fp(f)p Fu(s)25 b Fw(:)h Ff(L)2555 650 y Ft(s)2591 687 y Fw(\()p Fu(J)9 b Fw(\))26 b(=)f(0)p Fp(g)p Fu(:)486 843 y Fw(The)k(de\014nition)f(of)i(Hausdor\013)g(dimension)e(and)h(its) h(prop)s(erties)e(can)j(b)s(e)e(found)g(in)386 951 y([19)q(].)43 b(T)-8 b(o)32 b(denote)f(the)h(information)d(giv)m(en)i(b)m(y)g(the)h (t)m(w)m(o)g(dimensions,)d(ordinary)g(and)386 1071 y(logarithmic)i(the) h(dimension)e(of)i Fu(J)41 b Fw(will)30 b(b)s(e)h(written)h(as)g(the)h (ordered)e(pair)3111 1047 y Fu(~)3065 1071 y Fw(dim)o Fu(J)37 b Fw(=)386 1179 y(\(dim)14 b Fu(J)o(;)h Fw(Ldim)24 b Fu(J)9 b Fw(\).)486 1287 y(F)-8 b(or)31 b(the)f(homogeneous)h(case)h (the)e(follo)m(wing)f(theorem)i(will)d(b)s(e)h(pro)m(v)m(ed.)386 1450 y FE(Theorem)34 b(6.2.)46 b Fk(L)-5 b(et)33 b Fu(\033)28 b Fp(2)d Fs(R)1421 1417 y Fq(+)1486 1450 y Fk(.)42 b(Then)1839 1426 y Fu(~)1793 1450 y Fw(dim)n Fu(E)2011 1464 y Fq(0)2051 1450 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g(d)p Fw(\))26 b(=)f(\(\()p Fu(n)c Fp(\000)f Fw(1\))p Fu(d;)15 b(n\033)s Fw(\))p Fk(.)486 1606 y Fw(By)39 b(mo)s(difying)d(the)i(pro)s(of)g(of)h (Theorem)g(6.2)g(an)g(inhomogeneous)f(v)m(ersion)g(will)386 1714 y(also)30 b(b)s(e)g(pro)m(v)m(ed)h(for)f(the)g(case)i Fu(d)25 b Fw(=)g(1.)386 1877 y FE(Theorem)34 b(6.3.)46 b Fk(L)-5 b(et)31 b Fu(\033)e Fp(2)c Fs(R)1419 1844 y Fq(+)1515 1877 y Fk(and)32 b Fu(\014)f Fp(2)25 b Fs(R)r Fk(.)47 b(Then)2274 1853 y Fu(~)2228 1877 y Fw(dim)n Fu(E)2446 1892 y Ft(\014)2494 1877 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g Fw(1\))27 b(=)e(\()p Fu(n)16 b Fp(\000)h Fw(1)p Fu(;)e(n\033)s Fw(\))p Fk(.)386 2040 y FE(Remark)34 b(6.4.)46 b Fk(L)-5 b(et)39 b(us)e(identify)i Fs(R)1672 2007 y Ft(nd)1799 2040 y Fk(with)g Fw(\()p Fs(R)2097 2007 y Ft(n)2150 2040 y Fw(\))2185 2007 y Ft(d)2226 2040 y Fk(.)57 b(Then)39 b(we)f(note)g(that:)54 b(a\))38 b(the)386 2148 y(sets)25 b Fu(N)10 b(G)712 2115 y Ft(\033)759 2148 y Fw(\()p Fi(\014)s Fw(;)15 b 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(Hausdor\013)h(dimension)e(of)386 3180 y Fu(E)453 3195 y Fh(\014)506 3180 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g(d)p Fw(\))29 b(is)f(determined)e(for)i(all)f Fi(\014)s Fw(,)i Fu(n)f Fw(and)f Fu(d)i Fw(and)e(\014nally)f(lo)m(w)m(er)j(b)s(ounds)d (for)h(the)386 3288 y(logarithmic)j(Hausdor\013)g(dimensions)f(of)i (the)h(t)m(w)m(o)g(sets)g(are)f(found.)42 b(W)-8 b(e)33 b(recall)d(that)386 3395 y(in)f([14)q(])i(the)f(Hausdor\013)g (dimension)e(of)i(the)h(set)591 3541 y Fu(W)677 3555 y Fq(0)716 3541 y Fw(\()p Fu(\034)10 b Fw(\))84 b(=)e Fu(W)1159 3555 y Fq(0)1199 3541 y Fw(\()p Fu(n;)15 b(d)p Fw(;)g Fu(\034)10 b Fw(\))920 3683 y(=)82 b Fp(f)p Fu(X)33 b Fp(2)25 b Fs(R)1372 3645 y Ft(nd)1486 3683 y Fw(:)h Fp(j)p FE(q)p Fu(X)7 b Fp(j)26 b Fu(<)f Fp(j)p FE(q)p Fp(j)1951 3645 y Fm(\000)p Ft(\034)2080 3683 y Fw(for)30 b(in\014nitely)f(man)m(y)j FE(q)25 b Fp(2)g Fs(Z)3082 3645 y Ft(n)3125 3683 y Fp(g)386 3829 y Fw(w)m(as)37 b(sho)m(wn)f(to)i(b)s(e)e(\()p Fu(n)25 b Fp(\000)f Fw(1\))p Fu(d)h Fw(+)g Fu(n=)p Fw(\()p Fu(\034)35 b Fw(+)24 b(1\).)61 b(It)37 b(is)f(readily)f(v)m(eri\014ed)h(that)h(the)g(sets)386 3939 y Fu(R)455 3953 y Fg(q)529 3939 y Fw(=)26 b Fp(f)p Fu(X)34 b Fp(2)27 b Fu(I)915 3906 y Ft(nd)1024 3939 y Fw(:)g FE(q)p Fu(X)35 b Fw(=)26 b(0)p Fp(g)32 b Fw(are)f(\()p Fu(n)21 b Fp(\000)f Fw(1\))p Fu(d)p Fw({dimensional)30 b(h)m(yp)s(erplanes)f(con)m(tained)386 4047 y(in)38 b Fu(E)568 4061 y Fq(0)607 4047 y Fw(\()p Fu(\033)s Fw(\))j(for)e(all)f FE(q)i Fp(2)g Fs(Z)1317 4014 y Ft(n)1359 4047 y Fw(.)68 b(Also)38 b Fu(E)1732 4061 y Fq(0)1772 4047 y Fw(\()p Fu(\033)s Fw(\))j Fp(\032)f Fu(W)2135 4061 y Fq(0)2174 4047 y Fw(\()p Fu(\034)10 b Fw(\))40 b(for)f(all)f Fu(\034)49 b Fw(and)39 b(all)f Fu(\033)s Fw(.)68 b(Th)m(us)386 4155 y(dim)14 b Fu(E)620 4169 y Fq(0)659 4155 y Fw(\()p Fu(\033)s Fw(\))26 b(=)f(\()p Fu(n)20 b Fp(\000)g Fw(1\))p Fu(d)p Fw(.)486 4262 y(F)-8 b(or)41 b(the)g(inhomogeneous)f(case)i(the)f(sets) g Fu(R)2095 4277 y Ft(\014)s(;)p Fg(q)2247 4262 y Fw(=)i Fp(f)p FE(x)g Fp(2)f Fu(I)2654 4229 y Ft(n)2743 4262 y Fw(:)h FE(q)p Fu(:)p FE(x)28 b Fw(+)f Fu(\014)48 b Fw(=)42 b(0)p Fp(g)386 4370 y Fw(are)30 b(\()p Fu(n)19 b Fp(\000)g Fw(1\){dimensional)28 b(h)m(yp)s(erplanes)g(con)m(tained)i (in)f Fu(E)2449 4385 y Ft(\014)2496 4370 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g Fw(1\))31 b(implying)c(that)386 4478 y(dim)14 b Fu(E)620 4493 y Ft(\014)667 4478 y Fw(\()p Fu(\033)s Fw(;)h Fu(n;)g Fw(1\))50 b Fp(\025)f Fu(n)29 b Fp(\000)g Fw(1.)84 b(Using)43 b(the)i(upp)s(er)d(b)s(ound)h(argumen)m (t)i(in)e(the)h(next)386 4586 y(section)37 b(with)f Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(C)d Fw(\)\))38 b(replaced)f(b)m(y)g Fu(L)p Fw(\()p Fu(C)7 b Fw(\))1980 4553 y Ft(s)2054 4586 y Fw(it)37 b(is)f(readily)g(v)m(eri\014ed)g(that)i(for)f(an)m(y)386 4694 y Fu(s)25 b(>)g(n)19 b Fp(\000)g Fw(1)31 b(the)f(Hausdor\013)g Fu(s)p Fw({measure)g(of)g Fu(E)1973 4709 y Ft(\014)2020 4694 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g Fw(1\))32 b(is)d(zero)i(whic)m(h)e(implies)e(that)386 4802 y(dim)14 b Fu(E)620 4817 y Ft(\014)667 4802 y Fw(\()p Fu(\033)s Fw(;)h Fu(n;)g Fw(1\))27 b Fp(\024)e Fu(n)20 b Fp(\000)f Fw(1)31 b(completing)f(the)g(result.)486 4927 y(No)m(w)35 b(w)m(e)g(b)s(egin)e(to)j(deal)e(with)f(the)i(logarithmic)f(comp)s (onen)m(t)h(of)2876 4903 y Fu(~)2830 4927 y Fw(dim)n(.)54 b(In)34 b(what)386 5035 y(follo)m(w)40 b(the)h(Vinogrado)m(v)g (notation)f(will)e(b)s(e)i(used;)46 b(that)41 b(is)f Fu(a)i Fp(\034)g Fu(b)f Fw(\(resp)s(ectiv)m(ely)386 5143 y Fu(a)28 b Fp(\035)f Fu(b)p Fw(\))32 b(means)g(that)g(there)g(exists)f (a)i(constan)m(t)g Fu(c)27 b(>)h Fw(0)k(suc)m(h)f(that)i Fu(a)27 b Fp(\024)g Fu(cb)33 b Fw(\()p Fu(a)27 b Fp(\025)h Fu(cb)p Fw(\).)386 5251 y(If)i Fu(a)25 b Fp(\034)g Fu(b)31 b Fw(and)e Fu(a)d Fp(\035)f Fu(b)30 b Fw(then)g Fu(a)c Fp(\020)f Fu(b)p Fw(.)486 5359 y(First)k(the)i(upp)s(er)d(b)s(ounds)g (are)j(obtained)f(for)g(b)s(oth)g(theorems.)386 5515 y FE(Lemma)j(6.5.)46 b Fk(L)-5 b(et)42 b Fu(n;)15 b(d)42 b Fp(2)g Fs(N)7 b Fk(,)50 b Fu(n)42 b Fp(\025)g Fw(2)p Fk(,)i Fu(\033)i Fp(2)c Fs(R)2143 5482 y Fq(+)2250 5515 y Fk(and)h Fi(\014)i Fp(2)d Fs(R)s Fk(.)76 b(Then)42 b(we)g(have)386 5623 y Fw(Ldim)24 b Fu(E)687 5638 y Fh(\014)739 5623 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g(d)p Fw(\))27 b Fp(\024)e Fu(n\033)s Fk(.)p eop %%Page: 20 20 20 19 bop 386 239 a Fq(20)693 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g (YOSHINO)386 434 y Fk(Pr)-5 b(o)g(of.)43 b Fw(F)-8 b(or)31 b(future)e(reference)i(note)g(that)818 744 y Fu(f)10 b Fw(\()p Fu(e)950 707 y Fm(\000)p Ft(r)1039 683 y Fo(1)p FA(=\033)1147 744 y Fu(=r)s Fw(\))25 b(=)1508 549 y Fr(\020)1563 650 y Fu(e)1605 617 y Fm(\000)p Ft(r)1694 593 y Fo(1)p FA(=\033)1802 549 y Fr(\021)1856 571 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)p 1402 723 789 4 v 1402 810 a Fu(r)1446 784 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)1674 810 y Fw(\()p Fu(r)1753 784 y Ft(s=\033)1888 810 y Fw(+)20 b(log)c Fu(r)s Fw(\))2226 744 y Fp(\020)2331 549 y Fr(\020)2386 650 y Fu(e)2428 617 y Fm(\000)p Ft(r)2517 593 y Fo(1)p FA(=\033)2625 549 y Fr(\021)2679 571 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)p 2331 723 576 4 v 2401 810 a Fu(r)2445 784 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)p Fq(+)p Ft(s=\033)2917 744 y Fu(:)-2556 b Fw(\(6.7\))486 982 y(The)30 b(`resonan)m(t)i(sets')g Fu(R)1339 997 y Fh(\014)r Ft(;)p Fg(q)1485 982 y Fw(are)g(de\014ned)e(as)h Fu(R)2133 997 y Fh(\014)r Ft(;)p Fg(q)2275 982 y Fw(=)26 b Fp(f)p Fu(X)34 b Fp(2)26 b Fu(I)2660 949 y Ft(nd)2770 982 y Fw(:)h FE(q)p Fu(X)h Fw(+)21 b Fi(\014)29 b Fw(=)e(0)p Fp(g)p Fw(.)386 1090 y(Let)k Fu(C)7 b Fw(\()p FE(q)p Fw(\))30 b(denote)h(a)g(collection)f(of)h(h)m(yp)s(ercub)s(es)d(of)j (sidelength)e Fu(L)p Fw(\()p Fu(C)7 b Fw(\))386 1215 y(=)25 b(2)p Fu(n)582 1182 y Fq(1)p Ft(=)p Fq(2)692 1215 y Fu(e)734 1182 y Fm(\000j)p Fg(q)p Fm(j)872 1158 y Fo(1)p FA(=\033)979 1215 y Fu(=)p Fp(j)p FE(q)p Fp(j)i Fw(cen)m(tred)g(on)f Fu(R)1664 1230 y Fh(\014)r Ft(;)p Fg(q)1805 1215 y Fw(with)e(cen)m (tres)j(at)g(a)f(distance)g(of)g Fu(e)2978 1182 y Fm(\000j)p Fg(q)p Fm(j)3116 1158 y Fo(1)p FA(=\033)3224 1215 y Fu(=)p Fp(j)p FE(q)p Fp(j)386 1379 y Fw(apart.)70 b(There)40 b(are)g Fp(\034)1260 1278 y Fr(\020)1314 1379 y Fp(j)p FE(q)p Fp(j)p Fu(e)1461 1346 y Fq(\()p Fm(j)p Fg(q)p Fm(j)1571 1322 y Fo(1)p FA(=\033)1676 1346 y Fq(\))1707 1278 y Fr(\021)1761 1301 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)2030 1379 y Fw(suc)m(h)g(h)m(yp)s(ercub)s(es.)68 b(Denote)41 b(b)m(y)f Fu(C)3307 1393 y Ft(N)386 1514 y Fw(the)c(set)f Fu(C)759 1528 y Ft(N)860 1514 y Fw(=)f Fp(f)p Fu(C)7 b Fw(\()p FE(q)p Fw(\))34 b(:)g Fp(j)p FE(q)p Fp(j)g Fu(>)g(N)10 b Fp(g)p Fw(.)56 b(F)-8 b(or)36 b(eac)m(h)g Fu(N)46 b Fw(this)34 b(set)i(is)e(a)i(co)m(v)m(er)h(of)f Fu(E)3171 1529 y Fh(\014)3224 1514 y Fw(\()p Fu(\033)s Fw(\).)386 1622 y(Therefore)474 1820 y Ff(L)535 1783 y Ft(s)572 1820 y Fw(\()p Fu(E)674 1835 y Fh(\014)727 1820 y Fw(\()p Fu(\033)s Fw(\)\))84 b Fp(\024)1135 1734 y Fr(X)1125 1931 y Ft(r)r(>N)1311 1734 y Fr(X)1292 1935 y Fm(j)p Fg(q)p Fm(j)p Fq(=)p Ft(r)1539 1734 y Fr(X)1478 1935 y Ft(C)5 b Fm(2)p Ft(C)g Fq(\()p Fg(q)p Fq(\))1748 1820 y Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(C)d Fw(\)\))25 b Fp(\034)2228 1734 y Fr(X)2218 1931 y Ft(r)r(>N)2405 1734 y Fr(X)2385 1935 y Fm(j)p Fg(q)p Fm(j)p Fq(=)p Ft(r)2571 1720 y Fr(\020)2625 1820 y Fp(j)p FE(q)p Fp(j)p Fu(e)2772 1783 y Fm(j)p Fg(q)p Fm(j)2855 1759 y Fo(1)p FA(=\033)2963 1720 y Fr(\021)3018 1742 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)971 2219 y Fp(\002)1135 2024 y Fr(\020)1189 2124 y Fu(e)1231 2091 y Fm(\000j)p Fg(q)p Fm(j)1369 2068 y Fo(1)p FA(=\033)1476 2024 y Fr(\021)1531 2046 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)p 1135 2198 625 4 v 1197 2285 a Fp(j)p FE(q)p Fp(j)1302 2259 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)p Fq(+)p Ft(s=\033)1794 2219 y Fp(\034)1921 2133 y Fr(X)1910 2330 y Ft(r)r(>N)2077 2219 y Fu(r)2121 2181 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Fm(\000)p Ft(s=\033)2504 2219 y Fu(<)g(")386 2481 y Fw(for)35 b Fu(s)f(>)f(n\033)38 b Fw(and)d Fu(N)45 b Fw(su\016cien)m(tly)35 b(large.)56 b(Therefore)35 b Ff(L)2368 2448 y Ft(s)2404 2481 y Fw(\()p Fu(E)2506 2496 y Fh(\014)2559 2481 y Fw(\()p Fu(\033)s Fw(\)\))g(=)f(0)i(for)f Fu(s)e(>)h(n\033)386 2589 y Fw(whic)m(h)29 b(implies)f(that)j(Ldim)23 b Fu(E)1452 2604 y Fh(\014)1505 2589 y Fw(\()p Fu(\033)s Fw(\))j Fp(\024)f Fu(n\033)s Fw(.)p 3309 2589 4 62 v 3313 2531 55 4 v 3313 2589 V 3367 2589 4 62 v 386 2778 a Fk(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)34 b(6.2.)43 b Fw(In)29 b(view)h(of)g(the)h(lemma)f(in)f(the)h(ab)s(o)m(v)m(e)i(w)m(e)e(obtain) g(the)h(lo-)386 2886 y(w)m(er)37 b(b)s(ound)d(for)j(Theorem)f(6.2.)61 b(T)-8 b(o)37 b(sk)m(etc)m(h)h(the)f(ideas:)53 b(\014rstly)-8 b(,)37 b(a)g(Can)m(tor)g(subset)386 2994 y Fs(K)49 b Fw(of)28 b Fu(E)653 3008 y Fq(0)692 2994 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g(d)p Fw(\))30 b(and)e(a)g(probabilit)m(y)e(measure)i (supp)s(orted)e(on)i Fs(K)49 b Fw(are)28 b(constructed.)386 3102 y(Then)k(the)g(Mass)i(Distribution)c(Principle)f(is)j(used)g(to)h (sho)m(w)g(that)g(the)g(logarithmic)386 3210 y(Hausdor\013)h(measure)g (of)h Fs(K)55 b Fw(is)33 b(in\014nit)m(y)g(for)h Fu(s)d(<)h(n\033)s Fw(.)53 b(This)32 b(implies)g(that)j(the)g(loga-)386 3318 y(rithmic)e(Hausdor\013)i(dimension)e(of)i Fs(K)56 b Fw(and)34 b(hence)i Fu(E)2284 3332 y Fq(0)2323 3318 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g(d)p Fw(\))37 b(is)d(greater)j (than)e(or)386 3426 y(equal)30 b(to)h Fu(n\033)s Fw(.)486 3534 y(F)-8 b(or)35 b(the)g(rest)f(of)h(the)g(pro)s(of)f(it)g(is)f (assumed)h(that)h Fu(s)d(<)g(n\033)s Fw(.)53 b(Also)34 b(note)h(that)g(the)386 3641 y(follo)m(wing)29 b(inequalit)m(y)f(will)g (con)m(tin)m(ually)h(b)s(e)h(used:)1234 3827 y Fu(N)1317 3794 y Ft(n=d)p 1147 3867 377 4 v 1147 3951 a Fw(16)p Fu(n)15 b Fw(log)i Fu(N)1559 3888 y Fp(\024)1654 3733 y Fr(")1782 3827 y Fu(N)1865 3794 y Ft(n=d)p 1717 3867 331 4 v 1717 3951 a Fw(8)p Fu(n)e Fw(log)i Fu(N)2058 3733 y Fr(#)2136 3888 y Fp(\024)2307 3827 y Fu(N)2390 3794 y Ft(n=d)p 2242 3867 V 2242 3951 a Fw(8)p Fu(n)e Fw(log)i Fu(N)2583 3888 y(;)-2222 b Fw(\(6.8\))386 4140 y(where)30 b([)g Fu(:)h Fw(])f(denotes)h(in)m(teger)g(part.)486 4248 y(First)21 b(w)m(e)h(recall)f(the)h(follo)m(wing)e(assertion)h (\(Lemma)h(2)g(in)f([14)q(]\))h(\(Almost)g(all)f(means)386 4356 y(for)30 b(all)f(but)h(a)h(set)g(of)f(measure)g(zero\).)386 4534 y FE(Lemma)j(6.6.)46 b Fk(F)-7 b(or)47 b(every)g Fu(N)60 b(>)50 b Fw(2)d Fk(and)g(almost)h(every)f Fu(X)57 b Fp(2)50 b Fu(I)2788 4501 y Ft(nd)2918 4534 y Fk(ther)-5 b(e)47 b(exists)386 4642 y FE(q)25 b Fp(2)g Fs(Z)617 4609 y Ft(n)680 4642 y Fp(n)c FE(0)32 b Fk(with)386 4755 y Fu(N)r(=)15 b Fw(log)i Fu(N)36 b(<)25 b Fp(j)p FE(q)p Fp(j)h(\024)f Fu(N)42 b Fk(such)33 b(that)h Fw(dist)24 b(\()p Fu(X)r(;)15 b(R)1987 4769 y Fg(q)2035 4755 y Fw(\))26 b Fu(<)f(nN)2330 4722 y Fm(\000)p Ft(n=d)2518 4755 y Fw(log)16 b Fu(N)10 b Fk(.)486 4931 y Fw(Similar)41 b(Can)m(tor)j (constructions)g(can)h(b)s(e)e(found)g(in)g([8)q(,)48 b(p)43 b(347])j(and)e([16)q(])h(\(for)386 5039 y(general)30 b(dimension)e(functions\).)40 b(Cho)s(ose)30 b Fu(N)1981 5053 y Fq(1)2051 5039 y Fw(large)g(enough)g(so)h(that)1707 5291 y Fu(e)1749 5253 y Ft(N)1812 5218 y Fo(1)p FA(=\033)1805 5274 y Fo(1)2004 5291 y Fu(>)2168 5229 y Fw(5)p Fu(N)2296 5181 y Ft(n=d)p Fm(\000)p Fq(1)2286 5255 y(1)p 2168 5270 338 4 v 2183 5353 a Fw(2)15 b(log)i Fu(N)2449 5367 y Fq(1)2515 5291 y Fu(;)-2154 b Fw(\(6.9\))1370 5504 y(15)p Fu(N)1543 5456 y Ft(n)p Fm(\000)p Ft(s=\033)1533 5530 y Fq(1)p 1216 5544 696 4 v 1216 5638 a Fw(5)p Fu(n)1316 5612 y Ft(d)1356 5638 y Fw(2)1401 5612 y Fq(4)p Ft(nd)p Fq(+4)1625 5638 y Fw(log)1743 5601 y Ft(d)1798 5638 y Fu(N)1871 5652 y Fq(1)2004 5565 y Fu(>)83 b Fw(1)p Fu(:)-1842 b Fw(\(6.10\))p eop %%Page: 21 21 21 20 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f (FIELDS)h(ON)g(TORI)559 b(21)1175 1729 y Fd(\005)1188 1646 y(\005)1202 1563 y(\005)1216 1480 y(\005)1230 1397 y(\005)1244 1314 y(\005)1258 1231 y(\005)1271 1148 y(\005)1285 1065 y(\005)1299 982 y(\005)1313 899 y(\005)1327 816 y(\005)1341 733 y(\005)1355 650 y(\005)1368 567 y(\005)-14 b(\005)p 926 487 748 7 v 926 1732 V 1673 1732 42 7 v 1756 1732 V 1839 1732 V 1673 487 V 1756 487 V 1839 487 V 1092 650 a Fu(C)1164 617 y Fm(0)p 922 1729 7 1246 v 1527 633 a Fc(H)1444 592 y(H)1361 550 y(H)-83 b(Y)1656 633 y Fu(R)1725 647 y Fg(q)p 1158 1704 50 50 v 1175 1604 V 1191 1505 V 1208 1405 V 1224 1306 V 1241 1206 V 1258 1106 V 1274 1007 V 1291 907 V 1307 807 V 1324 708 V 1341 608 V 2169 1106 4 623 v 2171 567 a Fc(6)p 2169 1729 V 2171 1729 a(?)p 1248 1580 4 25 v 1249 1638 a(6)p 1248 1604 V 1249 1604 a(?)1291 1580 y Fu(e)1333 1547 y Fm(\000)p Ft(N)1451 1511 y Fo(1)p FA(=\033)1444 1568 y Fo(1)2254 1106 y Fw(1)p Fu(=)p Fw(2)q Fu(L)p Fw(\()p Fu(N)2560 1120 y Fq(1)2600 1106 y Fw(\))486 2063 y(Divide)29 b Fu(I)822 2030 y Ft(nd)935 2063 y Fw(in)m(to)i Fu(T)13 b Fw(\()p Fu(N)1294 2077 y Fq(1)1333 2063 y Fw(\))26 b(=)f([)p Fu(N)1598 2015 y Ft(n=d)1588 2089 y Fq(1)1717 2063 y Fu(=)p Fw(8)p Fu(n)15 b Fw(log)i Fu(N)2083 2077 y Fq(1)2122 2063 y Fw(])2147 2030 y Ft(nd)2261 2063 y Fw(h)m(yp)s(ercub)s(es)29 b Fu(C)36 b Fw(with)1162 2267 y Fu(L)1224 2281 y Ft(N)1280 2290 y Fo(1)1344 2267 y Fw(=)25 b Fu(L)p Fw(\()p Fu(C)7 b Fw(\))25 b(=)g([)p Fu(N)1873 2220 y Ft(n=d)1863 2294 y Fq(1)1992 2267 y Fu(=)p Fw(8)p Fu(n)15 b Fw(log)r(\()p Fu(N)2378 2281 y Fq(1)2418 2267 y Fw(\)])2478 2230 y Fm(\000)p Fq(1)2573 2267 y Fu(:)386 2453 y Fw(Let)42 b Fu(C)632 2420 y Fm(0)697 2453 y Fw(b)s(e)f(a)h(h)m(yp)s(ercub)s(e)f(with)f(the)i(same)h(cen)m (tre)g(as)f Fu(C)48 b Fw(and)41 b Fu(L)p Fw(\()p Fu(C)2872 2420 y Fm(0)2895 2453 y Fw(\))k(=)3100 2417 y Fq(1)p 3100 2432 36 4 v 3100 2485 a(2)3145 2453 y Fu(L)p Fw(\()p Fu(C)7 b Fw(\).)386 2561 y(Let)39 b Fu(R)626 2575 y Ft(N)733 2561 y Fw(=)g Fp(f)p Fu(R)957 2575 y Fg(q)1044 2561 y Fw(:)g Fu(N)49 b Fp(\024)39 b(j)p FE(q)p Fp(j)h(\024)f Fw(2)p Fu(N)10 b Fp(g)40 b Fw(and)e(let)h Fu(G)p Fw(\()p Fu(N)2313 2575 y Fq(1)2352 2561 y Fw(\))g(denote)g(the)g(set)h(of)f (cub)s(es)386 2669 y Fu(C)48 b Fw(for)42 b(whic)m(h)f(there)h(exists)f (a)i(resonan)m(t)f(set)h Fu(R)2117 2683 y Fg(q)2208 2669 y Fp(2)i Fu(R)2383 2683 y Ft(N)2439 2692 y Fo(1)2519 2669 y Fw(suc)m(h)d(that)h Fu(C)3017 2636 y Fm(0)3067 2669 y Fp(\\)28 b Fs(R)3215 2683 y Ft(q)3304 2669 y Fp(\025)386 2732 y Fr(\000)455 2770 y Fq(1)p 438 2785 71 4 v 438 2837 a(10)518 2806 y Fu(L)p Fw(\()p Fu(C)7 b Fw(\))722 2732 y Fr(\001)764 2754 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)992 2806 y Fw(.)47 b(F)-8 b(rom)32 b(Lemma)h(2)g(in)d([14)r(])i (it)g(can)h(b)s(e)e(v)m(eri\014ed)h(that)h(for)f Fu(N)3111 2820 y Fq(1)3182 2806 y Fw(su\016-)386 2914 y(cien)m(tly)e(large)1227 3192 y(#G\()p Fu(N)1482 3206 y Fq(1)1522 3192 y Fw(\))c Fp(\025)1679 3036 y Fr(")1821 3130 y Fu(N)1904 3082 y Ft(n=d)1894 3156 y Fq(1)p 1742 3171 361 4 v 1742 3254 a Fw(8)p Fu(n)15 b Fw(log)i Fu(N)2063 3268 y Fq(1)2112 3036 y Fr(#)2165 3059 y Ft(nd)2264 3192 y Fw(\(1)k Fp(\000)e Fu(")p Fw(\))-2146 b(\(6.11\))386 3450 y(for)30 b(an)m(y)h Fu(")25 b(>)g Fw(0.)486 3558 y(F)-8 b(or)39 b(eac)m(h)h Fu(C)46 b Fp(2)38 b Fw(G\()p Fu(N)1259 3572 y Fq(1)1299 3558 y Fw(\))h(pic)m(k)g(one)g(suc)m(h)f(in)m(tersection)h(and)f(call)g (it)g Fu(I)7 b Fw(\()p Fu(C)g Fw(\).)66 b(No)m(w)386 3666 y(dra)m(w)37 b(further,)i(smaller)d(h)m(yp)s(ercub)s(es)g(cen)m (tred)j(on)e Fu(I)7 b Fw(\()p Fu(C)g Fw(\))38 b(and)f(con)m(tained)h (in)f Fu(C)3241 3633 y Fm(0)3301 3666 y Fw(of)386 3802 y(sidelength)i Fu(e)868 3769 y Fm(\000)p Ft(N)986 3734 y Fo(1)p FA(=\033)979 3790 y Fo(1)1094 3802 y Fu(=)-5 b(N)1207 3816 y Fq(1)1287 3802 y Fw(with)40 b(cen)m(tres)h(at)g(a)g (distance)f(of)h(2)p Fu(e)2593 3769 y Fm(\000)p Ft(N)2711 3734 y Fo(1)p FA(=\033)2704 3790 y Fo(1)2820 3802 y Fu(=)-5 b(N)2933 3816 y Fq(1)3013 3802 y Fw(apart,)43 b(so)386 3910 y(that)36 b(they)f(are)h(disjoin)m(t,)f(\(see)h(\014gure)e(for)h Fu(n)e Fw(=)g(2\).)56 b(Using)36 b(\(6.9\))h(it)d(can)i(b)s(e)e(sho)m (wn)386 4018 y(that)488 4126 y Fr(\022)587 4193 y Fw(1)p 564 4234 91 4 v 564 4317 a(40)665 4255 y Fu(N)738 4269 y Fq(1)778 4255 y Fu(L)p Fw(\()p Fu(C)7 b Fw(\))p Fu(e)1024 4217 y Ft(N)1087 4182 y Fo(1)p FA(=\033)1080 4238 y Fo(1)1195 4126 y Fr(\023)1262 4149 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)1574 4255 y Fp(\024)1727 4126 y Fr(\022)1827 4193 y Fw(1)p 1804 4234 V 1804 4317 a(20)1905 4255 y Fu(L)p Fw(\()p Fu(C)g Fw(\))p Fu(N)2182 4269 y Fq(1)2222 4255 y Fu(e)2264 4217 y Ft(N)2327 4182 y Fo(1)p FA(=\033)2320 4238 y Fo(1)2455 4255 y Fp(\000)20 b Fw(1)2591 4126 y Fr(\023)2659 4149 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)2912 4255 y Fp(\024)25 b Fw(#)p Fu(I)7 b Fw(\()p Fu(C)g Fw(\))1574 4510 y Fp(\024)1727 4409 y Fr(\020)1782 4510 y Fu(N)1855 4524 y Fq(1)1894 4510 y Fu(e)1936 4472 y Ft(N)1999 4437 y Fo(1)p FA(=\033)1992 4494 y Fo(1)2108 4510 y Fu(L)p Fw(\()p Fu(C)g Fw(\))2312 4409 y Fr(\021)2366 4426 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)2609 4510 y Fu(;)386 4718 y Fw(where)32 b(#)p Fu(I)7 b Fw(\()p Fu(C)g Fw(\))33 b(is)f(the)h(n)m(um)m(b)s(er)e(of)i(cub)s(es)f(lying)f (on)i(a)g(segmen)m(t)h Fu(I)7 b Fw(\()p Fu(C)g Fw(\).)49 b(This)31 b(is)g(the)386 4826 y(\014rst)39 b(lev)m(el)g Fu(K)876 4840 y Fq(1)955 4826 y Fw(of)h(the)f(Can)m(tor)h(set.)69 b(F)-8 b(rom)40 b(\(6.8\))r(,)h(\(6.11\))h(\(with)d Fu(")i Fw(=)f(1)p Fu(=)p Fw(16\))h(and)386 4934 y(ab)s(o)m(v)m(e)801 5194 y(#)p Fu(K)954 5208 y Fq(1)1077 5194 y Fp(\025)1240 5133 y Fw(15)p 1240 5173 V 1240 5257 a(16)1356 5039 y Fr(")1499 5133 y Fu(N)1582 5085 y Ft(n=d)1572 5159 y Fq(1)p 1419 5173 361 4 v 1419 5257 a Fw(8)p Fu(n)15 b Fw(log)i Fu(N)1740 5271 y Fq(1)1790 5039 y Fr(#)1843 5061 y Ft(nd)1941 5066 y Fr(\022)2041 5133 y Fw(1)p 2018 5173 91 4 v 2018 5257 a(40)2119 5194 y Fu(N)2192 5208 y Fq(1)2231 5194 y Fu(L)p Fw(\()p Fu(C)7 b Fw(\))p Fu(e)2477 5157 y Ft(N)2540 5121 y Fo(1)p FA(=\033)2533 5178 y Fo(1)2649 5066 y Fr(\023)2716 5089 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)1077 5567 y Fp(\025)1287 5472 y Fw(15)p Fu(N)1460 5434 y Ft(nd)p Fm(\000)p Ft(d)p Fq(+)p Ft(n)1450 5498 y Fq(1)1749 5372 y Fr(\020)1803 5472 y Fu(e)1845 5439 y Ft(N)1908 5404 y Fo(1)p FA(=\033)1901 5461 y Fo(1)2016 5372 y Fr(\021)2071 5393 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)p 1240 5546 1106 4 v 1240 5633 a Fw(2)1285 5607 y Ft(nd)p Fq(+4)1459 5633 y Fw(\(8)p Fu(n)p Fw(\))1629 5607 y Ft(d)1671 5633 y Fw(40)1761 5607 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)1990 5633 y Fw(\(log)17 b Fu(N)2231 5647 y Fq(1)2270 5633 y Fw(\))2305 5607 y Ft(d)p eop %%Page: 22 22 22 21 bop 386 239 a Fq(22)693 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g (YOSHINO)386 456 y Fw(Let)i Fu(E)617 470 y Fq(1)688 456 y Fw(denote)g(a)g(h)m(yp)s(ercub)s(e)e(in)g Fu(K)1684 470 y Fq(1)1724 456 y Fw(,)i(so)g(that)g Fu(L)p Fw(\()p Fu(E)2256 470 y Fq(1)2295 456 y Fw(\))c(=)f Fu(e)2498 423 y Fm(\000)p Ft(N)2616 388 y Fo(1)p FA(=\033)2609 445 y Fo(1)2724 456 y Fu(=)-5 b(N)2837 470 y Fq(1)2877 456 y Fw(,)32 b(then)f(\(6.10\))386 564 y(yields)1088 790 y(#)p Fu(K)1241 804 y Fq(1)1280 790 y Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(E)1534 804 y Fq(1)1574 790 y Fw(\)\))26 b Fp(\025)1930 728 y Fw(15)p Fu(N)2103 680 y Ft(n)p Fm(\000)p Ft(s=\033)2093 754 y Fq(1)p 1776 769 696 4 v 1776 863 a Fw(5)p Fu(n)1876 836 y Ft(d)1916 863 y Fw(2)1961 836 y Fq(4)p Ft(nd)p Fq(+4)2185 863 y Fw(log)2303 826 y Ft(d)2358 863 y Fu(N)2431 877 y Fq(1)2506 790 y Fu(>)f Fw(1)p Fu(:)486 992 y Fw(T)-8 b(o)35 b(get)h(the)f(second)f(lev)m(el)h Fu(K)1527 1006 y Fq(2)1601 992 y Fw(precisely)e(the)i(same)g (construction)g(is)f(made)g(but)386 1100 y(instead)i(of)h Fu(I)863 1067 y Ft(nd)982 1100 y Fw(the)g(cub)s(es)f(from)g(the)h (previous)e(lev)m(el)h Fu(K)2444 1114 y Fq(1)2520 1100 y Fw(are)h(used.)59 b(Cho)s(ose)36 b Fu(N)3335 1114 y Fq(2)386 1208 y Fw(large)30 b(enough)h(so)f(that)1599 1371 y Fu(N)1682 1323 y Ft(n=d)1672 1397 y Fq(2)p 1497 1412 406 4 v 1497 1495 a Fw(16)p Fu(n)15 b Fw(log)j Fu(N)1864 1509 y Fq(2)1996 1433 y Fu(>)83 b(N)2223 1447 y Fq(1)2262 1433 y Fu(e)2304 1395 y Ft(N)2367 1360 y Fo(1)p FA(=\033)2360 1416 y Fo(1)2476 1433 y Fu(;)-2115 b Fw(\(6.12\))1352 1654 y(3)p Fu(N)1480 1606 y Ft(n)p Fm(\000)p Ft(s=\033)1470 1680 y Fq(2)p 1142 1694 761 4 v 1142 1788 a Fw(5)p Fu(n)1242 1762 y Ft(d)1303 1788 y Fp(\001)20 b Fw(2)1393 1762 y Fq(5)p Ft(nd)p Fq(+2)1618 1788 y Fw(log)1735 1751 y Ft(d)1791 1788 y Fu(N)1864 1802 y Fq(2)1996 1715 y Fu(>)2160 1654 y(e)2202 1621 y Ft(ndN)2344 1585 y Fo(1)p FA(=\033)2337 1642 y Fo(1)2452 1654 y Fu(N)2535 1621 y Ft(nd)2525 1678 y Fq(1)p 2160 1694 459 4 v 2293 1778 a Fw(#)p Fu(K)2446 1792 y Fq(1)386 1715 y Fw(\(6.13\))386 1917 y(and)30 b Fu(")25 b(>)g Fw(0)31 b(suc)m(h)f(that)1454 2079 y Fu(")25 b Fp(\024)g Fw(2)1662 2042 y Fm(\000)p Fq(1)1757 2079 y Fu(e)1799 2042 y Fm(\000)p Fq(2)p Ft(N)1952 2006 y Fo(1)p FA(=\033)1945 2063 y Fo(1)2061 2079 y Fu(N)2144 2041 y Fm(\000)p Ft(nd)2134 2105 y Fq(1)2282 2079 y Fu(:)-1921 b Fw(\(6.14\))386 2243 y(As)30 b(b)s(efore)g(divide)e Fu(I)1110 2210 y Ft(nd)1224 2243 y Fw(in)m(to)i Fu(T)13 b Fw(\()p Fu(N)1582 2257 y Fq(2)1622 2243 y Fw(\))31 b(h)m(yp)s(ercub)s(es)d(of)j(sidelength)e Fu(L)2759 2257 y Ft(N)2815 2266 y Fo(2)386 2373 y Fw(=)c([)p Fu(N)590 2325 y Ft(n=d)580 2399 y Fq(2)709 2373 y Fu(=)p Fw(8)p Fu(n)15 b Fw(log)i Fu(N)1075 2387 y Fq(2)1114 2373 y Fw(])1139 2340 y Fm(\000)p Fq(1)1234 2373 y Fw(.)41 b(Since)29 b Fu(N)1610 2387 y Fq(2)1675 2373 y Fu(>)c(N)1844 2387 y Fq(1)1913 2373 y Fw(from)30 b(\(6.8\))688 2626 y(#G\()p Fu(N)943 2640 y Fq(2)983 2626 y Fw(\))25 b Fp(\025)1139 2471 y Fr(")1282 2564 y Fu(N)1365 2517 y Ft(n=d)1355 2591 y Fq(2)p 1202 2605 361 4 v 1202 2688 a Fw(8)p Fu(n)15 b Fw(log)i Fu(N)1523 2702 y Fq(2)1573 2471 y Fr(#)1626 2493 y Ft(nd)1724 2626 y Fw(\(1)k Fp(\000)f Fu(")p Fw(\))26 b Fu(>)2364 2564 y(N)2447 2531 y Ft(n)2490 2508 y Fo(2)2437 2589 y Fq(2)p 2125 2605 643 4 v 2125 2699 a Fw(\(16)p Fu(n)p Fw(\))2340 2673 y Ft(nd)2440 2699 y Fw(log)2557 2662 y Ft(nd)2655 2699 y Fu(N)2728 2713 y Fq(2)2778 2626 y Fw(\(1)21 b Fp(\000)f Fu(")p Fw(\))p Fu(:)386 2856 y Fw(Let)31 b Fu(T)602 2870 y Ft(E)654 2879 y Fo(1)692 2856 y Fw(\()p Fu(N)800 2870 y Fq(2)840 2856 y Fw(\))g(denote)g(the)f (n)m(um)m(b)s(er)f(of)i(h)m(yp)s(ercub)s(es)d(con)m(tained)j(in)e Fu(E)2850 2870 y Fq(1)2920 2856 y Fw(and)386 2964 y(#G)533 2978 y Ft(E)585 2987 y Fo(1)623 2964 y Fw(\()p Fu(N)731 2978 y Fq(2)771 2964 y Fw(\))g(the)f(n)m(um)m(b)s(er)f(of)h(those)h (with)e(large)h(in)m(tersection.)40 b(It)28 b(is)f(readily)g(v)m (eri\014ed)386 3072 y(that)903 3152 y Fr( )985 3246 y Fu(e)1027 3213 y Fm(\000)p Ft(N)1145 3177 y Fo(1)p FA(=\033)1138 3234 y Fo(1)p 985 3286 270 4 v 985 3370 a Fu(N)1058 3384 y Fq(1)1097 3370 y Fu(L)1159 3384 y Ft(N)1215 3393 y Fo(2)1284 3307 y Fp(\000)20 b Fw(1)1420 3152 y Fr(!)1492 3166 y Ft(nd)1601 3307 y Fp(\024)25 b Fu(T)1750 3321 y Ft(E)1802 3330 y Fo(1)1840 3307 y Fw(\()p Fu(N)1948 3321 y Fq(2)1988 3307 y Fw(\))h Fp(\024)2145 3152 y Fr( )2227 3246 y Fu(e)2269 3213 y Fm(\000)p Ft(N)2387 3177 y Fo(1)p FA(=\033)2380 3234 y Fo(1)p 2227 3286 V 2227 3370 a Fu(N)2300 3384 y Fq(1)2339 3370 y Fu(L)2401 3384 y Ft(N)2457 3393 y Fo(2)2526 3307 y Fw(+)20 b(2)2662 3152 y Fr(!)2734 3166 y Ft(nd)2833 3307 y Fu(;)386 3537 y Fw(and)30 b(\(6.12\))i (implies)c(that)1014 3640 y Fr( )1119 3734 y Fu(e)1161 3701 y Fm(\000)p Ft(N)1279 3666 y Fo(1)p FA(=\033)1272 3722 y Fo(1)p 1096 3774 315 4 v 1096 3858 a Fw(2)p Fu(N)1214 3872 y Fq(1)1254 3858 y Fu(L)1316 3872 y Ft(N)1372 3881 y Fo(2)1421 3640 y Fr(!)1493 3654 y Ft(nd)1601 3795 y Fp(\024)d Fu(T)1750 3809 y Ft(E)1802 3818 y Fo(1)1841 3795 y Fw(\()p Fu(N)1949 3809 y Fq(2)1989 3795 y Fw(\))g Fp(\024)2145 3640 y Fr( )2227 3734 y Fw(2)p Fu(e)2314 3701 y Fm(\000)p Ft(N)2432 3666 y Fo(1)p FA(=\033)2425 3722 y Fo(1)p 2227 3774 314 4 v 2249 3858 a Fu(N)2322 3872 y Fq(1)2362 3858 y Fu(L)2424 3872 y Ft(N)2480 3881 y Fo(2)2551 3640 y Fr(!)2623 3654 y Ft(nd)2721 3795 y Fu(:)386 4025 y Fw(Equations)32 b(\(6.14\))j(and)d(\(6.11\))k(can)d(b)s (e)f(used)g(to)i(sho)m(w)f(that)g(#G\()p Fu(N)2828 4039 y Fq(2)2868 4025 y Fw(\))d Fp(\025)f Fu(T)13 b Fw(\()p Fu(N)3207 4039 y Fq(2)3247 4025 y Fw(\))22 b Fp(\000)386 4133 y Fu(T)439 4147 y Ft(E)491 4156 y Fo(1)530 4133 y Fw(\()p Fu(N)638 4147 y Fq(2)677 4133 y Fw(\))p Fu(=)p Fw(4)32 b(whence)737 4372 y(G)808 4386 y Ft(E)860 4395 y Fo(1)899 4372 y Fw(\()p Fu(N)1007 4386 y Fq(2)1047 4372 y Fw(\))25 b Fu(>)1213 4311 y Fw(3)p Fu(T)1311 4325 y Ft(E)1363 4334 y Fo(1)1402 4311 y Fw(\()p Fu(N)1510 4325 y Fq(2)1550 4311 y Fw(\))p 1213 4351 373 4 v 1377 4435 a(4)1621 4372 y Fu(>)2161 4311 y Fw(3)p Fu(e)2248 4278 y Fm(\000)p Ft(ndN)2445 4242 y Fo(1)p FA(=\033)2438 4299 y Fo(1)p 1727 4351 1262 4 v 1727 4451 a Fw(4)20 b Fp(\001)h Fw(2)1883 4425 y Ft(nd)1966 4451 y Fw(\(16)p Fu(n)p Fw(\))2181 4425 y Ft(nd)2266 4451 y Fu(N)2349 4420 y Ft(nd)2339 4477 y Fq(1)2432 4451 y Fu(N)2515 4413 y Fm(\000)p Ft(d)2606 4394 y Fo(2)2505 4477 y Fq(2)2660 4451 y Fw(log)2778 4415 y Ft(nd)2876 4451 y Fu(N)2949 4465 y Fq(2)2998 4372 y Fu(:)386 4594 y Fw(Do)26 b(exactly)f(the)h (same)f(as)g(b)s(efore)g(with)f(the)h(segmen)m(ts)h Fu(I)7 b Fw(\()p Fu(C)g Fw(\))25 b(to)h(construct)f Fu(K)3107 4608 y Fq(2)3147 4594 y Fw(,)h(then)386 4702 y(w)m(e)33 b(ha)m(v)m(e)h(#)p Fu(K)887 4716 y Fq(2)955 4702 y Fw(=)28 b(#)p Fu(K)1207 4716 y Fq(1)1247 4702 y Fw(#)p Fu(H)1399 4716 y Fq(2)1437 4702 y Fw(,)34 b(where)e(#)p Fu(H)1913 4716 y Fq(2)1984 4702 y Fw(is)f(the)i(n)m(um)m(b)s(er)e(of)i(h)m(yp)s (ercub)s(es)d(of)j Fu(K)3335 4716 y Fq(2)386 4810 y Fw(in)c(one)i Fu(E)726 4824 y Fq(1)795 4810 y Fw(with)445 4982 y(#G)592 4996 y Ft(E)644 5005 y Fo(1)682 4982 y Fw(\()p Fu(N)790 4996 y Fq(2)830 4982 y Fw(\)\()p Fu(L)962 4996 y Ft(N)1018 5005 y Fo(2)1057 4982 y Fu(N)1130 4996 y Fq(2)1170 4982 y Fu(e)1212 4944 y Ft(N)1275 4909 y Fo(1)p FA(=\033)1268 4966 y Fo(2)1383 4982 y Fu(=)p Fw(40\))1553 4944 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)1866 4982 y Fp(\024)83 b Fw(#)p Fu(H)2172 4996 y Fq(2)1866 5174 y Fp(\024)g Fw(#G)2167 5188 y Ft(E)2219 5197 y Fo(1)2258 5174 y Fw(\()p Fu(N)2366 5188 y Fq(2)2406 5174 y Fw(\))2456 5073 y Fr(\020)2510 5174 y Fu(L)2572 5188 y Ft(N)2628 5197 y Fo(2)2667 5174 y Fu(N)2740 5188 y Fq(2)2779 5174 y Fu(e)2821 5137 y Ft(N)2884 5101 y Fo(1)p FA(=\033)2877 5158 y Fo(2)2993 5073 y Fr(\021)3047 5090 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)3291 5174 y Fu(:)386 5345 y Fw(Th)m(us)832 5560 y(#)p Fu(K)985 5574 y Fq(2)1049 5560 y Fp(\025)25 b Fw(#)p Fu(K)1298 5574 y Fq(1)1518 5499 y Fw(3)p Fu(N)1646 5460 y Ft(nd)p Fm(\000)p Ft(d)p Fq(+)p Ft(n)1636 5525 y Fq(2)1918 5499 y Fu(e)1960 5466 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(dN)2246 5430 y Fo(1)p FA(=\033)2239 5487 y Fo(2)p 1348 5539 1179 4 v 1348 5633 a Fw(4)20 b Fp(\001)h Fw(2)1504 5607 y Fq(2)p Ft(nd)1623 5633 y Fw(\(8)p Fu(n)p Fw(\))1793 5607 y Ft(n)1861 5633 y Fp(\001)f Fw(40)1996 5607 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)2241 5633 y Fw(log)2358 5596 y Ft(d)2414 5633 y Fu(N)2487 5647 y Fq(2)2571 5499 y Fu(e)2613 5466 y Fm(\000)p Ft(ndN)2810 5430 y Fo(1)p FA(=\033)2803 5487 y Fo(1)p 2571 5539 348 4 v 2662 5628 a Fu(N)2745 5596 y Ft(nd)2735 5654 y Fq(1)p eop %%Page: 23 23 23 22 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f (FIELDS)h(ON)g(TORI)559 b(23)386 434 y Fw(and)750 653 y(#)p Fu(K)903 667 y Fq(2)942 653 y Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(E)1196 667 y Fq(2)1236 653 y Fw(\)\))26 b(=)1648 592 y(3)p Fu(N)1776 544 y Ft(n)p Fm(\000)p Ft(s=\033)1766 618 y Fq(2)p 1438 632 761 4 v 1438 726 a Fw(5)p Fu(n)1538 700 y Ft(d)1598 726 y Fp(\001)21 b Fw(2)1689 700 y Fq(5)p Ft(nd)p Fq(+2)1913 726 y Fw(log)2030 689 y Ft(d)2086 726 y Fu(N)2159 740 y Fq(2)2244 592 y Fu(e)2286 559 y Fm(\000)p Ft(ndN)2483 523 y Fo(1)p FA(=\033)2476 580 y Fo(1)p 2244 632 348 4 v 2334 720 a Fu(N)2417 689 y Ft(nd)2407 747 y Fq(1)2626 653 y Fw(#)p Fu(K)2779 667 y Fq(1)2844 653 y Fu(>)k Fw(1)p Fu(;)386 867 y Fw(from)30 b(\(6.13\))r(.)486 975 y(Similarly)23 b(for)k(the)g Fu(r)s Fw('th)g(lev)m(el.)39 b(Assume)26 b(that)i Fu(K)2185 989 y Ft(r)r Fm(\000)p Fq(1)2340 975 y Fw(has)f(b)s(een)g(constructed)g (with)386 1114 y(cub)s(es)i(of)i(sidelength)e Fu(e)1212 1076 y Fm(\000)p Ft(N)1330 1041 y Fo(1)p FA(=\033)1323 1097 y(r)r Fx(\000)p Fo(1)1440 1114 y Fu(=)-5 b(N)1553 1128 y Ft(r)r Fm(\000)p Fq(1)1682 1114 y Fw(.)40 b(Cho)s(ose)31 b Fu(N)2137 1128 y Ft(r)2205 1114 y Fw(su\016cien)m(tly)e(large)h(so)h (that)1581 1287 y Fu(N)1664 1239 y Ft(n=d)1654 1298 y(r)p 1480 1327 405 4 v 1480 1411 a Fw(16)p Fu(n)15 b Fw(log)i Fu(N)1846 1425 y Ft(r)1977 1348 y Fu(>)83 b(N)2204 1362 y Ft(r)2242 1348 y Fu(e)2284 1310 y Ft(N)2347 1275 y Fo(1)p FA(=\033)2340 1331 y(r)r Fx(\000)p Fo(1)2457 1348 y Fu(;)-2096 b Fw(\(6.15\))1334 1579 y(3)p Fu(N)1462 1531 y Ft(n)p Fm(\000)p Ft(s=\033)1452 1591 y(r)p 1124 1620 760 4 v 1124 1714 a Fw(5)p Fu(n)1224 1687 y Ft(d)1285 1714 y Fp(\001)20 b Fw(2)1375 1687 y Fq(5)p Ft(nd)p Fq(+2)1600 1714 y Fw(log)1717 1677 y Ft(d)1773 1714 y Fu(N)1846 1728 y Ft(r)1977 1641 y Fu(>)2140 1574 y(e)2182 1536 y Ft(ndN)2324 1501 y Fo(1)p FA(=\033)2317 1557 y(r)r Fx(\000)p Fo(1)2435 1574 y Fu(N)2518 1541 y Ft(nd)2508 1598 y(r)r Fm(\000)p Fq(1)p 2140 1620 496 4 v 2248 1703 a Fw(#)p Fu(K)2401 1717 y Ft(r)r Fm(\000)p Fq(1)386 1641 y Fw(\(6.16\))386 1847 y(and)30 b Fu(")25 b(>)g Fw(0)31 b(so)g(that)1465 2013 y Fu(")26 b Fp(\024)f Fw(2)1674 1975 y Fm(\000)p Fq(1)1769 2013 y Fu(e)1811 1975 y Fm(\000)p Fq(2)p Ft(N)1964 1940 y Fo(1)p FA(=\033)1957 1996 y(r)r Fx(\000)p Fo(1)2074 2013 y Fu(N)2157 1975 y Fm(\000)p Ft(nd)2147 2039 y(r)r Fm(\000)p Fq(1)386 2013 y Fw(\(6.17\))486 2170 y(Let)35 b Fu(T)706 2184 y Ft(E)758 2193 y FA(r)r Fx(\000)p Fo(1)875 2170 y Fw(\()p Fu(N)983 2184 y Ft(r)1021 2170 y Fw(\))g(denote)g(the)g(n)m(um)m(b)s(er)f(of)g(h)m(yp)s(ercub)s (es)f(con)m(tained)i(in)e Fu(E)3065 2184 y Ft(r)r Fm(\000)p Fq(1)3228 2170 y Fw(and)386 2277 y(#G)533 2291 y Ft(E)585 2300 y FA(r)r Fx(\000)p Fo(1)702 2277 y Fw(\()p Fu(N)810 2291 y Ft(r)848 2277 y Fw(\))c(the)g(n)m(um)m(b)s(er)e(whic)m(h)h(ha)m (v)m(e)i(large)e(in)m(tersection.)40 b(F)-8 b(ollo)m(wing)29 b(the)f(argu-)386 2385 y(men)m(t)35 b(exactly)h(as)f(for)g(the)g (second)g(lev)m(el)g(but)f(using)f(equations)i(\(6.15\))i(and)d (\(6.17\))386 2493 y(rather)c(than)g(equations)g(\(6.12\))j(and)d (\(6.14\))i(w)m(e)f(get)920 2740 y(#)p Fu(K)1073 2754 y Ft(r)1136 2740 y Fp(\025)25 b Fw(#)p Fu(K)1385 2754 y Ft(r)r Fm(\000)p Fq(1)1656 2679 y Fw(3)p Fu(N)1784 2646 y Ft(nd)p Fm(\000)p Ft(d)p Fq(+)p Ft(n)1774 2701 y(r)2057 2679 y Fu(e)2099 2646 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(dN)2385 2610 y Fo(1)p FA(=\033)2378 2656 y(r)p 1523 2719 1105 4 v 1523 2813 a Fw(4)c Fp(\001)f Fw(2)1679 2787 y Fq(2)p Ft(nd)1798 2813 y Fw(\(8)p Fu(n)p Fw(\))1968 2787 y Ft(d)2009 2813 y Fw(40)2099 2787 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)2344 2813 y Fw(log)2461 2776 y Ft(d)2517 2813 y Fu(N)2590 2827 y Ft(r)2673 2679 y Fu(e)2715 2641 y Fm(\000)p Ft(ndN)2912 2606 y Fo(1)p FA(=\033)2905 2662 y(r)r Fx(\000)p Fo(1)p 2673 2719 350 4 v 2747 2808 a Fu(N)2830 2776 y Ft(nd)2820 2834 y(r)r Fm(\000)p Fq(1)3033 2740 y Fu(;)-2672 b Fw(\(6.18\))386 2956 y(and)707 3180 y(#)p Fu(K)860 3194 y Ft(r)898 3180 y Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(E)1152 3194 y Ft(r)1190 3180 y Fw(\)\))26 b(=)1601 3118 y(3)p Fu(N)1729 3071 y Ft(n)p Fm(\000)p Ft(s=\033)1719 3144 y Fq(2)p 1391 3159 760 4 v 1391 3253 a Fw(5)p Fu(n)1491 3227 y Ft(d)1552 3253 y Fp(\001)21 b Fw(2)1643 3227 y Fq(5)p Ft(nd)p Fq(+2)1867 3253 y Fw(log)1984 3216 y Ft(d)2040 3253 y Fu(N)2113 3267 y Ft(r)2196 3118 y Fu(e)2238 3081 y Fm(\000)p Ft(ndN)2435 3045 y Fo(1)p FA(=\033)2428 3102 y(r)r Fx(\000)p Fo(1)p 2196 3159 350 4 v 2270 3247 a Fu(N)2353 3216 y Ft(nd)2343 3273 y(r)r Fm(\000)p Fq(1)2581 3180 y Fw(#)p Fu(K)2734 3194 y Ft(r)r Fm(\000)p Fq(1)2887 3180 y Fu(>)k Fw(1)p Fu(;)386 3401 y Fw(from)30 b(\(6.16\))r(.)486 3508 y(Let)37 b Fs(K)56 b Fw(=)36 b Fp(\\)929 3476 y Fm(1)929 3533 y Ft(r)r Fq(=1)1056 3508 y Fu(K)1133 3522 y Ft(r)1172 3508 y Fw(.)59 b(It)37 b(is)f(easy)h(to)g(sho)m(w)g(that)g Fs(K)57 b Fp(\032)35 b Fu(E)2497 3522 y Fq(0)2537 3508 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(n;)g(d)p Fw(\).)61 b(In)36 b(order)g(to)386 3616 y(construct)31 b(a)h(measure)f(on)g Fs(K)52 b Fw(w)m(e)31 b(need)g(the)h(follo)m(wing)d(lemma)i(whic)m(h)f (follo)m(ws)g(from)386 3724 y([19)q(,)h(page)g(55].)386 3890 y FE(Lemma)i(6.7)i Fw(\(The)30 b(Mass)h(distribution)c (principle\))p FE(.)42 b Fk(L)-5 b(et)38 b Fu(\026)g Fk(b)-5 b(e)38 b(a)g(pr)-5 b(ob)g(ability)40 b(me-)386 4000 y(asur)-5 b(e)31 b(supp)-5 b(orte)g(d)33 b(on)d(a)h(subset)f Fu(F)43 b Fk(of)30 b Fs(R)1763 3967 y Ft(k)1811 4000 y Fk(.)41 b(Supp)-5 b(ose)32 b(ther)-5 b(e)31 b(ar)-5 b(e)31 b(p)-5 b(ositive)31 b(c)-5 b(onstants)32 b Fu(\025)386 4108 y Fk(and)c Fu(\016)j Fk(such)d(that)g Fu(\026)p Fw(\()p Fu(C)7 b Fw(\))25 b Fp(\024)g Fu(\025f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(C)d Fw(\)\))p Fk(,)28 b(for)g(any)g(cub)-5 b(e)27 b Fu(C)33 b Fk(with)c(sidelength)f Fu(L)p Fw(\()p Fu(C)7 b Fw(\))25 b Fp(\024)g Fu(\016)s Fk(.)386 4216 y(Then)33 b(we)g(have)g Ff(L)1028 4183 y Ft(s)1064 4216 y Fw(\()p Fu(F)13 b Fw(\))26 b Fp(\025)f Fu(\025)1380 4183 y Fm(\000)p Fq(1)1474 4216 y Fk(.)486 4381 y Fw(Recall)31 b(that)i Fu(E)1027 4395 y Ft(i)1087 4381 y Fw(represen)m(ts)f(a)g(cub)s (e)f(in)g Fu(K)1995 4395 y Ft(i)2023 4381 y Fw(,)i(that)f(is)f(in)g (the)h Fu(i)p Fw('th)g(lev)m(el)g(of)g Fs(K)52 b Fw(and)386 4489 y(de\014ne)25 b(a)h(probabilit)m(y)d(measure)i(on)g Fs(K)47 b Fw(in)24 b(the)h(follo)m(wing)f(w)m(a)m(y:)40 b Fu(\026)p Fw(\()p Fu(E)2748 4503 y Ft(i)2776 4489 y Fw(\))26 b(=)f(\(#)p Fu(K)3121 4503 y Ft(i)3149 4489 y Fw(\))3184 4456 y Fm(\000)p Fq(1)3304 4489 y Fu(<)386 4597 y(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(E)640 4611 y Ft(i)668 4597 y Fw(\)\),)41 b Fu(i)e Fp(2)f Fs(N)7 b Fw(.)70 b(Th)m(us)37 b(the)i(sum)e(of)i(the)f(measures)g(of)h(all)e(h)m (yp)s(ercub)s(es)f(in)h(the)386 4705 y Fu(r)s Fw('th)27 b(lev)m(el)g(under)e(one)j(particular)d(b)s(o)m(x)i(on)g(the)h Fu(r)16 b Fp(\000)e Fw(1'th)27 b(lev)m(el)g(adds)g(up)f(to)i Fu(\026)p Fw(\()p Fu(E)3186 4719 y Ft(r)r Fm(\000)p Fq(1)3314 4705 y Fw(\),)386 4813 y(i.e.)44 b Fu(\026)p Fw(\()p Fu(E)703 4827 y Ft(r)741 4813 y Fw(\))28 b(=)f Fu(\026)p Fw(\()p Fu(E)1059 4827 y Ft(r)r Fm(\000)p Fq(1)1187 4813 y Fw(\))p Fu(=)p Fw(\(#)p Fu(H)1454 4827 y Ft(r)1492 4813 y Fw(\).)45 b(T)-8 b(o)32 b(use)f(Lemma)h(6.7)g(w)m(e)g(need)g(to) g(sho)m(w)f(that)h(the)386 4921 y(measure)d(of)g(an)g(arbitrary)f(cub)s (e)g Fu(B)33 b Fw(is)28 b Fp(\034)d Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(B)5 b Fw(\)\))29 b(where)g Fu(L)p Fw(\()p Fu(B)5 b Fw(\))29 b(is)f(the)h(sidelength)386 5029 y(of)45 b(the)h(cub)s(e.)84 b(This)43 b(ob)m(viously)h(holds)g(for)h(an)m(y)g (cub)s(e)g(in)e Fs(K)15 b Fw(.)91 b(Without)45 b(loss)f(of)386 5136 y(generalit)m(y)30 b(w)m(e)g(can)g(c)m(ho)s(ose)h(an)f(arbitrary)e (cub)s(e)h Fu(B)35 b Fw(con)m(tained)30 b(in)e(some)i(h)m(yp)s(ercub)s (e)386 5244 y Fu(E)453 5258 y Ft(r)r Fm(\000)p Fq(1)621 5244 y Fw(and)39 b(con)m(taining)g(at)h(least)g(t)m(w)m(o)h(cub)s(es)e (of)h(the)g Fu(r)s Fw('th)f(lev)m(el)g(so)h(that)g Fu(L)p Fw(\()p Fu(B)5 b Fw(\))41 b Fp(\024)386 5386 y Fu(e)428 5348 y Fm(\000)p Ft(N)546 5313 y Fo(1)p FA(=\033)539 5369 y(r)r Fx(\000)p Fo(1)656 5386 y Fu(=)-5 b(N)769 5400 y Ft(r)r Fm(\000)p Fq(1)898 5386 y Fw(.)38 b(There)22 b(are)h(t)m(w)m(o)h(cases)f(to)g(consider.)37 b(The)22 b(\014rst)g(is)g(when)f Fu(B)27 b Fw(in)m(tersects)386 5494 y(h)m(yp)s(ercub)s(es)34 b(lying)g(on)h(at)i(least)e(t)m(w)m(o)i (di\013eren)m(t)f(segmen)m(ts)g Fu(I)7 b Fw(\()p Fu(C)g Fw(\))36 b(of)g(resonan)m(t)g(sets.)386 5623 y(In)i(this)f(case)j Fu(L)p Fw(\()p Fu(B)5 b Fw(\))39 b Fp(\025)g Fu(L)1311 5637 y Ft(N)1367 5645 y FA(r)1444 5623 y Fw(=)g([)p Fu(N)1662 5575 y Ft(n=d)1652 5635 y(r)1781 5623 y Fu(=)p Fw(8)p Fu(n)15 b Fw(log)i Fu(N)2147 5637 y Ft(r)2185 5623 y Fw(])2210 5590 y Fm(\000)p Fq(1)2304 5623 y Fw(.)66 b(Let)39 b Fu(S)k Fw(b)s(e)38 b(the)h(n)m(um)m(b)s(er)e(of)p eop %%Page: 24 24 24 23 bop 386 239 a Fq(24)693 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g (YOSHINO)386 434 y Fw(segmen)m(ts)37 b Fu(I)7 b Fw(\()p Fu(C)g Fw(\))37 b(that)f Fu(B)41 b Fw(in)m(tersects.)59 b(Then)35 b(w)m(e)i(ha)m(v)m(e)g Fu(S)j Fp(\024)35 b Fw(\()p Fu(L)p Fw(\()p Fu(B)5 b Fw(\))p Fu(=L)2921 448 y Ft(N)2977 456 y FA(r)3016 434 y Fw(\))3051 401 y Ft(nd)3135 434 y Fw(.)58 b(The)386 542 y(maxim)m(um)22 b(n)m(um)m(b)s(er)h(of)h(h) m(yp)s(ercub)s(es)d(of)j Fu(K)1867 556 y Ft(r)1929 542 y Fw(that)g Fu(B)k Fw(can)c(in)m(tersect)g(on)g(eac)m(h)g(segmen)m(t) 386 697 y(is)477 596 y Fr(\020)532 697 y Fu(L)594 711 y Ft(N)650 719 y FA(r)688 697 y Fu(N)761 711 y Ft(r)799 697 y Fu(e)841 664 y Fq(\()p Ft(N)931 629 y Fo(1)p FA(=\033)924 674 y(r)1036 664 y Fq(\))1068 596 y Fr(\021)1122 618 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)1350 697 y Fw(.)41 b(Eviden)m(tly)631 1030 y Fu(\026)p Fw(\()p Fu(B)5 b Fw(\))26 b Fp(\024)961 935 y Fu(S)1053 834 y Fr(\020)1107 935 y Fu(L)1169 949 y Ft(N)1225 957 y FA(r)1264 935 y Fu(N)1337 949 y Ft(r)1375 935 y Fu(e)1417 902 y Ft(N)1480 867 y Fo(1)p FA(=\033)1473 913 y(r)1588 834 y Fr(\021)1642 856 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)p 961 1009 910 4 v 1321 1092 a Fw(#)p Fu(K)1474 1106 y Ft(r)1906 1030 y Fp(\024)2012 968 y Fu(L)p Fw(\()p Fu(B)5 b Fw(\))2218 935 y Ft(nd)2301 968 y Fu(N)2384 935 y Ft(nd)p Fm(\000)p Ft(d)p Fq(+)p Ft(n)2374 991 y(r)2656 968 y Fu(e)2698 935 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(dN)2984 900 y Fo(1)p FA(=\033)2977 946 y(r)p 2012 1009 1083 4 v 2316 1103 a Fw(log)2433 1066 y Ft(d)2489 1103 y Fu(N)2562 1117 y Ft(r)2600 1103 y Fw(#)p Fu(K)2753 1117 y Ft(r)3104 1030 y Fu(:)386 1239 y Fw(>F)-8 b(rom)31 b(\(6.18\))h(w)m(e)f(obtain) 1295 1489 y Fu(\026)p Fw(\()p Fu(B)5 b Fw(\))25 b Fp(\034)1645 1422 y Fu(L)p Fw(\()p Fu(B)5 b Fw(\))1851 1389 y Ft(nd)1935 1422 y Fu(e)1977 1384 y Ft(ndN)2119 1349 y Fo(1)p FA(=\033)2112 1405 y(r)r Fx(\000)p Fo(1)2229 1422 y Fu(N)2312 1389 y Ft(nd)2302 1446 y(r)r Fm(\000)p Fq(1)p 1645 1468 786 4 v 1897 1551 a Fw(#)p Fu(K)2050 1565 y Ft(r)r Fm(\000)p Fq(1)2440 1489 y Fu(:)386 1690 y Fw(Ho)m(w)m(ev)m(er,)33 b(#)p Fu(K)931 1704 y Ft(r)r Fm(\000)p Fq(1)1084 1690 y Fp(\025)25 b Fw(1)p Fu(=f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(E)1524 1704 y Ft(r)r Fm(\000)p Fq(1)1653 1690 y Fw(\)\))31 b(whic)m(h)e(implies)f(that)1078 1873 y Fu(\026)p Fw(\()p Fu(B)5 b Fw(\))26 b Fp(\034)f Fu(L)p Fw(\()p Fu(B)5 b Fw(\))1625 1836 y Ft(nd)1708 1873 y Fu(e)1750 1835 y Ft(ndN)1892 1800 y Fo(1)p FA(=\033)1885 1857 y(r)r Fx(\000)p Fo(1)2003 1873 y Fu(N)2086 1836 y Ft(nd)2076 1896 y(r)r Fm(\000)p Fq(1)2204 1873 y Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(E)2458 1887 y Ft(r)r Fm(\000)p Fq(1)2586 1873 y Fw(\)\))p Fu(:)386 2032 y Fw(F)-8 b(or)31 b Fu(r)d(<)d Fw(1)31 b(the)f(function)f Fu(f)10 b Fw(\()p Fu(r)s Fw(\))p Fu(=r)1560 1999 y Ft(nd)1673 2032 y Fw(decreases,)32 b(hence,)f(as)f Fu(L)p Fw(\()p Fu(B)5 b Fw(\))25 b Fp(\024)386 2161 y Fw(exp\()p Fp(\000)p Fu(N)714 2113 y Fq(1)p Ft(=\033)704 2187 y(r)r Fm(\000)p Fq(1)832 2161 y Fw(\))p Fu(=)-5 b(N)980 2175 y Ft(r)r Fm(\000)p Fq(1)1134 2161 y Fu(<)25 b Fw(1)31 b(w)m(e)g(obtain)f Fu(\026)p Fw(\()p Fu(B)5 b Fw(\))25 b Fp(\024)g Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(B)5 b Fw(\)\).)486 2269 y(F)-8 b(or)32 b(the)f(second)h(case)g (assume)f(that)h Fu(B)k Fw(in)m(tersects)c(cub)s(es)e(in)g Fu(K)2752 2283 y Ft(r)2822 2269 y Fw(from)h(only)f(one)386 2405 y(segmen)m(t)46 b Fu(I)7 b Fw(\()p Fu(C)g Fw(\).)85 b(As)45 b(b)s(efore)f Fu(L)p Fw(\()p Fu(B)5 b Fw(\))50 b Fp(\024)f Fu(e)1905 2367 y Fm(\000)p Ft(N)2023 2332 y Fo(1)p FA(=\033)2016 2388 y(r)r Fx(\000)p Fo(1)2134 2405 y Fu(=)-5 b(N)2247 2419 y Ft(r)r Fm(\000)p Fq(1)2375 2405 y Fw(,)49 b(and)c Fu(L)p Fw(\()p Fu(B)5 b Fw(\))49 b Fp(\025)h Fu(L)p Fw(\()p Fu(E)3181 2419 y Ft(r)3219 2405 y Fw(\))g(=)386 2542 y Fu(e)428 2509 y Fm(\000)p Ft(N)546 2474 y Fo(1)p FA(=\033)539 2519 y(r)654 2542 y Fu(=)-5 b(N)767 2556 y Ft(r)806 2542 y Fw(.)40 b(The)30 b(n)m(um)m(b)s(er)f(of)i(cub)s(es)e(that)i Fu(B)k Fw(can)c(in)m (tersect)g(is)386 2656 y Fp(\034)25 b Fw(\()p Fu(L)p Fw(\()p Fu(B)5 b Fw(\))p Fu(=L)p Fw(\()p Fu(E)952 2670 y Ft(r)991 2656 y Fw(\)\))1061 2623 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)1290 2656 y Fw(.)41 b(Therefore)890 2873 y Fu(\026)p Fw(\()p Fu(B)5 b Fw(\))25 b Fp(\034)1352 2811 y Fu(L)p Fw(\()p Fu(B)5 b Fw(\))1558 2778 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)p 1240 2852 657 4 v 1240 2939 a Fu(L)p Fw(\()p Fu(E)1404 2953 y Ft(r)1443 2939 y Fw(\))1478 2913 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)1706 2939 y Fw(#)p Fu(K)1859 2953 y Ft(r)1932 2873 y Fp(\024)2038 2811 y Fu(L)p Fw(\()p Fu(B)g Fw(\))2244 2778 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)2473 2811 y Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(E)2727 2825 y Ft(r)2765 2811 y Fw(\)\))p 2038 2852 798 4 v 2204 2939 a Fu(L)p Fw(\()p Fu(E)2368 2953 y Ft(r)2406 2939 y Fw(\))2441 2913 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)2845 2873 y Fu(:)386 3095 y Fw(As)31 b Fu(f)10 b Fw(\()p Fu(r)s Fw(\))p Fu(=r)779 3062 y Fq(\()p Ft(n)p Fm(\000)p Fq(1\))p Ft(d)1034 3095 y Fw(=)26 b(1)p Fu(=)15 b Fw(log)1354 3058 y Ft(s)1391 3095 y Fw(\(1)p Fu(=r)s Fw(\))32 b(is)f(an)g (increasing)f(function)f(and)i Fu(L)p Fw(\()p Fu(B)5 b Fw(\))27 b Fp(\025)f Fu(L)p Fw(\()p Fu(E)3301 3109 y Ft(r)3339 3095 y Fw(\))386 3203 y(it)k(is)f(easy)i(to)g(v)m(erify)f (that)h Fu(\026)p Fw(\()p Fu(B)5 b Fw(\))25 b Fp(\024)g Fu(f)10 b Fw(\()p Fu(L)p Fw(\()p Fu(B)5 b Fw(\)\).)486 3310 y(Th)m(us)30 b(for)h Fu(s)26 b(<)g(n\033)34 b Fw(from)d(Lemma)g (6.7)i(w)m(e)e(ha)m(v)m(e)i Ff(L)2259 3277 y Ft(s)2295 3310 y Fw(\()p Fu(F)13 b Fw(\))28 b Fp(\025)e Fw(1)32 b(whic)m(h)e(implies)e(that)386 3418 y(Ldim)c(\()p Fu(F)13 b Fw(\))26 b Fp(\025)f Fu(n\033)33 b Fw(completing)c(the)i(pro)m(v)m(e) g(of)g(Theorem)f(6.2.)p 3309 3418 4 62 v 3313 3360 55 4 v 3313 3418 V 3367 3418 4 62 v 486 3581 a(In)40 b(order)h(to)h(pro)m (v)m(e)h(Theorem)e(6.3)i(w)m(e)e(prepare)g(a)h(lemma.)74 b(Let)42 b Fu(I)2945 3548 y Ft(m)2938 3609 y(\016)3055 3581 y Fw(=)i Fp(f)p FE(x)g Fp(2)386 3689 y Fu(I)433 3656 y Ft(n)507 3689 y Fw(:)26 b Fp(j)p FE(x)p Fp(j)i Fu(>)e(\016)s Fp(g)32 b Fw(and)f(let)g Fp(jj)h Fu(:)f Fp(jj)h Fw(denote)g(the)f(distance)g(from)f(the)i(nearest)g(in)m (teger,)g(i.e.)386 3797 y Fp(jj)p FE(x)p Fp(jj)26 b Fw(=)f(min)815 3817 y Fg(k)p Fm(2)p Fl(Z)955 3798 y Fo(2)1003 3797 y Fp(j)p FE(x)c Fp(\000)f FE(k)p Fp(j)p Fw(.)386 3961 y FE(Lemma)33 b(6.8.)46 b Fk(F)-7 b(or)45 b(every)f Fu(N)57 b(>)46 b Fw(2)e Fk(and)i(almost)g(al)5 b(l)45 b FE(x)h Fp(2)g Fu(I)2622 3928 y Ft(n)2615 3989 y(\016)2714 3961 y Fk(ther)-5 b(e)45 b(exists)f FE(q)j Fp(2)386 4077 y Fs(Z)451 4044 y Ft(n)493 4077 y Fp(nf)p FE(0)p Fp(g)35 b Fk(with)e Fu(N)r(=)15 b Fw(log)1166 4040 y Fq(4)1220 4077 y Fu(N)35 b Fp(\024)25 b(j)p FE(q)p Fp(j)h(\024)f Fu(N)43 b Fk(such)32 b(that)i Fw(dist)25 b(\()p FE(x)p Fu(;)15 b(R)2531 4092 y Ft(\014)s(;)p Fg(q)2641 4077 y Fw(\))26 b Fu(<)f(N)2881 4044 y Fm(\000)p Fq(2)2990 4077 y Fw(log)3108 4040 y Fq(7)3162 4077 y Fu(N)10 b Fk(.)386 4241 y(Pr)-5 b(o)g(of.)43 b Fw(>F)-8 b(rom)39 b([29)q(],)i(pp.)63 b(74-75)41 b(it)c(can)i(b)s(e)f(v)m(erifed)g(that)g (for)h(ev)m(ery)g Fu(N)48 b(>)39 b Fw(2)f(and)386 4352 y(almost)c(all)e FE(x)f Fp(2)f Fu(I)1036 4319 y Ft(n)1116 4352 y Fw(there)k(exists)f FE(q)e Fp(2)f Fs(Z)1848 4319 y Ft(n)1891 4352 y Fp(nf)p FE(0)p Fp(g)35 b Fw(with)d Fu(N)r(=)15 b Fw(log)2577 4315 y Fq(4)2631 4352 y Fu(N)41 b Fp(\024)30 b(j)p FE(q)p Fp(j)h(\024)g Fu(N)43 b Fw(suc)m(h)386 4460 y(that)1370 4610 y Fp(jj)p FE(q)21 b Fp(\001)f FE(x)h Fp(\000)f Fu(\014)5 b Fp(jj)26 b(\024)f Fu(N)2019 4572 y Fm(\000)p Ft(n)2136 4610 y Fw(log)2253 4572 y Fq(3)2308 4610 y Fu(N)-1995 b Fw(\(6.19\))486 4759 y(The)36 b(follo)m(wing)g (argumen)m(t)h(is)f(done)h(in)f(detail)g(for)h Fu(n)f Fw(=)g(2)h(and)f(an)m(y)i(decreasing)386 4867 y(function)29 b Fu(\036)p Fw(\()p Fu(N)10 b Fw(\))31 b(and)f(is)f(not)i(di\016cult)d (to)j(extend)g(to)g Fu(n)25 b Fp(\025)g Fw(3.)486 4975 y(Consider)32 b(the)i(line)e Fu(x)1256 4989 y Fq(2)1327 4975 y Fw(=)f(1.)53 b(Then)33 b Fp(j)p FE(q)23 b Fp(\001)g FE(x)g Fw(+)f 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3313 1133 55 4 v 3313 1191 V 3367 1191 4 62 v 386 1357 a Fk(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)34 b(6.3.)43 b Fw(Only)f(the)i(lo)m(w)m(er)g(b) s(ound)d(is)i(left)g(to)i(pro)m(v)m(e.)82 b(Using)42 b(the)386 1465 y(ubiquit)m(y)35 b(argumen)m(t)i(from)g([29)q(])g(it)g (can)g(b)s(e)g(sho)m(wn)f(that)i(as)f(in)f(the)h(homogeneous)386 1575 y(case)31 b(with)e Fu(L)847 1589 y Ft(N)940 1575 y Fw(=)c([)p Fu(N)1144 1542 y Fq(2)1183 1575 y Fu(=)p Fw(8)p Fu(n)15 b Fw(log)1461 1538 y Fq(7)1516 1575 y Fu(N)10 b Fw(])1624 1542 y Fm(\000)p Fq(1)1749 1575 y Fw(w)m(e)31 b(ha)m(v)m(e)1235 1789 y(#)p Fu(G)p Fw(\()p Fu(N)10 b Fw(\))25 b Fp(\025)1657 1661 y Fr(\024)1852 1728 y Fu(N)1935 1695 y Fq(2)p 1715 1768 396 4 v 1715 1859 a Fw(8)p Fu(m)15 b Fw(log)1973 1822 y Fq(7)2028 1859 y Fu(N)2121 1661 y Fr(\025)2168 1678 y Ft(n)2231 1789 y Fw(\(1)21 b Fp(\000)f Fu(")p Fw(\))p Fu(:)386 1991 y Fw(Theorem)28 b(6.3)i(can)g(no)m(w)e(b)s(e)h(pro)m(v)m(ed)g(b)m (y)f(using)g(the)h(ab)s(o)m(v)m(e)h(and)e(follo)m(wing)f(the)i(pro)s (of)386 2099 y(of)h(Theorem)h(6.2)g(exactly)-8 b(.)p 3309 2099 4 62 v 3313 2041 55 4 v 3313 2099 V 3367 2099 4 62 v 486 2264 a(Theorems)44 b(6.2)h(and)f(6.3)i(sho)m(w)e(that)i(the) e(sets)h Fu(E)2312 2278 y Fq(0)2352 2264 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(m;)g(n)p Fw(\))45 b(and)f Fu(E)2995 2279 y Ft(\014)3043 2264 y Fw(\()p Fu(\033)s Fw(;)15 b Fu(m;)g Fw(1\))386 2372 y(consist)30 b(of)g(more)h(than)f(just)g(the)g(resonan) m(t)h(h)m(yp)s(erplanes)e Fu(R)2504 2387 y Fg(q)p Ft(;\014)2613 2372 y Fw(.)486 2480 y FE(Ac)m(kno)m(wledgemen)m(ts)p Fw(:)64 b(This)40 b(pap)s(er)h(w)m(as)h(set)h(forth)f(when)f(the)h (second)g(and)386 2588 y(the)g(third)f(authors)g(sta)m(y)m(ed)j(in)d (MF)m(O,)i(Ob)s(erw)m(olfac)m(h)e(in)g(the)h(framew)m(ork)g(of)h(RiP) 386 2696 y(program.)d(They)28 b(w)m(ould)g(lik)m(e)g(to)i(thank)f (Matthias)g(Krec)m(k)h(for)e(his)g(hospitalit)m(y)-8 b(.)39 b(The)386 2804 y(authors)i(are)g(also)g(grateful)f(to)i(Maurice) f(Do)s(dson)f(and)h(V)-8 b(asilii)39 b(Bernik)h(for)g(some)386 2912 y(enligh)m(tening)30 b(discussions)f(on)j(Diophan)m(tine)f(appro)m (ximation)f(and)h(applications)f(of)386 3020 y(Hausdor\013)g (dimension,)e(and)i(to)h(Tim)e(Steger)i(for)f(suggesting)g(the)h(use)f (of)g(theorems)386 3128 y(on)44 b(\014nitely)e(generated)k(ab)s(elian)c (groups)i(in)f(the)h(study)f(of)i(resonan)m(t)f(sets.)83 b(The)386 3236 y(second)31 b(author)g(thanks)f(Rafael)h(De)h(La)f(Lla)m (v)m(e)g(for)g(bringing)d(the)j(pap)s(er)f([11)q(])h(to)h(his)386 3344 y(atten)m(tion)i(and)e(for)h(useful)e(discussions)f(on)i(the)h (app)s(earance)g(of)g(Gevrey)h(spaces)f(in)386 3452 y(homological)d (equations.)1615 3651 y Fv(References)424 3796 y FD([1])40 b(V.I.)24 b(Arnol'd,)h Fn(Smal)t(l)h(denominators,)i(I.)e(Mappings)h (fr)l(om)f(the)i(cir)l(cle)f(onto)g(itself)p FD(,)e(Izv.)f(Ak)l(ad.)544 3888 y(Nauk)h(USSR,)f(Ser.)i(Mat.,)h Fb(25)p FD(:1)f(\(1961\),)i (21{86.)g(MR)d Fb(24)p FD(#4113)424 3979 y([2])40 b(V.I.)34 b(Arnol'd,)i Fn(Ge)l(ometric)l(al)h(metho)l(ds)g(in)e(the)h(the)l(ory)h (of)e(or)l(dinary)i(di\013er)l(ential)f(e)l(quations)p FD(,)544 4070 y(Springer,)26 b(New)f(Y)-6 b(ork)25 b(-)g(Heidelb)r(erg) i(-)e(Berlin,)i(1983.)g(MR)f Fb(84d)p FD(:58023)424 4162 y([3])40 b(D.)28 b(Bam)n(busi)f(and)h(N.N.)g(Nekhoroshev,)g Fn(A)i(pr)l(op)l(erty)i(of)d(exp)l(onential)i(stability)g(in)e(nonline) l(ar)544 4253 y(wave)34 b(e)l(quations)h(ne)l(ar)f(the)h(fundamental)f (line)l(ar)f(mo)l(de)p FD(,)i(Ph)n(ysica)d(D,)g Fb(122)h FD(\(1998\),)i(73{104.)544 4344 y(MR)25 b Fb(99h)p FD(:35137)424 4435 y([4])40 b(A.)31 b(Bergamasco,)k Fn(Perturb)l(ations)h(of)d(glob)l (al)t(ly)g(hyp)l(o)l(el)t(liptic)g(op)l(er)l(ators)p FD(,)j(J.)d(Di\013eren)n(tial)f(equ-)544 4527 y(ations,)27 b Fb(141)p FD(:2)g(\(1994\),)g(513{526.)i(MR)c Fb(95j)p FD(:35046)424 4618 y([5])40 b(A.)34 b(Bergamasco,)k Fn(R)l(emarks)f(ab) l(out)g(glob)l(al)e(analytic)h(hyp)l(o)l(el)t(lipticity)p FD(,)i(T)-6 b(rans.)35 b(Amer.)e(Math.)544 4709 y(So)r(c.,)26 b Fb(351)h FD(\(1999\),)g(4113{4126.)424 4801 y([6])40 b(A.)25 b(Bergamasco,)j(P)-6 b(.)25 b(Cordaro)i(and)f(P)-6 b(.)25 b(Malagutti,)j Fn(Glob)l(al)t(ly)g(hyp)l(o)l(el)t(liptic)f (systems)j(of)d(ve)l(ctor)544 4892 y(\014elds)p FD(,)f(J.)h(F)-6 b(unct.)25 b(Analysis,)h Fb(114)p FD(:2)h(\(1993\),)g(263{285.)i(MR)c Fb(94e)p FD(:35048)424 4983 y([7])40 b(L.)28 b(Boutet)g(de)g(Mon)n(v)n (el)g(and)f(P)-6 b(.)28 b(Kre)n(\023)-36 b(e,)29 b Fn(Pseudo-di\013er)l (ential)j(op)l(er)l(ators)g(and)f(Gevr)l(ey)g(classes)p FD(,)544 5075 y(Ann.)25 b(Inst.)g(F)-6 b(ourier,)26 b Fb(17)g FD(\(1967\),)i(295{323.)g(MR)e Fb(37)p FD(#1760)424 5166 y([8])40 b(J.D.)27 b(Bo)n(v)n(ey)f(and)g(M.M.)h(Do)r(dson,)g Fn(The)i(Hausdor\013)g(dimension)g(of)e(systems)j(of)e(line)l(ar)g (forms)p FD(,)544 5257 y(Acta)d(Arith.)h Fb(XL)-10 b(V)25 b FD(\(1986\),)i(337{358.)i(MR)d Fb(87h)p FD(:11074)424 5349 y([9])40 b(A.D.)33 b(Bruno,)i Fn(The)g(analytic)g(form)f(of)g (di\013er)l(ential)h(e)l(quations)p FD(,)i(T)-6 b(r.)33 b(Mosk.)h(Mat.)g(O-v)l(a,)g Fb(25)544 5440 y FD(\(1971\),)27 b(119{262)h(and)e Fb(26)g FD(\(1972\),)h(199{239)h(\(in)e(Russian\).)g (MR)f Fb(51)p FD(#13365)386 5531 y([10])40 b(A.D.)22 b(Bruno)g(and)g(S.)g(W)-6 b(alc)n(her,)23 b Fn(Symmetries)i(and)g(c)l (onver)l(genc)l(e)i(of)d(normalizing)h(tr)l(ansforma-)544 5623 y(tions)p FD(,)h(J.)h(Math.)f(Anal.)g(Appl.)f Fb(183)p FD(,)i(\(1994\),)g(571{576.)i(MR)c Fb(95i)p FD(:58160)p eop %%Page: 26 26 26 25 bop 386 239 a Fq(26)693 b(DICKINSON,)30 b(GRAMCHEV,)g(AND)g (YOSHINO)386 434 y FD([11])40 b(Chen)24 b(W)-6 b(en)n(yi)24 b(and)g(M.Y.)h(Chi,)h Fn(Hyp)l(o)l(el)t(liptic)h(ve)l(ctor)h(\014elds)f (and)g(almost)g(p)l(erio)l(dic)g(motions)g(on)544 525 y(the)h(torus)h Fb(T)930 493 y FA(n)972 525 y FD(,)d(to)g(app)r(ear)g (in)g Fn(Comm.)g(Partial)i(Di\013er)l(ential)g(Equations)p FD(.)386 616 y([12])40 b(R.)28 b(De)f(La)i(Lla)n(v)n(e,)g(KAM)f(theory) f(and)h(applications.)i(to)e(app)r(ear)g(in)g(Pro)r(c.)i(of)f(1999)g (Summer)544 708 y(Researc)n(h)34 b(Institute,)i(Smo)r(oth)e(Ergo)r(dic) h(Theory)g(and)f(Applications,)j(July)d(26-Augsut)h(13,)544 799 y(1999,)27 b(Univ.)e(of)i(W)-6 b(ashington,)26 b(Seattle.)386 890 y([13])40 b(D.)28 b(DeLatte,)g Fn(Diophantine)j(c)l(onditions)f (for)g(the)h(line)l(arization)f(of)f(c)l(ommuting)h(holomorphic)544 982 y(functions)p FD(,)d(Discr.)f(&)f(Con)n(t.)h(Dynam.)f(Sys.)g Fb(3)p FD(,)h(\(1997\),)h(317{332.)i(MR)d Fb(98c)p FD(:30035)386 1073 y([14])40 b(H.)c(Dic)n(kinson,)j Fn(The)f(Hausdor\013)h(dimension) e(of)g(systems)i(of)e(simultane)l(ously)i(smal)t(l)e(line)l(ar)544 1164 y(forms)p FD(,)26 b(Mathematik)l(a,)g Fb(40)g FD(\(1993\),)h (367{374.)h(MR)e Fb(94m)p FD(:11093)386 1256 y([15])40 b(D.)25 b(Dic)n(kinson,)g(T.)h(Gramc)n(hev)e(and)h(M.)h(Y)-6 b(oshino,)25 b Fn(First)j(or)l(der)h(pseudo)l(di\013er)l(ential)g(op)l (er)l(ators)544 1347 y(on)c(the)h(torus:)35 b(normal)25 b(forms,)g(Diophantine)h(phenomena)g(and)g(glob)l(al)f(hyp)l(o)l(el)t (lipticity)p FD(,)f(Pro)r(c.)544 1438 y(of)32 b(the)f(Conf.)h (\\Di\013eren)n(tial)g(Equations",)i(F)-6 b(errara,)34 b(June)d(1996,)j(Ann.)d(Univ.)f(F)-6 b(errara)32 b(Sez.)544 1530 y(VI)r(I)25 b(\(N.S.\))g Fb(41)p FD(,)h(suppl.)g(\(1997\),)h (51{64.)h(MR)d Fb(98k)p FD(:58221)386 1621 y([16])40 b(D.)27 b(Dic)n(kinson)h(and)f(S.)g(L.)h(V)-6 b(elani,)28 b Fn(Hausdor\013)j(me)l(asur)l(e)g(and)e(line)l(ar)h(forms)p FD(,)e(J.)g(reine)g(angew.)544 1712 y(Math.,)e Fb(490)h FD(\(1997\),)g(1{36.)g(MR)f Fb(98e)p FD(:11094)386 1804 y([17])40 b(M.M.)24 b(Do)r(dson,)g Fn(Hausdor\013)i(dimension,)f(lower) g(or)l(der)h(and)f(Khintchine's)g(the)l(or)l(em)i(in)d(metric)544 1895 y(Diophantine)g(appr)l(oximation)p FD(,)g(J.)e(reine)f(angew.)h (Math.)g Fb(432)g FD(\(1992\),)i(69{76.)f(MR)e Fb(94a)p FD(:11125)386 1986 y([18])40 b(M.M.)35 b(Do)r(dson)g(and)f(J.A.G.)i (Vic)n(k)n(ers,)g Fn(Exc)l(eptional)h(sets)g(in)e(Kolmo)l(gor)l(ov{A)n (rnol'd{Moser)544 2078 y(the)l(ory)p FD(,)28 b(J.)e(Ph)n(ys.)f(A,)h Fb(19)g FD(\(1986\),)h(349{374.)i(MR)c Fb(87m)p FD(:58054)386 2169 y([19])40 b(K.)33 b(F)-6 b(alconer,)37 b Fn(F)-6 b(r)l(actal)36 b(Ge)l(ometry:)51 b(Mathematic)l(al)36 b(F)-6 b(oundations)36 b(and)f(Applic)l(ations)p FD(,)i(John)544 2260 y(Wiley)26 b(&)f(sons,)i(1990.)g(MR)f Fb(92j)p FD(:28008)386 2352 y([20])40 b(T.)20 b(Gramc)n(hev,)g(P)-6 b(.)19 b(P)n(opiv)l(ano)n (v)g(and)h(M.)g(Y)-6 b(oshino,)21 b Fn(Glob)l(al)h(pr)l(op)l(erties)i (in)d(sp)l(ac)l(es)j(of)d(gener)l(alize)l(d)544 2443 y(functions)h(on)g(the)h(torus)g(for)f(se)l(c)l(ond)h(or)l(der)h (di\013er)l(ential)e(op)l(er)l(ators)i(with)e(variable)h(c)l(o)l (e\016cients)p FD(,)544 2534 y(Rend.)i(Sem.)f(Mat.)j(Univ.)e(P)n(ol.)i (T)-6 b(orino,)27 b Fb(51)p FD(:2)f(\(1993\),)i(144-174.)g(MR)d Fb(95k)p FD(:35047)386 2626 y([21])40 b(T.)26 b(Gramc)n(hev)f(and)g(G.) h(P)n(op)r(o)n(v,)g Fn(Nekhor)l(oshev)31 b(typ)l(e)d(estimates)i(for)d (bil)t(liar)l(d)g(b)l(al)t(l)g(maps)p FD(,)f(Ann.)544 2717 y(Inst.)f(F)-6 b(ourier,)27 b Fb(45)p FD(:3)f(\(1995\),)i (859{895.)g(MR)e Fb(97a)p FD(:58145)386 2808 y([22])40 b(T.)27 b(Gramc)n(hev)e(and)h(M.)h(Y)-6 b(oshino,)27 b Fn(R)l(apid)t(ly)h(c)l(onver)l(gent)j(iter)l(ative)e(metho)l(d)g(for) f(simultane)l(ous)544 2900 y(normal)f(forms)h(of)f(c)l(ommuting)h(maps) p FD(,)f(Math.)f(Z.,)g Fb(231)g FD(\(1999\),)i(745{770.)386 2991 y([23])40 b(A.)33 b(Giorgilli,)38 b(A.)33 b(Delshams,)i(E.)f(F)-6 b(on)n(tic)n(h,)34 b(L.)g(Galgani,)j(and)c(C.)h(Sim\023)-38 b(o,)34 b Fn(E\013e)l(ctive)i(stability)544 3082 y(for)d(a)g (Hamiltonian)f(system)j(ne)l(ar)e(an)h(el)t(liptic)e(e)l(quilibrium)g (p)l(oint,)i(with)f(an)h(applic)l(ation)f(to)544 3174 y(the)d(r)l(estricte)l(d)h(thr)l(e)l(e)g(b)l(o)l(dy)f(pr)l(oblem)p FD(,)e(J.)g(Di\013eren)n(tial)g(Equations,)g Fb(77)f FD(\(1989\),)j(167{198.)g(MR)544 3265 y Fb(90c)p FD(:70026)386 3356 y([24])40 b(M.)32 b(Herman,)g Fn(Sur)h(la)f(c)l(onjugaison)i (di\013)n(\023)-37 b(er)l(entiable)34 b(des)f(di\013)n(\023)-37 b(eomorphismes)34 b(du)f(c)l(er)l(cle)h(\022)-39 b(a)33 b(des)544 3448 y(r)l(otations)p FD(,)28 b(Publ.)e(IHES,)f Fb(49)h FD(\(1979\),)h(5{233.)h(MR)d Fb(81h)p FD(:58039)386 3539 y([25])40 b(M.)23 b(Herman,)g Fn(R)l(e)l(c)l(ent)j(r)l(esults)h (and)e(some)g(op)l(en)h(questions)h(on)e(Sie)l(gel's)g(line)l (arization)g(the)l(or)l(em)544 3630 y(of)k(germs)i(of)f(c)l(omplex)g (analytic)h(di\013e)l(omorphisms)f(of)g FB(C)2253 3598 y FA(n)2331 3630 y Fn(ne)l(ar)h(a)f(\014xe)l(d)h(p)l(oint)p FD(,)f(Pro)r(c.)f(of)g(the)544 3721 y(VI)r(I)r(Ith)36 b(In)n(ternational)i(Congress)h(on)e(Mathematical)i(Ph)n(ysics,)i (Marseille,)i(1986,)f(eds.)c(M.)544 3813 y(Mebkhout)f(and)h(R.)g(S)n (\023)-36 b(en)n(\023)g(eor,)41 b(W)-6 b(orld)39 b(Scien)n(ti\014c)e (Publ.,)42 b(Singap)r(ore,)h(1987,)g(138{184.)e(MR)544 3904 y Fb(89k)p FD(:32013)386 3995 y([26])f(L.)26 b(H\177)-38 b(ormander,)26 b Fn(The)i(analysis)h(of)f(line)l(ar)g(p)l(artial)h (di\013er)l(ential)f(op)l(er)l(ators,)j(vol.)d(III)p FD(,)d(Springer)544 4087 y(V)-6 b(erlag,)26 b(Berlin)h({)f(Heidelb)r (erg)g({)g(New)g(Y)-6 b(ork,)25 b(1985.)i(MR)f Fb(87d)p FD(:35002a)386 4178 y([27])40 b(J.)f(Hounie,)i Fn(Glob)l(al)t(ly)e(hyp) l(o)l(el)t(liptic)g(ve)l(ctor)i(\014elds)f(on)f(c)l(omp)l(act)i(surfac) l(es)p FD(,)i(Comm.)37 b(P)n(artial)544 4269 y(Di\013eren)n(tial)26 b(Equations,)g Fb(7)g FD(\(1982\),)h(343{370.)i(MR)d Fb(83k)p FD(:35025)386 4361 y([28])40 b(Y.)28 b(Katznelson,)i(B.)f(Kra) g(and)f(D.)g(Ornstein,)h Fn(A)h(glob)l(al)g(version)h(of)f(a)g(the)l (or)l(em)i(of)e(J.)f(Moser)p FD(,)544 4452 y(to)d(app)r(ear)g(in)f Fn(Isr)l(ael)j(J.)f(Math.)386 4543 y FD([29])40 b(J.)20 b(Lev)n(esley)-6 b(,)20 b Fn(A)h(gener)l(al)i(inhomo)l(gene)l(ous)g (Jarn)-8 b(\023)-31 b(\020k{Besic)l(ovitch)24 b(the)l(or)l(em)p FD(,)f(J.)c(Num)n(b)r(er)f(Theory)-6 b(,)544 4635 y Fb(71)26 b FD(\(1998\),)h(65{80.)h(MR)d Fb(99g)p FD(:11091)386 4726 y([30])40 b(S.)25 b(MacLane)i(and)e(G.)i(Birkho\013,)f Fn(A)n(lgebr)l(a)p FD(,)h(MacMillan,)h(New)e(Y)-6 b(ork)25 b(1979.)i(MR)f Fb(80d)p FD(:00002)386 4817 y([31])40 b(J.)22 b(Moser,)h Fn(A)h(r)l(apid)t(ly)f(c)l(onver)l(gent)j(iter)l (ation)f(metho)l(d)f(and)g(non-line)l(ar)g(di\013er)l(ential)g(e)l (quations,)544 4909 y(II)p FD(,)h(Ann.)g(Scuola)h(Norm.)f(Sup.)g(Pisa)i (\(3\),)f Fb(20)g FD(\(1966\),)h(499{535.)i(MR)c Fb(34)p FD(#6280)386 5000 y([32])40 b(J.)27 b(Moser,)i Fn(On)g(c)l(ommuting)g (cir)l(cle)g(mappings)h(and)f(simultane)l(ous)h(Diophantine)f(appr)l (oxima-)544 5091 y(tions)p FD(,)d(Math.)h(Z.,)f Fb(205)g FD(\(1990\),)h(105{121.)i(MR)d Fb(92e)p FD(:58120)386 5183 y([33])40 b(L.)34 b(Ro)r(dino,)i Fn(Line)l(ar)g(p)l(artial)g (di\013er)l(ential)g(op)l(er)l(ators)i(in)d(Gevr)l(ey)h(sp)l(ac)l(es)p FD(,)j(W)-6 b(orld)34 b(Scien)n(ti\014c,)544 5274 y(Singap)r(ore-New)26 b(Jersey-London-Hong)f(Kong)h(1993.)i(MR)d Fb(95c)p FD(:35001)386 5365 y([34])40 b(H.)c(R)r(\177)-41 b(ussmann,)38 b Fn(On)f(optimal)g (estimates)i(for)e(the)h(solutions)g(of)e(line)l(ar)h(p)l(artial)h (di\013er)l(ential)544 5457 y(e)l(quations)26 b(of)d(\014rst)j(or)l (der)f(with)f(c)l(onstant)j(c)l(o)l(e\016cients)f(on)e(the)h(torus)p FD(,)g(Lect.)d(Notes)g(in)g(Ph)n(ysics,)544 5548 y Fb(38)p FD(,)k(Springer)g(V)-6 b(erlag,)26 b(Berlin)h(-)e(Heidelb)r(erg)h(-)f (New)h(Y)-6 b(ork)25 b(\(1975\),)i(598{624.)i(MR)c Fb(57)p FD(#7675)p eop %%Page: 27 27 27 26 bop 1016 239 a Fq(PER)-6 b(TURBA)g(TIONS)30 b(OF)h(VECTOR)f (FIELDS)h(ON)g(TORI)559 b(27)386 434 y FD([35])40 b(W.)26 b(Sc)n(hmidt,)e Fn(Diophantine)29 b(appr)l(oximation)p FD(,)f(Lect.)e(Notes)g(in)g(Mathematics,)h Fb(785)g FD(Springer)544 525 y(V)-6 b(erlag,)26 b(Berlin)h(-)e(Heidelb)r(erg)h(-)f(New)h(Y)-6 b(ork)25 b(1980.)i(MR)f Fb(81j)p FD(:10038)386 616 y([36])40 b(L.)28 b(Stolo)n(vitc)n(h,)i Fn(Compl)n(\022)-37 b(ete)30 b(int)n(\023)-37 b(egr)l(abilit)n(\023)g(e)31 b(singuli)n(\022)-37 b(er)l(e)p FD(,)30 b(C.)f(R.)f(Acad.)h(Sci.)g(P)n(aris)g(S)n(\022)-36 b(er.)29 b(I.)f(Math,)544 708 y Fb(326:6)e FD(\(1998\),)h(733-736.)h (MR)d Fb(99h)p FD(:32041)386 799 y([37])40 b(J.-C.)28 b(Y)-6 b(o)r(ccoz,)27 b Fn(R)l(e)l(c)l(ent)k(developments)f(in)e (dynamics)p FD(,)g(Pro)r(c.)g(of)g(the)e(In)n(ternational)h(Congress) 544 890 y(of)i(Mathematicians,)i(Z)r(\177)-41 b(uric)n(h,)30 b(Switzerland,)h(August)d(1994,)j(Birkh\177)-38 b(auser,)31 b(\(1995\),)g(246{265.)544 982 y(MR)25 b Fb(98e)p FD(:58146)386 1073 y([38])40 b(M.)32 b(Y)-6 b(oshino,)33 b Fn(A)g(class)g(of)g(glob)l (al)t(ly)g(hyp)l(o)l(el)t(liptic)g(op)l(er)l(ators)j(on)d(the)g(torus)p FD(,)i(Math.)d(Z.)g(,)h Fb(201)544 1164 y FD(\(1989\),)27 b(1{11.)h(MR)d Fb(90m)p FD(:35046)486 1400 y FC(Dept)i(of)i(Ma)-5 b(ths,)28 b(University)f(of)h(York,)h(Heslington,)f(York)h(Y010)g(5DD,) g(UK)486 1491 y Fn(E-mail)e(addr)l(ess)6 b FD(:)36 b Fa(hd3@york.ac.uk)486 1646 y FC(Dip)-5 b(ar)g(timento)27 b(di)g(Ma)-5 b(tema)g(tica,)27 b(Universit)1992 1640 y(\022)1990 1646 y(a)g(di)g(Ca)n(gliari,)h(09124)h(Ca)n(gliari,)f(It)-5 b(alia)486 1737 y Fn(E-mail)27 b(addr)l(ess)6 b FD(:)36 b Fa(todor@unica.it)486 1893 y FC(Dept)27 b(of)i(Ma)-5 b(ths,)28 b(Chuo)g(University,)f(Tokyo)i(192-0393,)h(Jap)-5 b(an)486 1984 y Fn(E-mail)27 b(addr)l(ess)6 b FD(:)36 b Fa(yoshinom@tamacc.chuo-u.ac.jp)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0001101301876--