%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: kamfinal.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips kamfinal %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.04.26:1659 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 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2309 y Fi(o)1696 2367 y(i)1751 2344 y Fq(+)17 b Fj(\013)1886 2367 y Fi(i)1914 2344 y Fq(\()p Fj(t)p Fq(\))p Fj(;)d(')2099 2309 y Fi(o)2099 2367 y(i)2154 2344 y Fq(+)j Fj(!)2288 2367 y Fi(i)2315 2344 y Fj(t)h Fq(+)f Fj(\014)2492 2367 y Fi(i)2520 2344 y Fq(\()p Fj(t)p Fq(\)\))p Fl(g)27 b Fq(with)h Fl(j)p Fj(\013)2980 2367 y Fi(i)3008 2344 y Fl(j)f Fq(and)g Fl(j)p Fj(\014)3289 2367 y Fi(i)3317 2344 y Fl(j)g Fq(small)g(pro)n(vided)118 2460 y(that:)83 b(1\))30 b(subsets)f(of)g Fj(!)j Fq(lik)n(e)d Fj(!)1158 2425 y Fk(\()p Fi(l)p Fk(\))1282 2413 y Fj(:)1261 2460 y Fq(=)c(\()p Fj(!)1435 2483 y Fk(1)1473 2460 y Fj(;)14 b(!)1562 2483 y Fk(2)1598 2460 y Fj(;)g(:)g(:)g(:)g(!)1798 2483 y Fi(l)1823 2460 y Fq(\))30 b(satisfy)f(the)g(Diophan)n(tine)h (condition)f Fl(j)p Fj(!)3208 2425 y Fk(\()p Fi(l)p Fk(\))3304 2460 y Fl(\001)20 b Fj(\027)5 b Fl(j)3416 2425 y Fh(\000)p Fk(1)3532 2460 y Fl(\024)25 b Fj(\015)5 b(e)3709 2425 y Fh(j)p Fi(\027)t Fh(j)3786 2400 y Ff(\034)3822 2425 y Fi(=\034)118 2576 y Fq(for)35 b(all)f Fj(\027)41 b Fl(2)36 b Fn(Z)609 2541 y Fi(l)629 2576 y Fl(nf)p Fq(0)p Fl(g)p Fj(;)d Fq(0)i Fj(<)g(\034)45 b(<)35 b Fq(1)g(\(actually)g(w)n(e) f(will)i(tak)n(e)e(0)h Fj(<)g(\034)45 b(<)2493 2544 y Fk(1)p 2493 2558 34 4 v 2493 2605 a(2)2536 2576 y Fq(;)35 b Fj(\015)40 b Fq(is)35 b(a)g(p)r(ositiv)n(e)f(n)n(um)n(b)r(er\),)83 b(2\))35 b Fl(j)p Fj(!)3756 2599 y Fi(i)3784 2576 y Fl(j)g Fq(is)118 2692 y(su\016cien)n(tly)28 b(large.)118 2809 y(By)22 b(our)g(tec)n(hniques)g(w)n(e)h(cannot)f(relax)f(the)i (condition)f Fl(j)p Fj(h)1940 2832 y Fi(AA)2044 2809 y Fl(j)h(\024)g Fj(a)f Fq(allo)n(wing)g Fl(j)2598 2775 y Fi(h)2637 2783 y Ff(A)p 2598 2790 87 4 v 2616 2838 a Fi(A)2694 2809 y Fl(j)g Fq(to)h(b)r(e)g(un)n(b)r(ounded)f(for)g Fl(j)p Fj(A)p Fl(j)i(!)f Fq(+)p Fl(1)p Fj(:)118 2925 y Fq(The)j(same)g(mo)r(del)g(has)g(b)r(een)g(studied)h(in)f([P])g(but)h (a)f(m)n(uc)n(h)g(greater)e(lo)n(w)n(er)h(b)r(ound)h(on)g Fl(j)p Fj(!)2972 2948 y Fi(i)3000 2925 y Fl(j)g Fq(is)g(necessary)e(b)r (ecause)i(w)n(e)118 3041 y(imp)r(ose)i(a)f(di\013eren)n(t)h(Diophan)n (tine)f(condition)g(on)h(the)g(frequencies;)f(namely)g Fl(j)p Fj(!)2703 3006 y Fk(\()p Fi(l)p Fk(\))2799 3041 y Fl(\001)18 b Fj(\027)5 b Fl(j)2909 3006 y Fh(\000)p Fk(1)3022 3041 y Fl(\024)22 b Fj(\015)5 b Fl(j)p Fj(\027)g Fl(j)3249 3006 y Fi(l)3303 3041 y Fl(8)26 b Fj(\027)j Fl(2)23 b Fn(Z)3585 3006 y Fi(l)3605 3041 y Fl(nf)p Fq(0)p Fl(g)p Fj(:)118 3158 y Fq(F)-7 b(or)21 b(a)f(mo)r(del)i(represen)n(ted) d(b)n(y)i(\(1.1\))g(one)g(can)f(think)i(of)f(an)g(arra)n(y)e(of)i (rotators:)31 b(p)r(oin)n(t)22 b(particles)e(constrained)g(to)h(mo)n(v) n(e)118 3274 y(circularly)28 b(and)h(cen)n(tered)f(on)h(the)h(sites)f (of)g Fn(N)t Fj(:)35 b Fq(In)30 b(this)f(case)g(the)g(kinetic)g(energy) g(of)g(a)f(single)h(particle)g(is)g(quadratic)118 3390 y(in)f(the)g(action{v)-5 b(ariable)25 b Fj(h)p Fq(\()p Fj(A)p Fq(\))f(=)1228 3358 y Fk(1)p 1228 3372 34 4 v 1228 3419 a(2)1271 3390 y Fj(A)1333 3355 y Fk(2)1370 3390 y Fj(:)118 3507 y Fq(The)34 b(problem)f(considered)g(here)g(falls) g(within)h(the)h(family)e(of)h(the)g(problems)f(concerning)f(the)i (searc)n(h)e(of)i(almost{)118 3623 y(p)r(erio)r(dic)22 b(solutions)e(in)i(systems)f(with)h(in\014nitely)h(man)n(y)e(d.o.f.)35 b(\(originating)20 b(in)i(statistical)f(mec)n(hanics)g(for)g (instance\).)118 3739 y(Viewing)28 b(\(1.1\))f(as)g(the)h(equations)f (of)g(in\014nite)i(particles,)d(the)i(existence)g(of)f(the)h(solutions) f(describ)r(ed)g(in)h(this)g(pap)r(er)118 3855 y(w)n(ould)19 b(imply)h(that)f(energy)f(remains)h(con\014ned)g(in)g(suitable)g (subsets)g(of)h(phace{space)d(during)i(all)g(the)h(time{ev)n(olution.) 118 3971 y(The)h(dra)n(wbac)n(k)e(\(up)j(to)f(no)n(w\))g(is)g(that)h (the)f(energy{dep)r(endence)f(of)h(eac)n(h)g(d.o.f.)35 b(is)21 b(to)r(o)g(large)e(\(faster)i(than)h(an)n(y)e(p)r(o)n(w)n(er) 118 4087 y(but)27 b(slo)n(w)n(er)d(than)i(an)n(y)g(exp)r(onen)n (tial\).)36 b(W)-7 b(e)26 b(obtain)g Fj(e)1811 4052 y Fk(\()p Fh(j)p Fi(j)s Fh(j)11 b Fk(ln)1974 4027 y Fe(2)2017 4052 y Fh(j)p Fi(j)s Fh(j)p Fk(\))2113 4027 y Ff(\013)2186 4087 y Fq(0)22 b Fj(<)h(\013)h(<)e Fq(1)k(\(see)g(the)g(upp)r(er)h(b)r (ound)f(on)g Fj(\015)5 b Fq(\012)3589 4110 y Fi(l)3610 4119 y Ff(k)3676 4087 y Fq(b)r(efore)118 4203 y(Theorem)19 b(4.7\)\).)34 b(F)-7 b(or)20 b(more)f(realistic)g(rates)g(it)i(migh)n (t)f(b)r(e)g(appropriate)e(to)i(consider)f(a)h(particular)f(\(but)i (quite)f(general\))118 4319 y(in)n(teraction)k Fl(f)p Fj(f)616 4342 y Fi(i)643 4319 y Fl(g)p Fj(:)g Fq(An)h(example)f(is)h Fj(f)1309 4342 y Fi(i)1336 4319 y Fq(\()p Fj(')1422 4342 y Fi(i)p Fh(\000)p Fk(1)1535 4319 y Fj(;)14 b(')1626 4342 y Fi(i)1654 4319 y Fj(;)g(')1745 4342 y Fi(i)p Fk(+1)1857 4319 y Fq(\))23 b(=)g Fj(a)2044 4342 y Fi(i)2085 4319 y Fq(cos\()p Fj(')2283 4342 y Fi(i)2323 4319 y Fl(\000)13 b Fj(')2455 4342 y Fi(i)p Fh(\000)p Fk(1)2568 4319 y Fq(\))g(+)g Fj(a)2735 4342 y Fi(i)p Fk(+1)2859 4319 y Fq(cos)o(\()p Fj(')3056 4342 y Fi(i)p Fk(+1)3181 4319 y Fl(\000)g Fj(')3313 4342 y Fi(i)3340 4319 y Fq(\))25 b(\(or)f(something)118 4435 y(lik)n(e)g(it\))g(and)g(then)h(to)e(study) i(carefully)e(the)h(geometry)f(of)h(the)g(small)g(denominators)e(\(see) i(for)g(instance)f([BF)n(G])h(where)118 4551 y(a)33 b(Nekhoroshev{t)n (yp)r(e)f(theorem)h(is)g(obtained)h(for)f(in\014nitely)h(man)n(y)f (harmonic)f(oscillator)g(coupled)h(b)n(y)h(a)f(suitable)118 4668 y(p)r(oten)n(tial)21 b(or)g([W)-7 b(a1]{[W)g(a3])20 b(where)h(go)r(o)r(d)g(estimates)g(on)g(the)g(p)r(erturbativ)n(e)g (parameter)f(are)g(obtained)h(for)g(c)n(hains)g(of)g(a)118 4784 y(v)n(ery)h(large)f Fd(but)k(\014nite)d Fq(n)n(um)n(b)r(er)h(of)f (rotator)f(coupled)i(through)f(anelastic)g(forces)2675 4748 y Fk(1)2713 4784 y Fq(.)35 b(In\014nite{dimensional)22 b(extensions)p 118 4870 1200 4 v 197 4929 a Fk(1)276 4965 y(Nev)n(ertheless)j(the)i(results)f(of)h([W)-5 b(a1])27 b(can)f(b)r(e)g(hardly)h(extended)d(to)k(co)n(v)n(er)e(the)h (in\014nite{dimensional)c(theory)j(b)r(ecause)f(the)i(p)r(oten)n(tial)f (is)118 5081 y(needed)20 b(to)j(b)r(e)f(analytic)f(o)n(v)n(er)i(a)g (strip)f(around)g(the)g(real)g(axis)g(whose)h(size)f(is)h(prop)r (ortional)e(to)i(NlnN)f(where)g(N)g(is)h(the)f(n)n(um)n(b)r(er)f(of)h (particles)f(\(see)118 5197 y(\(1.14\))h(at)h(p.314\))1987 5396 y Fq(2)p eop %%Page: 3 3 3 2 bop 118 -35 a Fq(of)24 b(KAM)g(theorem)f(ha)n(v)n(e)g(b)r(een)h (already)f(studied)h(but)h(with)f(some)f(di\013erences)h(resp)r(ect)f (to)h(this)g(pap)r(er.)36 b([FSW],)24 b([VB])118 81 y(and)f([P\177)-42 b(o])23 b(consider)g(p)r(erturbations)f(\(the)i Fl(f)p Fj(f)1532 104 y Fi(i)1559 81 y Fl(g)1601 46 y Fh(0)1624 81 y Fj(s)p Fq(\))g(dep)r(ending)g(on)f(the)h(action{v)-5 b(ariables)21 b(\(sa)n(y)i(\()p Fj(I)3207 104 y Fk(1)3245 81 y Fj(;)14 b(I)3318 104 y Fk(2)3355 81 y Fj(;)g(I)3428 104 y Fk(3)3466 81 y Fj(;)g(:)g(:)g(:)p Fq(\)\))24 b(whic)n(h)118 198 y(are)33 b(extremely)g(lo)r(calized)g(in)h(phace{space)e(\()p Fl(j)p Fj(I)1668 220 y Fi(j)1703 198 y Fq(\(0\))p Fl(j)i(\030)e Fj(e)2002 162 y Fh(\000j)p Fi(j)s Fh(j)2124 137 y Fe(1+)p Ff(\013)2275 198 y Fj(\013)h(>)g Fq(0)g(\()p Fj(e)2605 162 y Fh(\000j)p Fi(j)s Fh(j)2727 137 y Ff(d)p Fe(+)p Ff(\013)2881 198 y Fj(\013)g(>)g Fq(0)g(for)g(a)g(d{dimensional\))118 314 y(problem)k(in)h([FSW])g(and)g([VB])g(while)f(at)h(least)f Fj(e)1785 279 y Fh(\000)11 b Fk(ln)1903 254 y Ff(\015)1953 279 y Fh(j)p Fi(j)s Fh(j)2065 314 y Fq(with)38 b Fj(\015)44 b(>)39 b Fq(1)e(in)h([P\177)-42 b(o]\).)66 b(Moreo)n(v)n(er)35 b(according)h(to)h(our)118 431 y(de\014nitions)25 b(\(see)g(\(2.3\))g (and)g(\(2.4\)\),)g(their)g(mo)r(dels)g(are)f(all)h(of)g(short{range)d (t)n(yp)r(e)2736 395 y Fk(2)2773 431 y Fq(.)36 b(A)25 b(\014rst)g(impro)n(v)n(emen)n(t)f(has)g(b)r(een)118 547 y(obtained)36 b(in)h([CP])f(where)g(the)h(short{range)d(condition)j (has)f(b)r(een)h(relaxed)e(allo)n(wing)g(eac)n(h)h(particle)g(to)g(in)n (teract)118 664 y(directly)31 b(with)g(eac)n(h)f(other)h(particle)f (\\of)h(the)g(w)n(orld.")f(Also)g(our)g(mo)r(dels)h(pro)n(vide)f(p)r (erturbation)g(functions)i(whic)n(h)118 780 y(are)k(indep)r(enden)n(t)i (on)e(the)i(actions{v)-5 b(ariables.)62 b(Although)37 b(closely)f(related,)j(the)e(result)g(in)g([P])f(is)h(sligh)n(tly)g (more)118 897 y(general)27 b(than)i([CP])g(b)r(ecause)f(the)h(kinetic)g (part)f(of)h(the)g(hamiltonian,)g(lik)n(e)f(here,)h(can)f(ha)n(v)n(e)f (a)i(quadratic)e(part)i(plus)118 1013 y(a)35 b Fd(c)l(orr)l(e)l(ction)g Fq(in)g(suc)n(h)g(a)g(w)n(a)n(y)e(that)j(the)f(leading)f(term)h(is)g (at)g(most)g(quadratric)e(for)i Fl(j)p Fj(A)p Fl(j)h(!)f Fq(+)p Fl(1)p Fj(:)g Fq(In)g([P])g(as)f(w)n(ell)118 1130 y(as)d(in)h([CP],)f(the)h(size)g(of)f(the)h(frequencies)f(is)h(of)f (order)g Fj(e)1972 1094 y Fi(p)p Fh(j)p Fi(j)s Fh(j)11 b Fk(ln)g Fh(j)p Fi(j)s Fh(j)2260 1130 y Fq(where)31 b Fj(p)f Fl(\030)f Fq(10)i(\()p Fj(e)2856 1094 y Fi(pd)p Fh(j)p Fi(j)s Fh(j)11 b Fk(ln)g Fh(j)p Fi(j)s Fh(j)3179 1130 y Fq(for)31 b(a)h Fj(d)p Fq({dimensional)118 1246 y(problem\).)i(Here)20 b(the)h(size)f(has)g(b)r(een)h(lo)n(w)n(ered)d (to)j Fj(e)1734 1211 y Fk(\()p Fh(j)p Fi(j)s Fh(j)11 b Fk(ln)1896 1186 y Fe(2)1940 1211 y Fh(j)p Fi(j)s Fh(j)p Fk(\))2036 1186 y Ff(\013)2082 1246 y Fj(;)21 b Fq(0)i Fj(<)f(\013)i(<)e Fq(1)e(using)g(a)g(di\013eren)n(t)h(Diophan)n(tine)f (condition)118 1363 y(describ)r(ed)k(at)g(p)r(oin)n(t)g(1\))f(b)r (efore.)36 b(T)-7 b(o)23 b(reac)n(h)g(a)h(p)r(o)n(w)n(er)e(la)n(w)h(\() p Fl(j)p Fj(j)5 b Fl(j)2072 1327 y Fi(p)2111 1363 y Fj(;)24 b(p)f Fl(\025)g Fq(1\))g(w)n(ould)h(b)r(e)g(a)g(v)n(ery)f(in)n (teresting)g(result)g(at)h(least)118 1479 y(for)k(t)n(w)n(o)g (applications:)39 b(1\))28 b(the)h(thermo)r(dynamic)g(limit)g(in)g(the) h(con)n(text)e(of)h(statistical)f(mec)n(hanics)g(2\))h(the)g(study)g (of)118 1596 y(almost{p)r(erio)r(dic)k(solutions)g(of)h(some)f (discretized)h(PDE's.)55 b(Another)33 b(di\013erence)h(resp)r(ect)g(to) g([CP])f(and)h([P])f(rests)118 1712 y(in)h(the)g(metho)r(d)g(of)f(pro)r (of.)54 b(In)34 b([CP])f(as)g(w)n(ell)g(as)g(in)g([P],)h(a)f (\\con\014gurational")d(KAM)j(theorem)g(has)g(b)r(een)h(pro)n(v)n(ed) 118 1829 y(\(tec)n(hnically)21 b(sp)r(eaking)f(one)h(solv)n(es)e(a)i (system)f(of)h(partial)f(di\013eren)n(tial)h(equations)f(whose)g(v)-5 b(ariables)20 b(are)g(the)h(functions)118 1945 y(de\014ning)29 b(the)f(torus)g(in)h(con\014guration{space\))d(while)i(here)g(w)n(e)h (p)r(erform)e(a)i(pro)r(of)e(based)h(on)h(coun)n(table)e(c)n(hanged)h (of)118 2062 y(action{angle)c(co)r(ordinates)g(in)i(phace{space)e(\(a)h (\\classic)f(pro)r(of"\).)36 b(Ma)n(y)24 b(b)r(e)i(analogous)e(results) h(could)g(b)r(e)h(obtained)118 2178 y(through)e(a)g(con\014gurational)e (pro)r(of.)36 b(W)-7 b(e)25 b(ha)n(v)n(e)e(used)h(a)g(\\classic")f(pro) r(of)h(b)r(ecause)g(w)n(e)g(think)h(it)g(allo)n(ws)e(the)i(treatmen)n (t)118 2295 y(of:)40 b(1\))29 b(p)r(erturbations)f(of)h(in\014nitely)h (man)n(y)f(harmonic)f(oscillators)f(\(where)i(con\014gurational)e(pro)r (ofs)h(can)h(b)r(e)g(hardly)118 2411 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3525 y Fq(\()p Fn(T)309 3489 y Fh(j)o Fi(I)5 b Fh(j)386 3525 y Fq(;)14 b Fn(R)p Fq(\))32 b Fb(function,)27 b Fj(V)955 3489 y Fk(\()p Fi(I)5 b Fk(\))1045 3525 y Fq(\()p Fj(')p Fq(\))p Fb(,)27 b(so)e(that)52 b Fj(f)1567 3482 y Fk([)p Fi(I)5 b Fk(])1558 3548 y Fi(i)1642 3525 y Fq(\()p Fj(')p Fq(\))24 b(=)f Fl(\000)p Fj(@)1981 3547 y Fi(')2025 3555 y Ff(i)2055 3525 y Fj(V)2122 3489 y Fk(\()p Fi(I)5 b Fk(\))2212 3525 y Fq(\()p Fj(')p Fq(\))p Fj(;)97 b Fl(8)25 b Fj(i)e Fl(2)g Fj(I)33 b(;)97 b Fl(8)25 b Fj(')f Fl(2)f Fn(T)3125 3489 y Fh(j)p Fi(I)5 b Fh(j)3202 3525 y Fj(:)26 b Fq(It)g(is)g(easy)f(to)h(c)n(hec)n(k)118 3640 y(that)i(the)g(examples)f(\(2.3\),)g(\(2.4\))h(in)f(Section)h(2)f (are)g(g{gradien)n(ts.)118 3827 y Fd(We)j(shal)t(l)h(sp)l(e)l(ak)g(of)f (Hamiltonian)h(e)l(quations)f(on)g Fl(T)1787 3850 y Fi(w)1870 3827 y Fd(whenever)h Fj(f)38 b Fd(in)30 b(\(1.1\))h(is)f(a)g(g{gr)l (adient.)118 4015 y Fq(Let)e(us)f(conclude)h(this)g(section)f(b)n(y)g (in)n(tro)r(ducing)g(the)h(\(strong\))f 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y(Section)c(1)f(is)h(uniformly)g(w)n(eakly)f(analytic)g(and)g(as)h (parameter)e Fj(\033)30 b Fq(one)d(could)g(tak)n(e)f(an)n(y)g(p)r (ositiv)n(e)g(n)n(um)n(b)r(er;)h(example)118 5197 y(\(2.3\))g(is)h (uniformly)f(w)n(eakly)g(analytic)g(for)g(some)g(\(small)g(enough\))h Fj(\033)e(>)d Fq(0.)1987 5396 y(5)p eop %%Page: 6 6 6 5 bop 118 -35 a Fr(3.)51 b(Almost{P)m(erio)s(dic)38 b(Solutions)f(and)g(Diophan)m(tine)f(Sequences)118 204 y Fp(De\014nition)27 b(3.1)d Fb(A)h(sequence)e Fj(!)j Fl(2)e Fn(R)1345 168 y Fm(Z)1418 204 y Fb(is)g(said)g(to)g(b)r(e)h Fq(rationally)e(indep)r(enden)n(t)i Fb(if)g(for)f(an)n(y)g(\014nite)h (subset)f Fj(I)32 b Fb(of)24 b Fn(Z)18 b Fb(and)118 321 y(for)27 b(an)n(y)g Fj(n)452 343 y Fi(i)503 321 y Fl(2)c Fn(Z)664 258 y Fg(P)752 345 y Fi(i)p Fh(2)p Fi(I)872 321 y Fj(!)924 343 y Fi(i)951 321 y Fj(n)1001 343 y Fi(i)1052 321 y Fl(6)p Fq(=)f(0)83 b(unless)g Fj(n)1615 343 y Fi(i)1665 321 y Fq(=)23 b(0)k Fl(8)g Fj(i)22 b Fl(2)i Fj(I)118 442 y Fq(In)31 b(other)g(w)n(ords,)f Fj(!)k Fq(is)c(rationally)g(indep) r(enden)n(t)i(if)f(an)n(y)f(\014nite)i(v)n(ector)d Fj(!)2499 406 y Fk(\()p Fi(I)5 b Fk(\))2612 442 y Fl(\021)23 b(f)p Fj(!)2794 464 y Fi(i)2820 442 y Fl(g)2862 464 y Fi(i)p Fh(2)p Fi(I)2999 442 y Fl(2)29 b Fn(R)3137 406 y Fh(j)p Fi(I)5 b Fh(j)3252 442 y Fq(is)30 b(rationally)g(inde-)118 558 y(p)r(enden)n(t.)118 726 y Fp(De\014nition)f(3.2)c Fb(A)h(con)n(tin)n(uous)f(real)g(function)h Fj(q)s Fq(\()p Fj(t)p Fq(\))g Fb(is)g(said)f(to)h(b)r(e)g Fq(almost{p)r(erio)r(dic)f (o)n(v)n(er)f Fl(T)3137 749 y Fi(w)3217 726 y Fb(\(with)i(frequency)g Fj(!)s Fb(\))118 843 y(if)h(there)g(exist)f(a)g(rationally)g(indep)r (enden)n(t)h(sequence)f Fj(!)g Fl(2)d Fn(R)2066 807 y Fm(Z)2142 843 y Fb(and)j(a)h(con)n(tin)n(uous)e(function)j Fj(Q)22 b Fq(:)h Fl(T)3289 865 y Fi(w)3366 843 y Fl(!)g Fn(R)33 b Fb(suc)n(h)26 b(that)118 960 y Fj(q)s Fq(\()p Fj(t)p Fq(\))g(=)g Fj(Q)p Fq(\([)p Fj(!)s(t)p Fq(]\))p Fb(.)41 b(A)30 b(solution)f Fq(\()p Fj(A)p Fq(\()p Fj(t)p Fq(\))p Fj(;)14 b(')p Fq(\()p Fj(t)p Fq(\)\))31 b Fb(of)e(\(1.1\))g(is) g(called)g Fq(maximal)g(almost{p)r(erio)r(dic)e Fb(if)j Fq(\()p Fj(A)3270 982 y Fi(i)3298 960 y Fq(\()p Fj(t)p Fq(\))p Fj(;)14 b(')3483 982 y Fi(i)3512 960 y Fq(\()p Fj(t)p Fq(\))20 b Fl(\000)f Fq([)p Fj(!)3785 982 y Fi(i)3812 960 y Fj(t)p Fq(]\))118 1076 y Fb(is,)28 b(for)f(all)g Fj(i)p Fb(,)g(almost{p)r(erio)r(dic)g(o)n(v)n(er)e Fl(T)1369 1099 y Fi(w)1451 1076 y Fb(with)j(frequency)f Fj(!)s Fb(.)118 1244 y Fq(As)g(usual)g(in)g(problems)f(in)n(v)n(olving)f (small)i(denominators,)e(w)n(e)i(ha)n(v)n(e)e(to)i(imp)r(ose)g(on)g (the)g(frequencies)f(a)g(quan)n(titativ)n(e)118 1361 y(irrationalit)n(y)20 b(condition.)35 b(Our)21 b(aim)h(is)g(to)g(get)g (the)g(lo)n(w)n(er)f(b)r(ound)h(of)g Fl(j)p Fj(!)2360 1383 y Fi(i)2388 1361 y Fl(j)g Fq(as)f(small)h(as)f(p)r(ossible)h(and)g (a)g(go)r(o)r(d)f(condition)118 1477 y(is)28 b(the)g(follo)n(wing)118 1646 y Fp(De\014nition)34 b(3.3)c Fb(A)g(rationally)f(indep)r(enden)n (t)i(sequence)f Fj(!)h Fl(2)d Fn(R)2231 1610 y Fm(Z)2310 1646 y Fb(is)i(called)g Fq(\()p Fj(\015)5 b(;)14 b(\034)9 b Fq(\))p Fb({)p Fq(Diophan)n(tine)31 b Fb(if)g(for)f(an)n(y)f (\014nite)118 1762 y(set)24 b Fj(I)30 b Fl(\032)23 b Fn(Z)o Fb(,)18 b(there)24 b(exist)g(constan)n(ts)e Fj(\015)28 b(>)23 b Fq(0)g Fb(and)g Fj(\034)33 b Fl(2)24 b Fq(\(0)p Fj(;)14 b Fq(1\))23 b Fb(suc)n(h)g(that)h(for)f(an)n(y)g(c)n(hoice)g (of)h Fj(n)3021 1785 y Fi(i)3071 1762 y Fl(2)g Fn(Z)17 b Fb(with)3414 1700 y Fg(P)3502 1787 y Fi(i)p Fh(2)p Fi(I)3622 1762 y Fl(j)p Fj(n)3695 1785 y Fi(i)3722 1762 y Fl(j)23 b Fj(>)g Fq(0)118 1879 y Fb(it)28 b(is:)1462 2006 y Fl(j)1499 1927 y Fg(X)1507 2105 y Fi(i)p Fh(2)p Fi(I)1632 2006 y Fj(!)1684 2029 y Fi(i)1712 2006 y Fj(n)1762 2029 y Fi(i)1789 2006 y Fl(j)23 b(\025)g Fj(\015)1971 1970 y Fh(\000)p Fk(1)2073 2006 y Fq(exp\()p Fl(\000j)p Fj(n)p Fl(j)2393 1970 y Fi(\034)2435 2006 y Fj(=\034)9 b Fq(\))1173 b(\(3)p Fj(:)p Fq(1\))150 2296 y(Of)31 b(course)f(\(3.1\)) i(implies)f(that)h(a)f(sequence)g Fj(!)h Fl(2)e Fn(R)1845 2260 y Fm(Z)1925 2296 y Fq(is)i(rationally)e(indep)r(enden)n(t:)45 b(for)31 b(an)n(y)g(\014nite)h(subset)f Fj(I)39 b Fq(of)31 b Fn(Z)118 2412 y Fq(and)h(for)g(an)n(y)f Fj(n)627 2435 y Fi(i)686 2412 y Fl(2)g Fn(Z)o Fj(;)882 2350 y Fg(P)970 2437 y Fi(i)p Fh(2)p Fi(I)1090 2412 y Fj(!)1142 2435 y Fi(i)1170 2412 y Fj(n)1220 2435 y Fi(i)1278 2412 y Fl(6)p Fq(=)f(0)83 b(unless)f Fj(n)1848 2435 y Fi(i)1907 2412 y Fq(=)30 b(0)i Fl(8)f Fj(i)g Fl(2)g Fj(I)7 b(:)32 b Fq(Diophan)n(tine)g(v)n(ectors)f(in)h(the)h(sense)f(of)g(\(3.1\))118 2529 y(form)26 b(a)h(set)f(of)h(full)g(Leb)r(esgue)f(measure)g(in)h Fn(R)p Fj(:)32 b Fq(The)27 b(pro)r(of)f(is)h(standard)e(and)i(w)n(e)f (omit.)37 b(It)27 b(can)f(b)r(e)h(p)r(erformed)f(along)118 2645 y(the)i(same)f(lines)g(of)h(the)f(pro)r(of)g(presen)n(t)g(in)h ([CP])f(p.194)f(although)h(the)g(irrationalit)n(y)f(condition)h (considered)g(there)g(is)118 2762 y Fl(j)155 2700 y Fg(P)243 2787 y Fi(i)p Fh(2)p Fi(I)363 2762 y Fj(!)415 2784 y Fi(i)442 2762 y Fj(n)492 2784 y Fi(i)520 2762 y Fl(j)f(\025)g Fj(\015)708 2726 y Fh(\000)p Fk(1)797 2762 y Fl(j)p Fj(n)p Fl(j)893 2726 y Fh(\000j)p Fi(I)5 b Fh(j)1022 2762 y Fj(:)30 b Fq(Also)f(in)h(in\014nite)g(dimensions)f(they)h(are)f(rather) f(abundan)n(t)i(as)e(the)i(follo)n(wing)f(Lemma)118 2879 y(sho)n(ws.)36 b(F)-7 b(or)27 b(an)n(y)g(v)n(ector)f Fj(v)g Fl(2)e Fn(R)1143 2843 y Fi(n)1221 2879 y Fq(let)k(b)r(e)g Fl(k)p Fj(v)s Fl(k)23 b Fq(=)f(\()1723 2816 y Fg(P)1811 2837 y Fi(n)1811 2904 y(i)p Fk(=1)1937 2879 y Fj(v)1980 2843 y Fk(2)1977 2901 y Fi(i)2017 2879 y Fq(\))2049 2843 y Fk(1)p Fi(=)p Fk(2)2182 2879 y Fq(while)27 b Fl(j)p Fj(v)s Fl(j)d Fq(=)2598 2816 y Fg(P)2686 2837 y Fi(n)2686 2904 y(i)p Fk(=1)2812 2879 y Fl(j)p Fj(v)2875 2901 y Fi(i)2903 2879 y Fl(j)p Fq(;)j Fj(E)5 b Fq(\()p Fj(t)p Fq(\))3181 2832 y Fj(:)3160 2879 y Fq(=)23 b Fj(e)3287 2843 y Fi(t)3312 2818 y Ff(\034)3349 2843 y Fi(=\034)118 3047 y Fp(Lemma)35 b(3.4)82 b Fb(Let)33 b Fj(!)h Fl(2)e Fn(R)1048 3011 y Fi(l)q Fh(\000)p Fk(1)1197 3047 y Fb(b)r(e)h Fq(\()p Fj(\015)5 b(;)14 b(\034)9 b Fq(\))p Fb({Diophan)n(tine)33 b(and)g Fq(\012)e Fl(\025)2395 3014 y Fk(1)p 2392 3028 39 4 v 2392 3076 a Fi(\015)2473 3047 y Fb(b)r(e)j(a)e(p)r(ositiv)n(e)g (n)n(um)n(b)r(er.)52 b(Let's)32 b(de\014ne)h(the)118 3163 y(subset)28 b(of)83 b Fq([\012)p Fj(;)14 b Fq(+)p Fl(1)p Fq(\))22 b(:)618 3390 y Fl(A)684 3413 y Fi(l)754 3343 y Fj(:)733 3390 y Fq(=)g Fl(A)886 3413 y Fi(l)912 3390 y Fq(\()p Fj(!)s(;)14 b Fq(\012\))23 b(=)1239 3323 y Fg(\010)1287 3390 y Fj(\013)h Fl(\025)f Fq(\012:)1562 3320 y Fg(\014)1562 3370 y(\014)1590 3390 y Fq(\()p Fj(!)s(;)14 b(\013)p Fq(\))19 b Fl(\001)f Fj(\027)1905 3320 y Fg(\014)1905 3370 y(\014)1933 3340 y Fh(\000)p Fk(1)2046 3390 y Fl(\024)k Fq(\012)2193 3355 y Fh(\000)p Fk(1)2282 3390 y Fj(E)5 b Fq(\()p Fl(j)p Fj(\027)g Fl(j)p Fq(\))84 b Fl(8)14 b Fj(\027)28 b Fl(2)23 b Fn(Z)2857 3355 y Fi(l)2960 3390 y Fj(\027)3001 3413 y Fi(l)3050 3390 y Fl(2)g 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Fb(and)27 b Fj(!)k Fb(indep)r(enden)n(t)118 3929 y Fd(Pr)l(o)l(of)1291 4056 y Fl(j)p Fj(!)21 b Fl(\001)e Fj(\027)5 b Fl(j)1498 4020 y Fh(\000)p Fk(1)1610 4056 y Fl(\024)23 b Fj(\015)5 b(E)g Fq(\()p Fl(j)p Fj(\027)g Fl(j)p Fq(\))p Fj(;)180 b Fl(8)27 b Fj(\027)i Fl(2)23 b Fn(Z)2454 4020 y Fi(l)p Fh(\000)p Fk(1)2559 4056 y Fl(nf)p Fq(0)p Fl(g)118 4241 y Fq(W)-7 b(e)28 b(need)g(the)g(follo)n(wing)e(relation)h(for)g Fj(\013)d Fl(\025)e Fq(\012)1065 4397 y Fg(\014)1065 4447 y(\014)1093 4468 y Fq(\()p Fj(!)1177 4490 y Fi(o)1214 4468 y Fj(;)14 b(\013)p Fq(\))19 b Fl(\001)f Fj(\027)1442 4397 y Fg(\014)1442 4447 y(\014)1470 4418 y Fh(\000)p Fk(1)1582 4468 y Fl(\024)23 b Fq(\012)1730 4432 y Fh(\000)p Fk(1)1819 4468 y Fj(E)5 b Fq(\()p Fl(j)p Fj(\027)g Fl(j)p Fq(\))167 b Fl(8)p Fj(\027)28 b Fl(2)23 b Fn(Z)2463 4432 y Fi(l)2483 4468 y Fl(nf)p Fq(0)p Fl(g)81 b Fj(\027)2773 4490 y Fi(l)2822 4468 y Fl(6)p Fq(=)22 b(0)776 b(\(3)p Fj(:)p Fq(3\))118 4728 y(If)28 b Fj(\027)g Fq(=)23 b(\()5 b(^)-47 b Fj(\027)6 b(;)14 b(\027)515 4750 y Fi(l)540 4728 y Fq(\))23 b(=)g(\()5 b(^)-47 b Fj(\027)6 b(;)14 b Fq(0\))27 b(w)n(e)g(ha)n(v)n(e)1214 4657 y Fg(\014)1214 4707 y(\014)1241 4728 y Fj(!)21 b Fl(\001)j Fq(^)-47 b Fj(\027)1402 4657 y Fg(\014)1402 4707 y(\014)1430 4678 y Fh(\000)p Fk(1)1542 4728 y Fl(\024)23 b Fq(\012)1690 4692 y Fh(\000)p Fk(1)1779 4728 y Fj(E)5 b Fq(\()p Fl(j)g Fq(^)-47 b Fj(\027)6 b Fl(j)p Fq(\))23 b Fl(\024)g Fj(\015)5 b(E)g Fq(\()p Fl(j)g Fq(^)-47 b Fj(\027)5 b Fl(j)p Fq(\))28 b(if)g(w)n(e)f(imp)r(ose)h(\012)23 b Fl(\025)3072 4695 y Fk(1)p 3069 4709 39 4 v 3069 4757 a Fi(\015)118 4849 y Fq(In)33 b(^)-47 b Fj(\027)28 b Fq(=)379 4831 y(^)379 4849 y(0)f(\(3.3\))g(b)r(ecomes)h Fl(j)p Fj(\027)1041 4871 y Fi(l)1066 4849 y Fj(\013)p Fl(j)1142 4813 y Fh(\000)p Fk(1)1255 4849 y Fl(\024)23 b Fq(\012)1403 4813 y Fh(\000)p Fk(1)1492 4849 y Fj(E)5 b Fq(\()p Fl(j)p Fj(\027)1654 4871 y Fi(l)1680 4849 y Fl(j)p Fq(\))28 b(whic)n(h)f(is)h(due)f(to)h Fj(\013)23 b Fl(\025)g Fq(\012)28 b(and)2797 4816 y Fk(1)p 2765 4830 97 4 v 2765 4878 a Fh(j)p Fi(\027)2818 4887 y Ff(l)2842 4878 y Fh(j)2895 4849 y Fl(\024)22 b Fj(E)5 b Fq(\()p Fl(j)p Fj(\027)3144 4871 y Fi(l)3170 4849 y Fl(j)p Fq(\))p Fj(:)118 4970 y Fl(A)184 4934 y Fi(c)184 4992 y(l)246 4970 y Fq(is)27 b(the)h(set)g(of)g Fj(\013)g Fq(for)f(whic)n(h)g(\(3.3\))h(is)f(false)883 5197 y Fl(A)949 5161 y Fi(c)949 5219 y(l)1006 5197 y Fq(=)1094 5130 y Fg(\010)1142 5197 y Fj(\013)d Fl(\025)e Fq(\012)14 b(:)28 b Fl(9)14 b Fj(\027)28 b Fl(2)c Fn(Z)1700 5161 y Fi(l)1803 5197 y Fj(\027)1844 5219 y Fi(l)1892 5197 y Fl(6)p Fq(=)f(0)p Fj(;)2141 5126 y Fg(\014)2141 5176 y(\014)2169 5197 y Fq(\()p Fj(!)s(;)14 b(\013)p Fq(\))19 b Fl(\001)f Fj(\027)2484 5126 y Fg(\014)2484 5176 y(\014)2512 5147 y Fh(\000)p Fk(1)2624 5197 y Fj(>)23 b Fq(\012)2772 5161 y Fh(\000)p Fk(1)2861 5197 y Fj(E)5 b Fq(\()p Fl(j)p Fj(\027)g Fl(j)p Fq(\))3083 5130 y Fg(\011)3727 5197 y Fq(\(3)p Fj(:)p Fq(4\))1987 5396 y(6)p eop %%Page: 7 7 7 6 bop 118 -35 a Fq(and)28 b(its)f(Leb)r(esgue)g(measure)g(is)g(b)r (ounded)h(b)n(y)853 219 y Fj(Leb)p Fq(\()p Fl(A)1083 183 y Fi(c)1083 241 y(l)1116 219 y Fq(\))c Fl(\024)1318 140 y Fg(X)1273 332 y Fk(^)-37 b Fi(\027)t Fh(2)p Fm(Z)1396 307 y Ff(l)p Fa(\000)p Fe(1)1321 387 y Fk(^)h Fi(\027)t Fh(6)p Fk(=0)1532 140 y Fg(X)1521 318 y Fi(\027)1554 327 y Ff(l)1579 318 y Fh(2)p Fm(Z)1521 378 y Fi(\027)1554 387 y Ff(l)1578 378 y Fh(6)p Fk(=0)1686 219 y Fj(Leb)1818 152 y Fg(\010)1865 219 y Fj(\013)24 b Fl(\025)e Fq(\012)37 b(:)2172 123 y Fg(\014)2172 173 y(\014)2172 223 y(\014)2200 219 y Fq(\()p Fj(\013)19 b Fq(+)2397 163 y Fj(!)i Fl(\001)j Fq(^)-47 b Fj(\027)p 2397 200 161 4 v 2444 276 a(\027)2485 298 y Fi(l)2568 219 y Fq(\))2600 123 y Fg(\014)2600 173 y(\014)2600 223 y(\014)2651 219 y Fj(<)2775 163 y Fq(\012)p 2749 200 113 4 v 2749 276 a Fl(j)p Fj(\027)2813 298 y Fi(l)2838 276 y Fl(j)2972 163 y Fq(1)p 2881 200 224 4 v 2881 276 a Fj(E)5 b Fq(\()p Fl(j)p Fj(\027)g Fl(j)p Fq(\))3114 152 y Fg(\011)3727 219 y Fq(\(3)p Fj(:)p Fq(5\))118 604 y(T)-7 b(o)30 b(get)h(\(3.5\))f(let's)h(consider)e(a)h(cen)n(tered) g(in)n(terv)-5 b(al)30 b(of)h(lengh)n(t)2186 571 y Fk(\012)p 2161 585 97 4 v 2161 632 a Fh(j)p Fi(\027)2214 641 y Ff(l)2238 632 y Fh(j)2352 571 y Fk(1)p 2278 585 181 4 v 2278 632 a Fi(E)s Fk(\()p Fh(j)p Fi(\027)t Fh(j)p Fk(\))2499 604 y Fq(around)f(eac)n(h)f(v)-5 b(alue)3201 571 y Fi(!)r Fh(\001)t Fk(^)-37 b Fi(\027)p 3201 585 101 4 v 3222 632 a(\027)3255 641 y Ff(l)3342 604 y Fq(\()p Fj(\027)3415 626 y Fi(l)3469 604 y Fl(6)p Fq(=)27 b(0\).)46 b(F)-7 b(or)30 b(a)118 756 y(giv)n(en)e(v)n(ector)g Fj(\027)i Fl(2)c Fn(Z)800 720 y Fi(l)819 756 y Fj(;)j(Leb)1003 689 y Fg(\010)1051 756 y Fj(\013)c Fl(\025)g Fq(\012)39 b(:)1366 685 y Fg(\014)1366 735 y(\014)1394 756 y Fq(\()p Fj(\013)20 b Fq(+)1593 723 y Fi(!)r Fh(\001)t Fk(^)-37 b Fi(\027)p 1593 737 V 1614 785 a(\027)1647 794 y Ff(l)1703 756 y Fq(\))1735 685 y Fg(\014)1735 735 y(\014)1788 756 y Fj(<)1913 723 y Fk(\012)p 1888 737 97 4 v 1888 785 a Fh(j)p Fi(\027)1941 794 y Ff(l)1965 785 y Fh(j)2078 723 y Fk(1)p 2005 737 181 4 v 2005 785 a Fi(E)s Fk(\()p Fh(j)p Fi(\027)t Fh(j)p Fk(\))2195 689 y Fg(\011)2273 756 y Fq(is)28 b(an)h(upp)r(er)g(b)r(ound)g(on)g(the)g(measure)f(of)g (the)i(set)118 905 y(of)k(the)g Fj(\013)421 869 y Fh(0)445 905 y Fj(s)f Fq(suc)n(h)h(that)g(the)g(v)n(ector)f(\()p Fj(!)s(;)14 b(\013)p Fq(\))p Fj(;)34 b Fq(m)n(ultiplied)h(b)n(y)e Fl(j)p Fj(\027)5 b Fl(j)p Fj(;)34 b Fq(has)g(pro)5 b(jection)32 b(o)n(v)n(er)g Fj(\027)39 b Fq(b)r(ounded)c(b)n(y)3602 872 y Fk(\012)p 3535 886 V 3535 933 a Fi(E)s Fk(\()p Fh(j)p Fi(\027)t Fh(j)p Fk(\))3759 905 y Fq(\(i.e.)118 1054 y(\()p Fj(!)s(;)14 b(\013)p Fq(\))32 b(ma)n(y)f(b)r(e)h(resonan)n (t)e(with)i Fj(\027)37 b Fq(or)31 b(non{resonan)n(t)e(up)j(to)g(a)f (size)g(giv)n(en)g(b)n(y)2771 1021 y Fk(\012)p 2704 1035 V 2704 1082 a Fi(E)s Fk(\()p Fh(j)p Fi(\027)t Fh(j)p Fk(\))2894 1054 y Fq(\).)50 b(If)32 b(for)f(for)g(some)g Fj(\027)37 b Fq(w)n(e)31 b(had)118 1147 y Fg(\014)118 1197 y(\014)146 1217 y Fq(\()p Fj(!)s(;)14 b(\013)p Fq(\))22 b Fl(\001)g Fj(\027)468 1147 y Fg(\014)468 1197 y(\014)496 1167 y Fh(\000)p Fk(1)616 1217 y Fl(\024)31 b Fq(\012)772 1182 y Fh(\000)p Fk(1)861 1217 y Fj(E)5 b Fq(\()p Fl(j)p Fj(\027)g Fl(j)p Fq(\))1083 1150 y Fg(\011)1132 1217 y Fj(;)33 b Fq(then)g(summing)f(o)n(v)n(er)f(all)h Fj(\027)37 b Fl(2)31 b Fn(Z)2270 1182 y Fi(l)p Fh(\000)p Fk(1)2375 1217 y Fj(;)i(\027)2472 1240 y Fi(l)2528 1217 y Fl(6)p Fq(=)e(0)p Fj(;)h Fq(w)n(e)g(are)g(considering)f(more)g(v)n(ectors)118 1334 y(than)d(those)f(satis\014yng)g(\(3.4\))g(so)g(that)h(the)g(sum)g (in)f(\(3.5\))h(is)f(an)h(upp)r(er)f(b)r(ound)h(of)g Fj(Leb)p Fq(\()p Fl(A)3008 1298 y Fi(c)3008 1356 y(l)3041 1334 y Fq(\))717 1593 y Fj(Leb)p Fq(\()p Fl(A)947 1557 y Fi(c)947 1615 y(l)981 1593 y Fq(\))23 b Fl(\024)1183 1514 y Fg(X)1137 1706 y Fk(^)-36 b Fi(\027)t Fh(2)p Fm(Z)1260 1681 y Ff(l)p Fa(\000)p Fe(1)1186 1761 y Fk(^)f Fi(\027)t Fh(6)p Fk(=0)1463 1514 y Fg(X)1385 1696 y Fi(\027)1418 1705 y Ff(l)1443 1696 y Fh(2)p Fm(Z)-11 b Fh(nf)p Fk(0)p Fh(g)1423 1749 y Ff(!)q Fa(\001)t Fe(^)-32 b Ff(\027)p 1423 1758 90 4 v 1441 1792 a(\027)1471 1807 y(l)1522 1772 y Fh(\025)1584 1749 y Fe(\012)p 1584 1758 40 4 v 1590 1792 a(2)1701 1537 y Fq(2\012)p 1695 1574 113 4 v 1695 1650 a Fl(j)p Fj(\027)1759 1672 y Fi(l)1785 1650 y Fl(j)1818 1593 y Fj(e)1857 1557 y Fh(\000j)p Fi(\027)t Fh(j)1986 1532 y Ff(\034)2022 1557 y Fi(=\034)2121 1593 y Fl(\024)2267 1514 y Fg(X)2222 1706 y Fk(^)-37 b Fi(\027)t Fh(2)p Fm(Z)2345 1681 y Ff(l)p Fa(\000)p Fe(1)2270 1761 y Fk(^)h Fi(\027)t Fh(6)p Fk(=0)2548 1514 y Fg(X)2470 1696 y Fi(\027)2503 1705 y Ff(l)2527 1696 y Fh(2)p Fm(Z)-10 b Fh(nf)p Fk(0)p Fh(g)2507 1749 y Ff(!)q Fa(\001)t Fe(^)-32 b Ff(\027)p 2507 1758 90 4 v 2525 1792 a(\027)2555 1807 y(l)2607 1772 y Fh(\025)2669 1749 y Fe(\012)p 2669 1758 40 4 v 2675 1792 a(2)2785 1537 y Fq(2\012)p 2780 1574 113 4 v 2780 1650 a Fl(j)p Fj(\027)2844 1672 y Fi(l)2869 1650 y Fl(j)2902 1593 y Fj(e)2941 1557 y Fh(\000k)p Fi(\027)t Fh(k)3098 1532 y Ff(\034)3135 1557 y Fi(=\034)3234 1593 y Fl(\024)740 1982 y(\024)887 1903 y Fg(X)842 2096 y Fk(^)-37 b Fi(\027)t Fh(2)p Fm(Z)965 2071 y Ff(l)p Fa(\000)p Fe(1)890 2163 y Fk(^)g Fi(\027)5 b Fh(6)p Fk(=)975 2151 y(^)975 2163 y(0)1142 1903 y Fg(X)1131 2082 y Fi(\027)1164 2091 y Ff(l)1188 2082 y Fh(\025)p Fk(1)1100 2133 y Fa(j)p Ff(\027)1149 2145 y Fe(1)1181 2133 y Fa(j)p 1100 2153 101 4 v 1123 2185 a Ff(\027)1153 2200 y(l)1210 2165 y Fh(\025)1272 2143 y Ff(a)p 1272 2152 33 4 v 1274 2185 a Fe(2)1360 1926 y Fq(4)p 1347 1963 67 4 v 1347 2039 a Fj(\027)1388 2062 y Fi(l)1424 1982 y Fq(\012)p Fj(e)1523 1947 y Fh(\000k)p Fi(\027)t Fh(k)1680 1921 y Ff(\034)1717 1947 y Fi(=\034)1815 1982 y Fl(\024)1962 1903 y Fg(X)1917 2096 y Fk(^)-37 b Fi(\027)t Fh(2)p Fm(Z)2040 2071 y Ff(l)p Fa(\000)p Fe(1)1965 2163 y Fk(^)g Fi(\027)5 b Fh(6)p Fk(=)2050 2151 y(^)2050 2163 y(0)2209 1903 y Fg(X)2198 2082 y Fi(\027)2231 2091 y Ff(l)2255 2082 y Fh(\025)p Fk(1)2165 2146 y Fh(j)p Fi(\027)2218 2154 y Fe(1)2250 2146 y Fh(j\025)2332 2124 y Ff(a)p 2332 2133 33 4 v 2334 2166 a Fe(2)2420 1926 y Fq(4)p 2408 1963 67 4 v 2408 2039 a Fj(\027)2449 2062 y Fi(l)2484 1982 y Fq(\012)p Fj(e)2583 1947 y Fh(\000k)p Fi(\027)t Fh(k)2740 1921 y Ff(\034)2777 1947 y Fi(=\034)3727 1865 y Fq(\(3)p Fj(:)p Fq(6\))118 2405 y(The)28 b(condition)664 2373 y Fi(!)r Fh(\001)t Fk(^)-37 b Fi(\027)p 664 2387 101 4 v 685 2434 a(\027)718 2443 y Ff(l)798 2405 y Fl(\025)896 2373 y Fk(\012)p 896 2387 48 4 v 903 2434 a(2)981 2405 y Fq(comes)27 b(from)h Fj(\013)23 b Fl(\025)g Fq(\012)p Fj(:)28 b Fq(Imp)r(osing)g (in)g(fact)g Fl(\000)2395 2373 y Fi(!)r Fh(\001)t Fk(^)-37 b Fi(\027)p 2394 2387 101 4 v 2416 2434 a(\027)2449 2443 y Ff(l)2524 2405 y Fq(+)2732 2373 y Fk(\012)p 2617 2387 278 4 v 2617 2434 a Fh(j)p Fi(\027)2670 2443 y Ff(l)2694 2434 y Fh(j)p Fi(E)s Fk(\()p Fh(j)t Fk(^)g Fi(\027)s Fh(j)p Fk(\))2927 2405 y Fl(\025)23 b Fq(\012)28 b(w)n(e)g(\014nd)g(a)f (bigger)g(v)-5 b(alue)118 2555 y(of)25 b(the)h(measure.)35 b Fl(\000)781 2522 y Fi(!)r Fh(\001)t Fk(^)-37 b Fi(\027)p 780 2536 101 4 v 802 2583 a(\027)835 2592 y Ff(l)914 2555 y Fl(\025)23 b(\000)1191 2522 y Fk(\012)p 1076 2536 278 4 v 1076 2583 a Fh(j)p Fi(\027)1129 2592 y Ff(l)1154 2583 y Fh(j)p Fi(E)s Fk(\()p Fh(j)t Fk(^)-37 b Fi(\027)s Fh(j)p Fk(\))1377 2555 y Fq(+)14 b(\012)25 b(is)g(implied)h(b)n(y)f Fl(\000)2103 2522 y Fi(!)r Fh(\001)t Fk(^)-37 b Fi(\027)p 2102 2536 101 4 v 2124 2583 a(\027)2157 2592 y Ff(l)2236 2555 y Fl(\025)23 b Fq(\012\(1)14 b Fl(\000)2611 2522 y Fk(1)p 2559 2536 137 4 v 2559 2583 a Fi(E)s Fk(\(1\))2706 2555 y Fq(\))24 b Fl(\025)2859 2522 y Fk(\012)p 2859 2536 48 4 v 2866 2583 a(2)2917 2555 y Fj(:)h Fq(Moreo)n(v)n(er)d (rotating)j(in)g(suc)n(h)118 2710 y(a)30 b(w)n(a)n(y)f(that)i Fj(R)608 2675 y Fi(T)660 2710 y Fj(e)d Fq(=)f Fj(e)858 2675 y Fk(\(1\))975 2710 y Fq(=)g(\(1)p Fj(;)14 b Fq(0)p Fj(;)g(:)g(:)g(:)f(;)h Fq(0\))28 b Fl(2)g Fn(R)1643 2675 y Fi(l)p Fh(\000)p Fk(1)1759 2710 y Fj(;)j(e)d Fq(=)2016 2676 y Fi(!)2058 2684 y Ff(o)p 1982 2691 144 4 v 1982 2739 a Fh(k)p Fi(!)2058 2747 y Ff(o)2091 2739 y Fh(k)2166 2710 y Fq(and)i(calling)g Fj(a)2688 2663 y(:)2668 2710 y Fq(=)2818 2678 y Fk(\012)p 2770 2692 V 2770 2739 a Fh(k)p Fi(!)2846 2747 y Ff(o)2879 2739 y Fh(k)2923 2710 y Fj(;)h Fq(w)n(e)f(obtain)g(the)h(last)f(b)r(ound)118 2869 y(in)j(\(3.6\).)51 b(Being)31 b(\()735 2807 y Fg(P)823 2827 y Fi(l)823 2894 y(i)p Fk(=1)949 2869 y Fj(k)995 2833 y Fk(2)992 2891 y Fi(i)1032 2869 y Fq(\))1064 2833 y Fi(\034)1137 2869 y Fl(\025)f(j)p Fj(k)1298 2891 y Fk(1)1336 2869 y Fl(j)1359 2833 y Fi(\034)1422 2869 y Fq(+)21 b Fj(C)6 b Fq(\()1605 2807 y Fg(P)1693 2827 y Fi(l)1693 2894 y(i)p Fk(=2)1819 2869 y Fj(k)1865 2833 y Fk(2)1862 2891 y Fi(i)1902 2869 y Fq(\))1934 2833 y Fi(\034)2008 2869 y Fq(for)32 b(all)g Fj(k)2303 2891 y Fk(1)2372 2869 y Fl(\025)e Fq(0)p Fj(;)2564 2807 y Fg(P)2652 2827 y Fi(l)2652 2894 y(i)p Fk(=2)2777 2869 y Fl(j)p Fj(k)2843 2891 y Fi(i)2871 2869 y Fl(j)h(\025)g Fq(1)h Fj(i)f Fq(=)f(2)p Fj(;)14 b(:)g(:)g(:)f(;)h(l)r(;)32 b Fq(and)g(for)g(a)118 2985 y(suitable)c Fj(C)q(;)g Fq(w)n(e)f(ha)n(v)n (e)g(\([)918 2952 y Fi(a)p 918 2966 37 4 v 919 3014 a Fk(2)964 2985 y Fq(])c(=)g Fj(A)p Fq(\))1068 3242 y Fj(Leb)p Fq(\()p Fl(A)1298 3206 y Fi(c)1298 3264 y(l)1331 3242 y Fq(\))h Fl(\024)1533 3163 y Fg(X)1488 3355 y Fk(^)-37 b Fi(\027)t Fh(2)p Fm(Z)1611 3330 y Ff(l)p Fa(\000)p Fe(1)1536 3422 y Fk(^)g Fi(\027)5 b Fh(6)p Fk(=)1621 3410 y(^)1621 3422 y(0)1779 3163 y Fg(X)1768 3341 y Fi(\027)1801 3350 y Ff(l)1825 3341 y Fh(\025)p Fk(1)1736 3406 y Fh(j)p Fi(\027)1789 3414 y Fe(1)1821 3406 y Fh(j\025)p Fi(A)1967 3242 y Fq(4\012)p Fj(e)2108 3202 y Fh(\000)p Fk(\()p Fh(j)p Fi(\027)2239 3210 y Fe(1)2270 3202 y Fh(j)2290 3177 y Ff(\034)2327 3202 y Fk(+)p Fi(C)t Fk(\()2456 3146 y Fg(P)2543 3166 y Ff(l)2543 3233 y(i)p Fe(=2)2652 3202 y Fh(j)p Fi(\027)2705 3210 y Ff(i)2731 3202 y Fh(j)2751 3177 y Fe(2)2783 3202 y Fk(\))2809 3177 y Ff(\034)2846 3202 y Fk(\))p Fi(=\034)3727 3242 y Fq(\(3)p Fj(:)p Fq(7\))581 3605 y Fg(X)536 3787 y Fh(j)p Fi(\027)589 3795 y Fe(1)621 3787 y Fh(j\025)703 3764 y Ff(a)p 703 3773 33 4 v 705 3807 a Fe(2)759 3684 y Fj(e)798 3648 y Fh(\000j)p Fi(\027)903 3656 y Fe(1)935 3648 y Fh(j)955 3623 y Ff(\034)992 3648 y Fi(=\034)1090 3684 y Fl(\024)23 b Fq(2)1234 3571 y Fg(Z)1316 3591 y Fh(1)1280 3759 y Fi(A)1401 3684 y Fj(dxe)1530 3648 y Fh(\000)p Fi(x)1620 3623 y Ff(\034)1657 3648 y Fi(=\034)1755 3684 y Fq(=)g(2)p Fj(\034)1940 3621 y Fe(1)p Fa(\000)p Ff(\034)p 1940 3635 107 4 v 1977 3668 a(\034)2061 3684 y Fj(e)2100 3648 y Fh(\000)p Fi(A)2202 3623 y Ff(\034)2238 3648 y Fi(=\034)2328 3571 y Fg(Z)2411 3591 y Fh(1)2374 3759 y Fk(0)2495 3684 y Fj(dx)p Fq(\()p Fj(x)d Fq(+)2779 3627 y(1)p 2777 3664 46 4 v 2777 3741 a Fj(\034)2832 3684 y(A)2894 3648 y Fi(\034)2936 3684 y Fq(\))2978 3621 y Fe(1)p Fa(\000)p Ff(\034)p 2978 3635 107 4 v 3015 3668 a(\034)3099 3684 y Fj(e)3138 3648 y Fh(\000)p Fi(x)3254 3684 y Fl(\024)545 4009 y(\024)j Fq(2)p Fj(\034)730 3947 y Fe(1)p Fa(\000)p Ff(\034)p 730 3961 V 767 3993 a(\034)851 4009 y Fj(e)890 3973 y Fh(\000)p Fi(A)992 3948 y Ff(\034)1028 3973 y Fi(=\034)1104 3917 y Fg(n)1159 3896 y(Z)1242 3917 y Fi(A)1292 3892 y Ff(\034)1329 3917 y Fi(=\034)1205 4085 y Fk(0)1419 4009 y Fj(dx)p Fq(\()1553 3953 y(2)p 1551 3990 46 4 v 1551 4066 a Fj(\034)1607 4009 y(A)1669 3973 y Fi(\034)1711 4009 y Fq(\))1753 3947 y Fe(1)p Fa(\000)p Ff(\034)p 1753 3961 107 4 v 1790 3993 a(\034)1874 4009 y Fj(e)1913 3973 y Fh(\000)p Fi(x)2024 4009 y Fq(+)2107 3896 y Fg(Z)2191 3917 y Fh(1)2154 4085 y Fi(A)2204 4068 y Ff(\034)2241 4085 y Fi(=\034)2316 4009 y Fq(\(2)p Fj(x)p Fq(\))2479 3947 y Fe(1)p Fa(\000)p Ff(\034)p 2480 3961 V 2516 3993 a(\034)2600 4009 y Fj(e)2639 3973 y Fh(\000)p Fi(x)2733 3917 y Fg(o)2811 4009 y Fl(\024)p 2899 3942 66 4 v 23 w Fj(C)6 b(A)3026 3973 y Fk(1)p Fh(\000)p Fi(\034)3153 4009 y Fj(e)3192 3973 y Fh(\000)p Fi(A)3294 3948 y Ff(\034)3330 3973 y Fi(=\034)3429 4009 y Fq(=)545 4249 y(=)23 b Fj(C)692 4272 y Fk(1)730 4249 y Fq([)763 4193 y Fj(a)p 763 4230 44 4 v 764 4306 a Fq(2)816 4249 y(])839 4214 y Fk(1)p Fh(\000)p Fi(\034)966 4249 y Fj(e)1005 4214 y Fh(\000)1069 4191 y Fe(1)p 1066 4200 34 4 v 1066 4234 a Ff(\034)1109 4214 y Fk([)1138 4191 y Ff(a)p 1138 4200 33 4 v 1140 4234 a Fe(2)1180 4214 y Fk(])1199 4189 y Ff(\034)118 4498 y Fq(the)37 b(rest)f(of)g(\(3.7\))g(is)h(b)r(ounded)g(b)n(y)1316 4436 y Fg(P)1403 4523 y Fi(k)q Fh(2)p Fm(Z)1530 4506 y Ff(l)o Fa(\000)p Fe(1)1639 4498 y Fj(e)1678 4458 y Fh(\000)p Fi(C)t Fk(\()1808 4402 y Fg(P)1895 4423 y Ff(l)p Fa(\000)p Fe(1)1895 4489 y Ff(i)p Fe(=1)2003 4458 y Fh(j)p Fi(k)2058 4466 y Ff(i)2085 4458 y Fh(j)2105 4433 y Fe(2)2137 4458 y Fk(\))2163 4433 y Ff(\034)2200 4458 y Fi(=\034)2313 4498 y Fl(\024)h Fj(C)2475 4521 y Fk(2)2512 4498 y Fq(2)2554 4462 y Fi(c)2600 4440 y Ff(l)p 2593 4449 34 4 v 2593 4482 a(\034)2651 4452 y Fk(\000\()2727 4426 y Ff(l)p Fa(\000)p Fe(1)p 2727 4440 94 4 v 2757 4472 a Ff(\034)2831 4452 y Fk(\))p 2651 4479 207 4 v 2651 4538 a(\000\()2727 4511 y Ff(l)p Fa(\000)p Fe(1)p 2727 4525 94 4 v 2760 4558 a(2)2831 4538 y Fk(\))2866 4498 y Fj(:)f Fq(Reuniting)g(all)f(the) h(previous)118 4648 y(b)r(ounds)28 b(and)f(estimates)h(w)n(e)f(get)822 4923 y Fj(Leb)p Fq(\()p Fl(A)1052 4887 y Fi(c)1052 4945 y(l)1085 4923 y Fq(\))c Fl(\024)g Fq(4\012)p Fj(C)1389 4945 y Fk(1)1426 4923 y Fq([)1459 4866 y Fj(a)p 1459 4904 44 4 v 1460 4980 a Fq(2)1513 4923 y(])1536 4887 y Fk(1)p Fh(\000)p Fi(\034)1662 4923 y Fj(e)1701 4887 y Fh(\000)1765 4865 y Fe(1)p 1763 4874 34 4 v 1763 4907 a Ff(\034)1806 4887 y Fk([)1834 4865 y Ff(a)p 1834 4874 33 4 v 1836 4907 a Fe(2)1877 4887 y Fk(])1896 4862 y Ff(\034)1937 4923 y Fj(C)1996 4945 y Fk(2)2033 4923 y Fq(2)2075 4887 y Fi(c)2121 4865 y Ff(l)p 2115 4874 34 4 v 2115 4907 a(\034)2172 4862 y Fq(\000\()2266 4829 y Fi(l)p Fh(\000)p Fk(1)p 2266 4843 107 4 v 2300 4890 a Fi(\034)2382 4862 y Fq(\))p 2172 4904 243 4 v 2172 4987 a(\000\()2266 4954 y Fi(l)p Fh(\000)p Fk(1)p 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3071 y Fi(l)1071 3079 y Fe(0)1241 3046 y Fq(+)1342 3025 y(~)1324 3046 y Fj(P)39 b Fq(are)27 b(of)h(order)e(\012)1927 3011 y Fh(\000)p Fk(1)2044 3046 y Fq(and)i(\012)g(is)g(a)f(\(fast\))h(increasing)f (function)h(of)g(the)g(n)n(um)n(b)r(er)118 3200 y Fj(l)i Fq(of)e(d.o.f..)38 b(Then)28 b(a)g(further)g(canonical)e (transformation)h(reduces)g(the)h(p)r(erturbation)g(to)g(order)e(\012) 3315 3164 y Fh(\000)p Fk(2)3400 3131 y Ff(n)3437 3146 y(l)3457 3158 y Fe(1)3502 3200 y Fj(:)i Fq(Of)g(course)118 3317 y(the)35 b(pro)r(ceedure)e(w)n(orks)f(if)j(\012)f(is)g(su\016cien) n(tly)g(large.)56 b(A)n(t)34 b(this)h(p)r(oin)n(t)f(w)n(e)g(can)g(con)n (tin)n(ue)f(the)i(pro)r(cedure)e(\\adding")118 3435 y(more)24 b(and)g(more)f(d.o.f..)36 b(In)25 b(the)g(limit)g Fj(k)h Fl(!)d Fq(+)p Fl(1)h Fq(w)n(e)g(get)g(the)h(in\014nite{dimensional)f (torus)f(whose)h(v)n(ector)f(frequency)118 3552 y(\(Diophan)n(tine\))34 b(is)g(\()792 3518 y Fi(!)834 3526 y Ff(o)p 792 3533 76 4 v 810 3581 a Fi(\026)877 3552 y Fj(;)14 b(!)966 3575 y Fi(l)992 3552 y Fj(;)g(!)1081 3575 y Fi(l)1102 3583 y Fe(1)1138 3552 y Fj(;)g(:)g(:)g(:)o Fq(\);)35 b Fl(j)p Fj(!)1436 3575 y Fi(l)1461 3552 y Fl(j)e(\025)g Fq(\012)1675 3575 y Fi(l)1734 3552 y Fq(\(\012)1826 3575 y Fi(l)1886 3552 y Fq(is)h(the)g(previous)f(\012\),)i Fl(j)p Fj(!)2687 3575 y Fi(l)2708 3583 y Fe(1)2744 3552 y Fl(j)f(\025)f Fq(\012)2959 3575 y Fi(l)2980 3583 y Fe(1)3016 3552 y Fj(;)h Fl(j)p Fj(!)3148 3575 y Fi(l)3169 3583 y Fe(2)3206 3552 y Fl(j)f(\025)g Fq(\012)3420 3575 y Fi(l)3441 3583 y Fe(2)3511 3552 y Fq(and)h(so)f(on.)118 3670 y(The)27 b(v)n(ector)f(\()579 3636 y Fi(!)621 3644 y Ff(o)p 579 3651 V 597 3698 a Fi(\026)664 3670 y Fj(;)14 b Fq(\012)761 3692 y Fi(l)787 3670 y Fj(;)g Fq(\012)884 3692 y Fi(l)905 3700 y Fe(1)941 3670 y Fj(;)g(:)g(:)g(:)p Fq(\))27 b Fd(may)j(not)f(b)l(e)e Fq(a)g(Diophan)n(tine)g(v)n(ector)e (b)r(ecause)i(it)g(ma)n(y)g(not)g(satisfy)f(the)i(irrationalit)n(y)118 3787 y(conditions)h(\(see)g(De\014nition)g(3.3\))g(ev)n(en)f(if)i(its)f (comp)r(onen)n(ts)g(are)f(su\016cien)n(tly)h(large.)40 b(A)29 b(priori)f(nothing)h(forbids)f(the)118 3905 y(v)n(ector)j(\()414 3871 y Fi(!)456 3879 y Ff(o)p 414 3886 V 431 3933 a Fi(\026)499 3905 y Fj(;)14 b Fq(\012)596 3927 y Fi(l)621 3905 y Fj(;)g Fq(\012)718 3927 y Fi(l)739 3935 y Fe(1)776 3905 y Fj(;)g(:)g(:)g(:)p Fq(\))32 b(from)f(b)r(eing)h(Diophan)n(tine.)50 b(What)32 b(w)n(e)f(kno)n(w)g(b)n(y)h(Lemma)f(3.4)g(\(applied)i(for)e(eac)n(h)g (v)-5 b(alue)118 4057 y(of)30 b Fj(k)g Fq(=)c(0)p Fj(;)14 b Fq(1)p Fj(;)g Fq(2)p Fj(;)g(:)g(:)g(:)n Fq(\))30 b(is)g(that)g (outside)g(the)g(in)n(terv)-5 b(al)30 b([\012)1861 4080 y Fi(l)1882 4089 y Ff(k)1922 4057 y Fj(;)14 b Fq(\012)2019 4080 y Fi(l)2040 4089 y Ff(k)2100 4057 y Fq(+)20 b Fj(L)p Fq(\()p Fj(\034)5 b(;)14 b(l)2377 4080 y Fi(k)2417 4057 y Fq(\))p Fl(k)p Fj(!)2546 4021 y Fk(\()p Fi(l)2593 4030 y Ff(k)q Fa(\000)p Fe(1)2702 4021 y Fk(\))2732 4057 y Fl(k)p Fq(])p Fj(;)30 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b(what)f(increases)g(faster)g(is)h Fj(L)p Fq(\()p Fj(\034)5 b(;)14 b(l)2020 4796 y Fi(k)2060 4774 y Fq(\))p Fl(k)p Fj(!)2189 4738 y Fk(\()p Fi(l)2236 4747 y Ff(k)q Fa(\000)p Fe(1)2345 4738 y Fh(k)p Fk(\))2438 4774 y Fq(but)30 b(this)f Fd(do)l(es)i(not)g(forbid)g Fq(the)e(comp)r(onen)n(ts)118 4891 y Fj(!)170 4914 y Fi(l)191 4923 y Ff(k)259 4891 y Fq(from)e(b)r(eing)h(close)f(to)g(\012)1039 4914 y Fi(l)1060 4923 y Ff(k)1128 4891 y Fq(\(p)r(ossibly)h(equal\).)36 b(Let's)28 b(de\014ne)g Fj(M)9 b Fq(\()p Fj(\016)s Fq(\))2460 4844 y Fj(:)2439 4891 y Fq(=)36 b(sup)2665 4914 y Fi(t)p Fh(\025)p Fk(1)2793 4891 y Fj(E)5 b Fq(\()p Fj(t)p Fq(\))p Fj(e)2992 4855 y Fh(\000)p Fi(t\016)3129 4891 y Fq(=)23 b(exp)3344 4824 y Fg(\010)3402 4858 y Fk(1)p Fh(\000)p Fi(\034)p 3402 4872 123 4 v 3444 4920 a(\034)3534 4891 y Fj(\016)3598 4829 y Fa(\000)p Ff(\034)p 3584 4843 107 4 v 3584 4875 a Fe(1)p Fa(\000)p Ff(\034)3705 4824 y Fg(\011)3753 4891 y Fj(:)118 5069 y Fp(Lemma)29 b(4.2)165 b Fb(W)-7 b(e)28 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Fj(l)20 b Fl(\000)e Fq(1\))2316 1415 y Fk(16)p 2316 1429 67 4 v 2316 1477 a Fi(\032\016)2392 1448 y Fj(\015)h Fl(j)p Fj(\026)p Fl(j)14 b(V)7 b Fj(M)i Fq(\()p Fj(\016)s Fq(\))23 b Fl(\024)f Fq(1)929 1694 y(max)p Fl(f)1137 1638 y Fq(4)p 1136 1675 43 4 v 1136 1751 a Fj(\032)1188 1694 y Fl(k)p Fq(\004)p Fl(k)1337 1689 y Ff(\032)p 1337 1704 31 4 v 1338 1737 a Fe(4)1377 1717 y Fi(;\030)r Fh(\000)p Fk(3)p Fi(\016)1564 1694 y Fj(;)28 b Fl(k)p Fq(\001)p Fl(k)1778 1689 y Ff(\032)p 1777 1704 V 1778 1737 a Fe(4)1817 1717 y Fi(;\030)r Fh(\000)p Fk(3)p Fi(\016)1990 1694 y Fl(g)23 b(\024)f Fq(8\()p Fj(l)e Fl(\000)e Fq(1\))2429 1638 y(1)p 2428 1675 43 4 v 2428 1751 a Fj(\032)2481 1694 y Fl(V)7 b Fj(M)i Fq(\()p Fj(\016)s Fq(\))p Fj(\015)c Fl(j)p Fj(\026)p Fl(j)2934 1647 y Fj(:)2913 1694 y Fq(=)23 b Fj(T)3050 1717 y Fi(o)3727 1694 y Fq(\(4)p Fj(:)p Fq(4\))118 2038 y Fd(R)l(emarks)83 b Fq(i\))27 b(The)g(lemma)g(mak)n(es)f(clearer)f(\(in)j(an)e(tractable) 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3635 2482 76 4 v 3652 2530 a Fi(\026)3720 2501 y Fq(\))h(are)118 2617 y(forced)27 b(to)h(b)r(ecome)f(greater)f(and)h (greater.)118 2781 y Fd(Pr)l(o)l(of)104 b Fq(It)20 b(is)f(quite)h (standard)e(so)h(w)n(e)g(just)h(sk)n(etc)n(h)f(it.)35 b(As)19 b(usual)h(one)f(lo)r(oks)f(for)h(the)h(solution)f(of)g(the)h (Hamilton{Jacobi)118 2896 y(equation)1404 3076 y(\010\()p Fj(A)1558 3040 y Fh(0)1582 3076 y Fj(;)14 b(')p Fq(\))24 b(=)e Fl(\000)1964 2997 y Fg(X)1915 3190 y Fi(\027)t Fh(2)p Fm(Z)2041 3164 y Ff(l)p Fa(\000)p Fe(1)1905 3237 y(0)p Ff(<)p Fa(j)p Ff(\027)s Fa(j\024)p Ff(N)2178 3020 y Fj(e)2217 2984 y Fi(i\027)t Fh(\001)p Fi(')2344 3020 y Fj(V)2392 3042 y Fi(\027)p 2176 3057 260 4 v 2176 3133 a Fj(i\027)i Fl(\001)18 b Fj(h)2359 3155 y Fi(A)2409 3139 y Fa(0)3727 3076 y Fq(\(4)p Fj(:)p Fq(5\))118 3406 y(kno)n(wing)24 b(that)h Fj(h)668 3429 y Fi(A)722 3406 y Fq(\()p Fj(A)816 3371 y Fi(o)853 3406 y Fq(\))f(=)e(\()1038 3374 y Fi(!)1082 3349 y Ff(o)p 1038 3388 78 4 v 1057 3435 a Fi(\026)1126 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Fq(;)1448 1373 y Fk(~)-39 b Fi(\032)p 1442 1391 V 1443 1438 a Fk(4)1487 1410 y Fq(\))18 b Fl(\002)f Fq(\001)1695 1420 y Fk(~)1688 1436 y Fi(\030)s Fh(\000)p Fk(3)1809 1420 y(~)1806 1436 y Fi(\016)1865 1410 y Fl(3)24 b Fq(\()p Fj(A)2038 1374 y Fh(0)2062 1410 y Fj(;)14 b(')2153 1374 y Fh(0)2176 1410 y Fq(\))p Fj(;)2267 1388 y Fq(~)2259 1410 y Fj(\030)22 b Fl(\000)17 b Fq(3)2445 1388 y(~)2441 1410 y Fj(\016)26 b(>)d Fq(0)p Fj(;)k(N)k Fq(=)2896 1377 y Fk(1)p 2880 1391 66 4 v 2880 1448 a(2)2916 1433 y(~)2913 1448 y Fi(\016)2969 1410 y Fq(ln)3038 1342 y Fg(\002)3136 1377 y Fk(\012)p 3083 1391 154 4 v 3083 1448 a Fi(M)6 b Fk(\()3181 1433 y(~)3178 1448 y Fi(\016)s Fk(\))3360 1373 y(~)-39 b Fi(\032)p 3257 1391 230 4 v 3257 1442 a Fk(8)p Fi(C)3338 1450 y Ff(o)3371 1442 y Fh(V)3418 1425 y Fe(\()p Ff(l)p Fe(\))3496 1342 y Fg(\003)3558 1410 y Fb(suc)n(h)27 b(that)1096 1643 y Fq(~)1074 1664 y Fj(H)1143 1687 y Fi(l)1164 1695 y Fe(1)1201 1664 y Fq(\()1248 1643 y(~)1233 1664 y Fl(C)5 b Fq(\()p Fj(A)1376 1629 y Fh(0)1400 1664 y Fj(;)14 b(')1491 1629 y Fh(0)1514 1664 y Fq(\)\))1623 1617 y Fj(:)1602 1664 y Fq(=)22 b Fj(H)1765 1621 y Fk(\(0\))1758 1690 y Fi(l)1779 1698 y Fe(1)1854 1664 y Fq(\()p Fj(A)1948 1629 y Fh(0)1972 1664 y Fj(;)14 b(')2063 1629 y Fh(0)2087 1664 y Fq(\))23 b(=)2230 1643 y(~)2230 1664 y Fj(h)2278 1687 y Fi(l)2299 1695 y Fe(1)2335 1664 y Fq(\()p Fj(A)2429 1629 y Fh(0)2453 1664 y Fq(\))c(+)2599 1643 y(~)2587 1664 y Fj(V)2653 1629 y Fh(0)2677 1664 y Fq(\()p Fj(A)2771 1629 y Fh(0)2795 1664 y Fj(;)14 b(')2886 1629 y Fh(0)2909 1664 y Fq(\))744 b(\(4)p Fj(:)p Fq(17\))891 1890 y(16)p 891 1927 84 4 v 919 2003 a(~)-49 b Fj(\032)984 1946 y Fl(k)1039 1925 y Fq(~)1026 1946 y Fj(V)1093 1911 y Fh(0)1074 1969 y Fi(')1118 1952 y Fa(0)1145 1946 y Fl(k)28 b Fe(~)-34 b Ff(\032)p 1196 1961 57 4 v 1196 1993 a Fe(16)1263 1973 y Fi(;)1290 1958 y Fk(~)1283 1973 y Fi(\030)r Fh(\000)p Fk(4)1403 1958 y(~)1400 1973 y Fi(\016)1454 1946 y Fq(+)18 b Fl(k)1592 1925 y Fq(~)1579 1946 y Fj(V)1646 1911 y Fh(0)1627 1969 y Fi(A)1677 1952 y Fa(0)1704 1946 y Fl(k)28 b Fe(~)-33 b Ff(\032)p 1755 1961 V 1755 1993 a Fe(16)1822 1973 y Fi(;)1849 1958 y Fk(~)1842 1973 y Fi(\030)r Fh(\000)p Fk(4)1962 1958 y(~)1959 1973 y Fi(\016)2018 1946 y Fl(\024)2137 1890 y Fq(1)p 2116 1927 83 4 v 2120 1993 a(~)2116 2014 y Fj(\016)10 b Fq(~)-49 b Fj(\032)2209 1946 y Fq(2)2251 1911 y Fk(11)2321 1946 y Fq(\()p Fj(C)2412 1969 y Fi(o)2449 1946 y Fl(V)2507 1911 y Fk(\()p Fi(l)p Fk(\))2584 1946 y Fq(\))2616 1911 y Fk(2)2664 1890 y Fj(M)9 b Fq(\()2790 1868 y(~)2786 1890 y Fj(\016)s Fq(\))p 2664 1927 195 4 v 2717 2003 a(~)-50 b Fj(\032)p Fq(\012)2912 1899 y Fj(:)2891 1946 y Fq(=)2991 1925 y(~)2979 1946 y Fj(V)3045 1911 y Fk(\()p Fi(o)p Fk(\))3027 1969 y Fi(x)3685 1946 y Fq(\(4)p Fj(:)p Fq(18\))118 2172 y Fb(pro)n(vided)27 b(that)581 2436 y Fq(2)623 2400 y Fk(6)669 2380 y Fj(M)9 b Fq(\()795 2358 y(~)791 2380 y Fj(\016)s Fq(\))p 669 2417 V 750 2482 a(~)746 2504 y Fj(\016)922 2380 y(l)p 883 2417 103 4 v 891 2493 a Fq(~)-50 b Fj(\032)p Fq(\012)996 2436 y(\(4)p Fj(C)1129 2459 y Fi(o)1166 2436 y Fl(V)1224 2400 y Fk(\()p Fi(l)p Fk(\))1301 2436 y Fq(\))24 b Fl(\024)e Fq(1)p Fj(;)1712 2380 y(a)1756 2402 y Fi(o)1801 2380 y Fq(~)-50 b Fj(\032)p 1699 2417 151 4 v 1703 2482 a Fq(~)1699 2504 y Fj(\016)s(\033)s Fq(\012)1859 2369 y Fg(\000)2097 2380 y Fq(~)g Fj(\032)p Fq(\012)p 1907 2417 468 4 v 1907 2504 a(8)p Fj(M)9 b Fq(\()2075 2482 y(~)2071 2504 y Fj(\016)s Fq(\))p Fj(C)2202 2527 y Fi(o)2239 2504 y Fl(V)2297 2480 y Fk(\()p Fi(l)p Fk(\))2384 2369 y Fg(\001)2422 2386 y Fi(\033)2481 2436 y Fq(exp)2621 2319 y Fg(\032)2816 2380 y Fq(2)p 2694 2417 286 4 v 2694 2504 a(\()p Fj(\033)2780 2482 y Fq(~)2776 2504 y Fj(\016)s Fq(\))2895 2446 y Ff(\034)p 2859 2455 107 4 v 2859 2488 a Fe(1)p Fa(\000)p Ff(\034)2999 2380 y Fq(1)18 b Fl(\000)g Fj(\034)p 2999 2417 189 4 v 3071 2493 a(\034)3198 2319 y Fg(\033)3283 2436 y Fl(\024)23 b Fq(1)p Fj(;)581 2718 y Fq(2)623 2683 y Fk(6)669 2662 y Fj(M)9 b Fq(\()795 2640 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Fl(!)h Fq(+)p Fl(1)k Fq(implies)g Fj(n)23 b Fl(!)h Fq(+)p Fl(1)j Fq(to)r(o)h(and)f(w)n(e)h (\014nd)g(the)118 5047 y(in)n(v)-5 b(arian)n(t)29 b(torus)g(with)h(v)n (ector{frequency)1526 5013 y Fi(!)1568 5021 y Ff(o)p 1526 5028 76 4 v 1544 5076 a Fi(\026)1641 5047 y Fq(whic)n(h)g(exists)f (b)n(y)h(Corollary)d(4.5.)42 b(The)30 b(Hamilton{Jacobi)e(equation)118 5197 y(\(4.20\))f(do)r(es)g(not)h(in)n(v)n(olv)n(e)985 5176 y(~)972 5197 y Fj(V)1020 5219 y Fk(1)1085 5197 y Fq(b)r(ecause)f(of)h(its)g(sup)r(erexp)r(onen)n(tial)f(deca)n(y)-7 b(.)1966 5396 y(12)p eop %%Page: 13 13 13 12 bop 118 -35 a Fq(W)-7 b(e)29 b(are)f(no)n(w)h(in)g(the)g(same)f (situation)h(after)f(Lemma)h(4.2)f(but)i(with)f(one)g(more)f(d.o.f.)41 b(\()p Fj(l)r Fq(\))28 b(and)h(pro)r(ceed)f(as)h(follo)n(ws.)118 114 y(F)-7 b(or)32 b(eac)n(h)h Fj(k)h Fl(\025)e Fq(0)g(w)n(e)h(ha)n(v)n (e)e(\(the)j(n)n(um)n(b)r(er)f(of)g(d.o.f.)53 b(is)32 b Fj(l)24 b Fl(\000)d Fq(1)h(+)f Fj(k)s Fq(\))33 b(:)84 b(1\))32 b(a)h(sequence)f(of)h(hamiltonians)f Fl(f)p Fj(H)3699 62 y Fk(\()p Fi(n)3766 71 y Ff(l)3786 86 y(k)3826 62 y Fk(\))3692 139 y Fi(l)3713 148 y Ff(k)3856 114 y Fl(g)118 277 y Fq(whose)d(domain)g(is)g Fj(D)r Fq(\()p Fj(A)912 225 y Fk(\()p Fi(n)979 234 y Ff(l)999 249 y(k)1040 225 y Fk(\))912 287 y Fi(o)1070 277 y Fq(;)14 b Fj(\032)1150 241 y Fk(\()p Fi(l)1197 250 y Ff(k)1234 241 y Fk(\))1150 299 y Fi(n)1191 308 y Ff(l)1211 323 y(k)1264 277 y Fq(\))20 b Fl(\002)f Fq(\001)1469 314 y Fi(\030)1501 280 y Fe(\()p Ff(l)1543 295 y(k)1580 280 y Fe(\))1499 325 y Ff(n)1536 340 y(l)1556 355 y(k)1611 277 y Fj(;)29 b(H)1739 225 y Fk(\()p Fi(n)1806 234 y Ff(l)1826 249 y(k)1867 225 y Fk(\))1732 302 y Fi(l)1753 311 y Ff(k)1897 277 y Fq(\()p Fj(A)1991 241 y Fk(\()p Fi(n)2058 250 y Ff(l)2078 265 y(k)2119 241 y Fk(\))2149 277 y Fj(;)14 b(')2240 241 y Fk(\()p Fi(n)2307 250 y Ff(l)2327 265 y(k)2367 241 y Fk(\))2397 277 y Fq(\))27 b(=)f Fj(h)2595 241 y Fk(\()p Fi(n)2662 250 y Ff(l)2682 265 y(k)2722 241 y 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Fh(j)p Fi(V)2905 3957 y Ff(\027)2942 3949 y Fk(\()p Fi(A)p Fk(\))p Fh(j)p 2846 3970 218 4 v 2864 4018 a(j)p Fi(h)2923 4026 y Ff(A)2969 4018 y Fh(\001)p Fi(\027)t Fh(j)3100 3989 y Fl(\024)25 b Fj(\015)5 b(M)k Fq(\()p Fj(\016)s Fq(\))p Fl(k)p Fj(V)3522 4011 y Fi(')3570 3989 y Fl(k)3612 4011 y Fi(\032;\030)3702 3989 y Fj(:)30 b Fq(The)118 4178 y(pro)r(ofs)d(are)f(standard)h (calculations)g(of)g(complex)g(analysis.)1966 5396 y(14)p eop %%Page: 15 15 15 14 bop 1739 -35 a Fr(References)12 201 y Fq([BF)n(G])42 b(Benettin)32 b(G.,)g(F)-7 b(r\177)-42 b(olic)n(h,)32 b(J.,)g(Giorgilli)f(A.:)45 b Fd(A)32 b(Nekhor)l(oshev{T)-6 b(yp)l(e)37 b(The)l(or)l(em)d(for)g(Hamiltonian)g(Systems)f(with)276 317 y(In\014nitely)c(Many)i(De)l(gr)l(e)l(es)e(of)i(F)-6 b(r)l(e)l(e)l(dom,)30 b Fq(Comm)n(un.)37 b(Math.)g(Ph)n(ys.)f Fp(119)27 b Fq(\(1988\),)f(95-108)27 434 y([CC1])41 b(Celletti,)31 b(A.,)g(Chierc)n(hia,)f(L.:)42 b Fd(Construction)31 b(of)i(A)n(nalytic) 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b(Or)l(der)g(Hamiltonian)g(Equations)h(on) f Fn(T)2696 1332 y Fh(1)2787 1367 y Fd(and)g(A)n(lmost-Perio)l(dic)i (Solutions)p Fq(,)276 1484 y(Journal)i(of)i(Di\013eren)n(tial)f (Equations)g(V)-7 b(ol.)p Fp(116)26 b Fq(No.1)h(1995,)f(172-201)3 1601 y([FSW])42 b(F)-7 b(r\177)-42 b(olic)n(h,)40 b(J.,)h(Sp)r(encer,)g (T.,)g(W)-7 b(a)n(yne,)40 b(C.E.:)58 b Fd(L)l(o)l(c)l(alization)41 b(in)f(Disor)l(der)l(e)l(d)g(Nonline)l(ar)g(Dynamic)l(al)h(Systems,)276 1717 y Fq(Jour.)36 b(of)27 b(Stat.)37 b(Ph)n(ys.)f Fp(42)27 b Fq(\(1986\),)g(247-274)123 1834 y([G])42 b(Galla)n(v)n(otti,)26 b(G.,)i Fd(The)j(Elements)f(of)g(Me)l(chanics)g Fq(Springer{V)-7 b(erlag)109 1951 y([Je])41 b(Jessen,)26 b(B.:)37 b Fd(The)30 b(The)l(ory)g(of)g(Inte)l(gr)l(ation)f(in)g(a)h(sp)l(ac)l(e)g(of)g(an)f (In\014nite)f(Numb)l(er)g(of)i(Dimensions)p Fq(,)e(Acta)f(mathe-)276 2067 y(matica)g Fp(63)g Fq(1934,)f(250{323)87 2184 y([Ke])41 b(Kelley)-7 b(,)27 b(J.:)37 b Fd(Gener)l(al)30 b(T)-6 b(op)l(olo)l(gy,)32 b Fq(V)-7 b(an)28 b(Nostrand{Reinhold,)e (Princeton,)h(New)h(Jersey)-7 b(,)26 b(1955)132 2301 y([P])41 b(P)n(erfetti,)25 b(P)-7 b(.:)35 b Fd(Hamiltonian)29 b(Equations)e(on)h Fn(T)1764 2266 y Fh(1)1861 2301 y Fd(and)g(almost{p)l(erio)l(dic)i(solutions)p Fq(,)c(pro)r(ceedings)e (of)h(the)g(\\In)n(ter-)276 2418 y(national)j(Conference)g(on)h (Dynamical)f(Systems)h(and)g(Di\013eren)n(tial)f(Equations")g(Ma)n(y)g (18{21)e(2000)h(Kennesa)n(w)276 2534 y(State)37 b(Univ)n(ersit)n(y)e (\(GA\))j(USA,)f(An)h(Added)f(V)-7 b(olume)37 b(T)-7 b(o)36 b Fb(Discrete)g(and)h(Con)n(tin)n(uous)f(Dynamical)g(Systems)276 2650 y Fq(2001,)26 b(303{309)49 2767 y([P\177)-42 b(o1])40 b(P\177)-42 b(osc)n(hel,)40 b(J.:)58 b Fd(Smal)t(l)41 b(Divisors)f(with)h(Sp)l(atial)f(Structur)l(e)e(in)i(In\014nite)f (Dimensional)i(Hamiltonian)g(Systems,)276 2883 y Fq(Comm)n(un.)c(Math.) g(Ph)n(ys.)e Fp(127)27 b Fq(\(1990\))g(351-393)91 3000 y([SZ])42 b(Salamon,)d(D.,)h(Zehnder,)g(E.:)56 b Fd(KAM)38 b(the)l(ory)i(in)e(c)l(on\014gur)l(ation)h(sp)l(ac)l(e,)j Fq(Commen)n(t.Math.)66 b(Helv.)p Fp(64)37 b Fq(\(1989\))276 3117 y(84-132)27 3234 y([W)-7 b(a1])41 b(W)-7 b(a)n(yne,)36 b(C.,)h(E.:)52 b Fd(The)37 b(KAM)g(The)l(ory)h(of)f(Systems)f(with)i (Short)e(R)l(ange)h(Inter)l(actions,)h(I)f(and)g(II,)g Fq(Comm)n(un.)276 3350 y(Math.)g(Ph)n(ys.)f Fp(96)27 b Fq(\(1984\))f(311-329)f(and)i(331-344)27 3467 y([W)-7 b(a2])41 b(W)-7 b(a)n(yne,)39 b(C.,)h(E.:)56 b Fd(On)38 b(the)h(Elimination)i(of)e(Non{R)l(esonant)g(Harmonics,)j Fq(Comm)n(un.)66 b(Math.)g(Ph)n(ys.)f Fp(103)276 3583 y Fq(451{386)24 b(\(1986\))27 3700 y([W)-7 b(a3])41 b(W)-7 b(a)n(yne,)25 b(C.,)h(E.:)36 b Fd(Bound)28 b(on)g(the)g(T)-6 b(r)l(aje)l(ctories)29 b(of)g(a)f(System)f(of)i(We)l(akly)g(Couple)l(d) g(R)l(otators)f Fq(Comm)n(un.)36 b(Math.)276 3817 y(Ph)n(ys.)g Fp(104)27 b Fq(21{36)e(\(1986\))67 3934 y([VB])42 b(Vittot,)28 b(M.,)g(Bellissard,)f(J.:)37 b Fd(Invariant)30 b(tori)g(for)h(an)f (in\014nite)f(lattic)l(e)h(of)h(c)l(ouple)l(d)g(r)l(otators,)f Fq(Rep)r(ort)d(Cen)n(tre)h(de)276 4050 y(Ph)n(ysique)e(Th)n(\023)-39 b(eorique)26 b(CNRS-Lumin)n(y)-7 b(,)28 b(Marseille)e(\(1985\))1966 5396 y(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF