This is a multi-part message in MIME format. ---------------9910011429919 Content-Type: text/plain; name="99-369.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-369.keywords" localization, almost Mathieu operator, quasiperiodic, Lyapunov exponents ---------------9910011429919 Content-Type: application/postscript; name="bab12.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bab12.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: bab12.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips bab12 %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.10.01:1120 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 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b(Th)m(us)38 b(\\a.e.)53 b Fo(!)t(;)17 b(\022)s Fp(")35 b(part)h(is)f(necessary)k(for)c(1)2986 3956 y Fn(o)3024 3992 y Fp(.)54 b(W)-8 b(e)36 b(b)s(eliev)m(e)328 4113 y(ho)m(w)m(ev)m(er)f(that)d(3)962 4077 y Fn(o)1033 4113 y Fp(holds)g(for)g(all)f Fo(!)t(;)17 b(\022)s(;)31 b Fp(and)i(2)2027 4077 y Fn(o)2098 4113 y Fp(for)f(all)e Fo(\022)36 b Fp(and)c(all)f(irrational)e Fo(!)t(:)474 4233 y Fp(What)d(Aubry-Andre)g(actually)e(pro)m(v)m(ed)j(for)d Fo(\025)k(>)f Fp(2)p Fo(;)e Fp(w)m(as)i(p)s(ositivit)m(y)c(of)i(Ly)m (apuno)m(v)328 4354 y(exp)s(onen)m(ts)32 b(whic)m(h,)f(b)m(y)g(the)g (general)e(Ishii-P)m(astur)h(theorem)g(\(see,)h(e.g.,)g([11]\))f (implies)328 4474 y(absence)f(of)e(absolutely)g(con)m(tin)m(uous)h(sp)s (ectrum)g(\(for)e(a.e.)42 b Fo(\022)s Fp(,)29 b(and,)g(b)m(y)f(a)f (recen)m(t)i(result)328 4594 y(of)22 b(Last-Simon)f([38)o(],)k(for)d (all)f Fo(\022)s(:)p Fp(\).)40 b(Their)23 b(pro)s(of)e(w)m(as)j(made)e (rigorous)g(b)m(y)h(Avron-Simon)328 4715 y([3])34 b(and)g(P)m (astur-Figotin[17)m(],)h(and)e(a)h(di\013eren)m(t)g(pro)s(of)f(follo)m (w)m(ed)g(from)g(Herman)g([27].)328 4835 y(Using)j(the)h (Dinaburg-Sinai)32 b(tec)m(hnique)38 b([13])e(Bellissard-Lima-T)-8 b(estard)33 b([6])j(pro)m(v)m(ed)328 4955 y(the)45 b(existence)h(of)e (some)g(pure)h(p)s(oin)m(t)f(sp)s(ectrum)h(for)f Fo(\025)g Fp(v)m(ery)i(large)d(\(and)i(of)f(some)1922 5214 y(1)p eop %%Page: 2 3 2 2 bop 328 631 a Fp(absolutely)31 b(con)m(tin)m(uous)h(sp)s(ectrum)g (for)f Fo(\025)h Fp(v)m(ery)h(small\))c(for)i(a.e.)43 b Fo(!)t(:)31 b Fp(The)i(\014rst)f(pro)s(ofs)328 751 y(of)40 b(complete)g(lo)s(calization)d(\(pure)k(p)s(oin)m(t)e(sp)s (ectrum)i(with)g(exp)s(onen)m(tially)e(deca)m(ying)328 872 y(eigenfunctions\))44 b(for)g Fo(\025)h Fp(large)f(\(a.e.)80 b Fo(!)t(;)17 b(\022)s Fp(\))43 b(are)i(due)g(to)g(Sinai)e([42)o(])i (and)g(F)-8 b(r\177)-49 b(ohlic)m(h-)328 992 y(Sp)s(encer-Witt)m(w)m (er)42 b([20)o(].)68 b(Later,)43 b(alternativ)m(e)c(KAM-t)m(yp)s(e)j (argumen)m(ts)e(for)g Fo(\025)h Fp(large)328 1112 y(w)m(ere)34 b(dev)m(elop)s(ed)g(b)m(y)g(Eliasson)e([16])h(and)g(Goldsh)m(tein)f ([22)o(].)45 b(F)-8 b(or)32 b Fo(\025)h Fp(v)m(ery)h(small,)d(pure)328 1233 y(absolutely)23 b(con)m(tin)m(uous)h(sp)s(ectrum)g(w)m(as)g (obtained)f(b)m(y)i(Ch)m(ulaevsky-Dely)m(on)g([8])e(using)328 1353 y(dualit)m(y)45 b(and)h(the)h(construction)f(of)f(Sinai)g([42)o (].)84 b(Alternativ)m(e)46 b(pro)s(ofs)f(w)m(ere)j(giv)m(en)328 1474 y(b)m(y)h(Elliasson)d([15])h(and)h(Goldsh)m(tein)f([22].)90 b(All)46 b(those)i(results)h(use)f(p)s(erturbativ)m(e)328 1594 y(argumen)m(ts)33 b(and)f(most)g(hold)g(for)g(larger)g(classes)h (of)f(quasip)s(erio)s(dic)f(p)s(oten)m(tials)h(\(1.1\))474 1714 y(The)40 b(\014rst)f(nonp)s(erturbativ)m(e)h(results)f(for)f Fo(\025)h(<)f Fp(2)h(w)m(ere)h(obtained)e(b)m(y)i(Last)f([36],)328 1835 y(who)28 b(pro)m(v)m(ed)i(the)e(existence)i(of)d(a)h(large)f (absolutely)g(con)m(tin)m(uous)i(comp)s(onen)m(t)f(for)f(an)m(y)328 1955 y Fo(!)g Fp(and)e(an)m(y)g Fo(\025)i(<)h Fp(2)c(\(and)g(all)e Fo(\022)s(;)i Fp(as)h(sho)m(wn)g(b)m(y)g(Gestesy-Simon)f([21])g(or)g (as)g(follo)m(ws)f(from)328 2076 y([38]\).)48 b(In)34 b([28])g(a)g(nonp)s(erturbativ)m(e)g(approac)m(h)h(to)e(lo)s (calization)e(w)m(as)k(dev)m(elop)s(ed)g(and)328 2196 y(pure)29 b(p)s(oin)m(t)e(sp)s(ectrum)h(w)m(as)h(pro)m(v)m(ed)g(for)f Fo(\025)g Fm(\025)g Fp(15)p Fo(:)f Fp(In)i([25)o(])f(Gordon)g(et)g(al.) 41 b(established)328 2316 y(the)24 b(dual)f(result:)39 b(pure)25 b(absolutely)e(con)m(tin)m(uous)h(sp)s(ectrum)h(for)e Fo(\025)28 b Fm(\024)g Fp(4)p Fo(=)p Fp(15)p Fo(:)23 b Fp(In)h([29)o(])g(the)328 2437 y(tec)m(hnique)i(of)e([28)o(])h(w)m (as)g(com)m(bined)f(with)g(dualit)m(y)f(and)i(the)f(result)h(of)f([36)o (])h(to)f(pro)m(v)m(e)h(the)328 2557 y(existence)34 b(of)d(a)h(large)f (pure)h(p)s(oin)m(t)f(comp)s(onen)m(t)h(for)g(an)m(y)h Fo(\025)27 b(>)h Fp(2)p Fo(;)k Fp(a.e.)43 b Fo(!)t(;)17 b(\022)s(:)31 b Fp(Ho)m(w)m(ev)m(er,)328 2677 y(neither)f([36)o(])g (nor)f([29])h(ruled)f(out)g(the)i(existence)g(of)e(singular-con)m(tin)m (uous)f(sp)s(ectrum.)328 2798 y(Th)m(us,)h(the)d(absence)h(of)e (singular)g(con)m(tin)m(uous)h(sp)s(ectrum)g(for)f(4)p Fo(=)p Fp(15)i Fo(<)h(\025)f(<)h Fp(15)p Fo(;)44 b(\025)27 b Fm(6)p Fp(=)h(2)p Fo(;)328 2918 y Fp(remained)k(an)g(issue,)h(and)g (it)f(is)g(solv)m(ed)h(in)f(the)h(presen)m(t)h(pap)s(er.)474 3039 y(W)-8 b(e)31 b(note)f(that)f(3)1111 3002 y Fn(o)1179 3039 y Fp(immediately)e(follo)m(ws)i(from)f(1)2326 3002 y Fn(o)2394 3039 y Fp(b)m(y)j(the)f(strong)g(v)m(ersion)g(of)g(du-)328 3159 y(alit)m(y)c([25],)j(whic)m(h)f(states)h(that)f(for)f(an)m(y)i (irrational)24 b Fo(!)t(;)j Fp(if)g Fo(H)2539 3184 y Fn(!)r(;)2618 3157 y Fe(4)p 2615 3169 37 4 v 2615 3210 a Fd(\025)2661 3184 y Fn(;\022)2748 3159 y Fp(has)h(only)f(pure)i(p)s (oin)m(t)328 3279 y(sp)s(ectrum)37 b(for)e(a.e.)55 b Fo(\022)s(;)36 b Fp(then)h Fo(H)1529 3294 y Fn(!)r(;\025;\022)1731 3279 y Fp(has)f(purely)h(absolutely)e(con)m(tin)m(uous)i(sp)s(ectrum) 328 3400 y(for)32 b(a.e.)44 b Fo(\022)s(:)32 b Fp(Therefore,)i(w)m(e)g (will)c(concen)m(trate)k(on)f(the)g(pro)s(of)e(of)h(1)2812 3364 y Fn(o)2850 3400 y Fp(.)474 3520 y(W)-8 b(e)24 b(in)m(tro)s(duce)f (the)g(arithmetic)e(conditions)h(on)h Fo(!)t(;)17 b(\022)26 b Fp(that)c(will)f(enable)i(us)h(to)e(pro)m(v)m(e)328 3641 y(p)s(oin)m(t)38 b(sp)s(ectrum.)62 b(W)-8 b(e)39 b(will)d(sa)m(y)k(that)f Fo(!)i Fp(is)e(Diophan)m(tine)e(if)g(there)j (exist)f Fo(c)p Fp(\()p Fo(!)t Fp(\))e Fo(>)h Fp(0)328 3761 y(and)33 b(1)27 b Fo(<)h(r)s Fp(\()p Fo(!)t Fp(\))e Fo(<)i Fm(1)k Fp(suc)m(h)i(that)1538 4016 y Fm(j)17 b Fp(sin)f(2)p Fo(\031)t(j)6 b(!)t Fm(j)26 b Fo(>)2135 3949 y(c)p Fp(\()p Fo(!)t Fp(\))p 2105 3993 241 4 v 2105 4085 a Fm(j)p Fo(j)6 b Fm(j)2207 4056 y Fn(r)r Fi(\()p Fn(!)r Fi(\))3365 4016 y Fp(\(1.2\))328 4267 y(for)33 b(all)f Fo(j)k Fm(6)p Fp(=)30 b(0)p Fo(:)k Fp(It)g(is)g(w)m(ell)f(kno)m (wn)i(that)f(a.e.)48 b Fo(!)37 b Fp(is)d(Diophan)m(tine.)46 b(W)-8 b(e)34 b(de\014ne)i(the)e(set)328 4387 y(of)e(resonan)m(t)h (phases:)328 4604 y(\002)27 b(=)h Fm(f)p Fo(\022)j Fp(:)55 b(the)33 b(relation)26 b Fm(j)17 b Fp(sin)f(2)p Fo(\031)t Fp(\()p Fo(\022)s Fp(+\()p Fo(j)6 b(=)p Fp(2\))p Fo(!)t Fp(\))p Fm(j)25 b Fo(<)j Fp(exp)q(\()p Fm(\000)p Fo(k)2583 4527 y Fe(1)p 2523 4539 150 4 v 2523 4580 a(2)p Fd(r)r Fe(\()p Fd(!)r Fe(\))2687 4604 y Fp(\))g(holds)k(for)g(in\014nitely)f (man)m(y)d Fo(k)3881 4562 y Fc(0)3904 4604 y Fp(s)q Fm(g)p Fo(:)3365 4724 y Fp(\(1.3\))328 4844 y(It)j(follo)m(ws)e(from)g(the)i (Borel-Can)m(telli)d(lemma)g(that)j(\002)f(has)h(zero)g(Leb)s(esgue)h (measure.)328 4965 y(It)g(is,)g(ho)m(w)m(ev)m(er,)j(a)d(dense)i Fo(G)1383 4980 y Fn(\016)1421 4965 y Fo(:)e Fp(No)m(w)h(w)m(e)h(can)e (form)m(ulate)f(a)h(more)g(detailed)f(v)m(ersion)i(of)1922 5214 y(2)p eop %%Page: 3 4 3 3 bop 328 631 a Fp(part)32 b(1)588 595 y Fn(o)659 631 y Fp(of)g(Theorem)h(1.)328 818 y Fg(THEOREM)j(2)49 b Ff(Supp)-5 b(ose)38 b Fo(!)j Ff(is)e(Diophantine,)e Fo(\022)49 b(=)-60 b Fm(2)34 b Fp(\002)p Fo(;)51 b(\025)34 b(>)g Fp(2)p Fo(:)39 b Ff(Then)e Fo(H)3222 833 y Fn(!)r(;\025;\022)3426 818 y Ff(has)328 938 y(only)e(pur)-5 b(e)35 b(p)-5 b(oint)34 b(sp)-5 b(e)g(ctrum)35 b(with)g(exp)-5 b(onential)5 b(ly)33 b(de)-5 b(c)g(aying)34 b(eigenfunctions.)328 1125 y Fp(Moreo)m(v)m(er,) f(it)d(is)g(p)s(ossible)g(to)h(sho)m(w)h(that)f(the)g(eigenfunctions)g (deca)m(y)h(at)f(exactly)g(the)328 1246 y(Ly)m(apuno)m(v)e(rate.)42 b(With)26 b(the)i(Ly)m(apuno)m(v)h(exp)s(onen)m(t)g Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))27 b(de\014ned)i(b)m(y)f(\(2.2\),)g(w)m(e)g(ha)m (v)m(e)328 1366 y(the)33 b(follo)m(wing)328 1553 y Fg(THEOREM)j(3)49 b Ff(With)38 b Fo(!)t(;)17 b(\025;)g(\022)40 b Ff(as)e(in)f(The)-5 b(or)g(em)37 b(2,)h(for)g(any)g(eigenvalue)f Fo(E)44 b Ff(of)37 b(op-)328 1674 y(er)-5 b(ator)35 b Fo(H)698 1689 y Fn(!)r(;\025;\022)898 1674 y Ff(the)g(c)-5 b(orr)g(esp)g(onding) 33 b(eigenfunction)h Fp(\011)2354 1689 y Fn(E)2448 1674 y Ff(satis\014es)1159 1937 y Fp(lim)1115 2000 y Fc(j)p Fn(n)p Fc(j!1)1365 1869 y Fp(ln)o(\(\011)1560 1833 y Fi(2)1560 1894 y Fn(E)1619 1869 y Fp(\()p Fo(n)p Fp(\))23 b(+)f(\011)1950 1833 y Fi(2)1950 1894 y Fn(E)2009 1869 y Fp(\()p Fo(n)g Fp(+)g(1\)\))p 1365 1913 986 4 v 1776 2005 a(2)p Fm(j)p Fo(n)p Fm(j)2388 1937 y Fp(=)27 b Fm(\000)p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))587 b(\(1.4\))328 2188 y Fg(Remarks.)447 2375 y Fp(1.)49 b(As)34 b(discussed)i(ab)s(o)m(v)m (e,)g(Theorem)e(2)g(is)g(not)g(true)g(for)g(Liouville)d Fo(!)t(;)i Fp(nor)h(do)s(es)h(it)572 2495 y(hold)d(for)g(ev)m(ery)j Fo(\022)h Fp(for)d(a)f(giv)m(en)i(Diophan)m(tine)d Fo(!)t(:)h Fp(F)-8 b(urthermore,)33 b(it)f(is)g(p)s(ossible)572 2616 y(to)46 b(obtain)g(a)g(complete)g(result)h(on)f(lo)s(calization)d (that)k(is)f(true)h(for)f(all)f(irra-)572 2736 y(tional)25 b Fo(!)32 b Fp(\(new)c(critical)e(constan)m(ts)j(where)g(sp)s(ectrum)f (c)m(hanges)h(from)e(singular-)572 2857 y(con)m(tin)m(uous)45 b(to)g(pure)g(p)s(oin)m(t)f(will)f(app)s(ear,)k(dep)s(ending)e(on)g (the)g(exp)s(onen)m(tial)572 2977 y(rate)32 b(of)g(appro)m(ximation)f (of)h Fo(!)k Fp(b)m(y)d(rationals)e([31)o(]\).)447 3175 y(2.)49 b(The)39 b(set)h(\002)p Fo(;)e Fp(although)f(quite)i(small,)f (is)g(not)g(optimal)e(here.)62 b(One)39 b(can)g(mak)m(e)572 3322 y(it)32 b(smaller)f(b)m(y)i(replacing)f Fo(e)1605 3286 y Fc(\000)p Fn(k)1724 3235 y Fe(1)p 1709 3244 61 4 v 1709 3279 a(2)p Fd(r)1821 3322 y Fp(in)g(\(1.3\))g(b)m(y)i Fo(e)2349 3286 y Fc(\000)p Fn(k)r(\017)2508 3322 y Fp(for)f(small)d Fo(\017;)k Fp(ho)m(w)m(ev)m(er)h(that)572 3443 y(requires)24 b(a)g(more)f(delicate)f(argumen)m(t.)41 b(On)23 b(the)h(other)g(hand,)i (replacing)c(it)h(with)572 3563 y Fo(e)617 3527 y Fc(\000)p Fn(k)r(\017)782 3563 y Fp(for)38 b(su\016cien)m(tly)h(large)f Fo(\017;)h Fp(will)d(lead)j(to)f(singular-con)m(tin)m(uous)f(sp)s (ectrum)572 3683 y([35)o(].)48 b(It)34 b(w)m(as)h(conjectured)g(in)e ([30])g(that)h(there)h(is)e(a)h(sharp)g(treshhold)g(in)f Fo(\017)h Fp(for)572 3804 y(the)28 b(transition)e(from)h(pure)h(p)s (oin)m(t)f(to)g(singular-con)m(tin)m(uous)g(sp)s(ectrum.)42 b(In)28 b(an)m(y)572 3924 y(case,)33 b(w)m(e)h(did)e(not)g(try)h(to)g (optimize)d(\002)j(in)e(the)i(presen)m(t)i(pro)s(of.)447 4122 y(3.)49 b(W)-8 b(e)25 b(pro)m(v)m(e)h(Theorems)g(2)e(and)h(3)g (for)f(an)m(y)i Fo(\025)f Fp(suc)m(h)h(that)f(the)g(Ly)m(apuno)m(v)h (exp)s(onen)m(t)572 4242 y Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))44 b Fo(>)g Fp(0)e(for)g(all)e Fo(E)6 b(:)42 b Fp(W)-8 b(e)43 b(b)s(eliev)m(e)f(that)g(under)h(suc)m(h)h(condition)c (\(and)j Fo(!)t(;)17 b(\022)572 4363 y Fp(as)42 b(in)g(Theorem)g(2\))g (this)g(result)g(should)g(b)s(e)h(true)g(for)e(op)s(erators)h(\(1.1\))g (with)572 4483 y(rather)35 b(general)g(quasip)s(erio)s(dic)f(p)s(oten)m (tials)g Fo(f)11 b Fp(\()p Fo(\022)s Fp(\))p Fo(;)36 b Fp(just)g(lik)m(e)e(p)s(ositivit)m(y)g(of)h(the)572 4604 y(Ly)m(apuno)m(v)27 b(exp)s(onen)m(t)g(implies)d(zero)s (dimensionalit)m(y)f(of)i(sp)s(ectral)h(measures)g(for)572 4724 y(all)k Fo(!)t(;)17 b(\022)34 b Fp([34)o(].)43 b(While)31 b(p)s(ositivit)m(y)g(of)g(the)h(Ly)m(apuno)m(v)i(exp)s(onen)m(ts)f(can) f(easily)g(b)s(e)572 4844 y(extracted)j(from)e(the)h(presen)m(t)i(pro)s (of,)d(w)m(e)i(only)f(use)h(it,)e(rather)h(than)g(pro)m(v)m(e)i(it.)572 4965 y(Nor)c(do)h(w)m(e)g(use)h(the)f Fo(\025)27 b(>)h Fp(2)k(condition)f(otherwise.)1922 5214 y(3)p eop %%Page: 4 5 4 4 bop 474 631 a Fp(In)22 b(Section)g(2)f(w)m(e)i(pro)m(v)m(e)g (Theorems)g(2)e(and)h(3)f(up)h(to)g(the)g(main)e(tec)m(hnical)h(Lemma)g (4.)328 751 y(W)-8 b(e)39 b(pro)m(v)m(e)h(Lemma)d(4)h(in)g(Section)g(3) h(based)g(on)g(t)m(w)m(o)g(other)g(statemen)m(ts,)i(Lemma)c(5)328 872 y(and)f(Lemma)e(7.)51 b(The)37 b(latter)d(Lemmas)h(are)g(pro)m(v)m (ed,)j(corresp)s(ondingly)-8 b(,)36 b(in)e(Sections)328 992 y(4)29 b(and)g(5.)42 b(In)30 b(Sections)g(4)f(and)g(5)g(w)m(e)h (also)f(form)m(ulate)e(and)j(pro)m(v)m(e)g(some)g(Lemmas)e(that)328 1112 y(are)i(more)g(general)g(than)g(needed)i(for)e(the)g(presen)m(t)i (pro)s(of.)42 b(W)-8 b(e)31 b(b)s(eliev)m(e)f(they)h(ma)m(y)g(b)s(e)328 1233 y(useful)i(in)e(certain)i(other)f(situations.)474 1353 y(The)46 b(author)e(is)g(grateful)f(to)h(the)h(referee)h(for)e (the)h(suggestions)g(that)f(led)g(to)g(a)328 1474 y(signi\014can)m(t)32 b(impro)m(v)m(emen)m(t)g(of)g(this)g(pap)s(er.)328 1806 y Fj(2)161 b(General)52 b(setup.)70 b(Pro)t(of)52 b(of)g(Theorems)e(2)i (and)g(3)328 2025 y Fg(De\014nition.)44 b Fp(A)33 b(formal)e(solution)h (\011)1749 2040 y Fn(E)1808 2025 y Fp(\()p Fo(x)p Fp(\))i(of)e(the)i (equation)f Fo(H)2733 2040 y Fn(!)r(;\025;\022)2899 2025 y Fp(\011)2975 2040 y Fn(E)3063 2025 y Fp(=)28 b Fo(E)6 b Fp(\011)3321 2040 y Fn(E)3414 2025 y Fp(will)328 2145 y(b)s(e)40 b(called)f Fg(a)46 b(generalized)g(eigenfunction)39 b Fp(if)g Fm(j)p Fp(\011)2339 2160 y Fn(E)2398 2145 y Fp(\()p Fo(x)p Fp(\))p Fm(j)h(\024)h Fo(C)7 b Fp(\(1)27 b(+)g Fm(j)p Fo(x)p Fm(j)p Fp(\))39 b(for)h(some)328 2266 y Fo(C)35 b Fp(=)27 b Fo(C)7 b Fp(\(\011)727 2281 y Fn(E)786 2266 y Fp(\))28 b Fo(<)g Fm(1)p Fp(.)474 2476 y(It)34 b(is)f(w)m(ell)f(kno)m(wn)j(that)e(to)g(pro)m(v)m(e)i(Theorem)f (2)f(it)f(su\016ces)k(to)d(pro)m(v)m(e)h(that)g(gener-)328 2596 y(alized)d(eigenfunctions)i(deca)m(y)h(exp)s(onen)m(tially)d([7,)i (41)o(].)474 2717 y(W)-8 b(e)36 b(will)e(use)i(the)g(notation)f Fo(G)1646 2732 y Fi([)p Fn(x)1706 2741 y Fe(1)1739 2732 y Fn(;x)1799 2741 y Fe(2)1833 2732 y Fi(])1857 2717 y Fp(\()p Fo(E)6 b Fp(\)\()p Fo(x;)17 b(y)t Fp(\))35 b(for)g(the)h (Green's)g(function)g(\()p Fo(H)31 b Fm(\000)328 2837 y Fo(E)6 b Fp(\))444 2801 y Fc(\000)p Fi(1)538 2837 y Fp(\()p Fo(x;)17 b(y)t Fp(\))35 b(of)f(the)i(op)s(erator)e Fo(H)1560 2852 y Fn(!)r(;\025;\022)1761 2837 y Fp(restricted)i(to)e (the)i(in)m(terv)-5 b(al)34 b([)p Fo(x)2927 2852 y Fi(1)2967 2837 y Fo(;)17 b(x)3066 2852 y Fi(2)3105 2837 y Fp(])36 b(with)e(zero)328 2958 y(b)s(oundary)44 b(conditions)e(at)h Fo(x)1440 2973 y Fi(1)1509 2958 y Fm(\000)30 b Fp(1)44 b(and)f Fo(x)1964 2973 y Fi(2)2033 2958 y Fp(+)30 b(1.)75 b(W)-8 b(e)44 b(no)m(w)g(\014x)g Fo(E)6 b(;)33 b(\025)46 b Fm(2)h Fh(R)54 b Fp(and)43 b Fo(!)328 3078 y Fp(satisfying)29 b(\(1.2\).)43 b(T)-8 b(o)30 b(simplify)e(notations,)i(in)g(some)g (cases)i(the)f Fo(E)6 b(;)17 b(\025;)g(!)t Fp(-dep)s(endence)328 3198 y(of)26 b(v)-5 b(arious)26 b(quan)m(tities)g(will)e(b)s(e)j (omitted.)40 b(W)-8 b(e)27 b(denote)g(the)g(one-step)g(transfer-matrix) 328 3319 y(of)32 b Fo(H)8 b Fp(\011)27 b(=)h Fo(E)6 b Fp(\011)32 b(b)m(y)1162 3577 y Fo(B)5 b Fp(\()p Fo(\022)s(;)17 b(E)6 b Fp(\))28 b(=)1618 3456 y Fb(\022)1696 3517 y Fo(E)g Fm(\000)23 b Fo(\025)17 b Fp(cos)f(2)p Fo(\031)t(\022)101 b Fm(\000)p Fp(1)1960 3637 y(1)400 b(0)2513 3456 y Fb(\023)2607 3577 y Fo(;)328 3835 y Fp(The)33 b Fo(k)s Fm(\000)p Fp(step)i (transfer-matrix)30 b(is)j(giv)m(en)f(b)m(y)779 4051 y Fo(M)873 4066 y Fn(k)917 4051 y Fp(\()p Fo(\022)s(;)17 b(E)6 b Fp(\))27 b(=)h Fo(B)5 b Fp(\()p Fo(\022)25 b Fp(+)d(\()p Fo(k)j Fm(\000)e Fp(1\))p Fo(!)t(;)17 b(E)6 b Fp(\))17 b Fm(\001)g(\001)g(\001)c Fo(B)5 b Fp(\()p Fo(\022)25 b Fp(+)d Fo(!)t(;)17 b(E)6 b Fp(\))p Fo(B)f Fp(\()p Fo(\022)s(;)17 b(E)6 b Fp(\))p Fo(:)328 4267 y Fp(Let)33 b(us)g(denote)1171 4425 y Fo(P)1234 4440 y Fn(k)1276 4425 y Fp(\()p Fo(\022)s(;)17 b(E)6 b Fp(\))28 b(=)f(det)1806 4279 y Fb(")1854 4425 y Fp(\()p Fo(H)1973 4440 y Fn(!)r(;\025;\022)2161 4425 y Fm(\000)22 b Fo(E)6 b Fp(\))2376 4301 y Fb(\014)2376 4350 y(\014)2376 4400 y(\014)2376 4450 y(\014)2404 4504 y Fi([0)p Fn(;k)r Fc(\000)p Fi(1])2631 4279 y Fb(#)2696 4425 y Fo(:)328 4655 y Fp(Then)34 b(the)f Fo(k)s Fm(\000)p Fp(step)h(transfer-matrix)d(can)i(b)s(e)f (written)h(as)912 4913 y Fo(M)1006 4928 y Fn(k)1049 4913 y Fp(\()p Fo(\022)s(;)17 b(E)6 b Fp(\))27 b(=)1426 4792 y Fb(\022)1549 4853 y Fo(P)1612 4868 y Fn(k)1654 4853 y Fp(\()p Fo(\022)s(;)17 b(E)6 b Fp(\))143 b Fm(\000)p Fo(P)2183 4868 y Fn(k)r Fc(\000)p Fi(1)2316 4853 y Fp(\()p Fo(\022)25 b Fp(+)d Fo(!)t(;)17 b(E)6 b Fp(\))1504 4973 y Fo(P)1567 4988 y Fn(k)r Fc(\000)p Fi(1)1699 4973 y Fp(\()p Fo(\022)s(;)17 b(E)6 b Fp(\))98 b Fm(\000)p Fo(P)2183 4988 y Fn(k)r Fc(\000)p Fi(2)2316 4973 y Fp(\()p Fo(\022)25 b Fp(+)d Fo(!)t(;)17 b(E)6 b Fp(\))2763 4792 y Fb(\023)2857 4913 y Fo(:)481 b Fp(\(2.1\))1922 5214 y(4)p eop %%Page: 5 6 5 5 bop 328 631 a Fp(The)33 b(Ly)m(apuno)m(v)i(exp)s(onen)m(t)f Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))32 b(is)g(de\014ned)i(as)619 908 y Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))27 b(=)52 b(lim)960 958 y Fn(n)p Fc(!1)1170 770 y Fb(R)1226 796 y Fi(1)1210 866 y(0)1282 841 y Fp(ln)16 b Fm(k)p Fo(M)1524 856 y Fn(k)1567 841 y Fp(\()p Fo(\022)s(;)h(E)6 b Fp(\))p Fm(k)p Fo(d\022)p 1170 885 791 4 v 1539 977 a(k)1999 908 y Fp(=)27 b(inf)2142 968 y Fn(k)2247 770 y Fb(R)2302 796 y Fi(1)2286 866 y(0)2358 841 y Fp(ln)16 b Fm(k)p Fo(M)2600 856 y Fn(k)2643 841 y Fp(\()p Fo(\022)s(;)h(E)6 b Fp(\))p Fm(k)p Fo(d\022)p 2247 885 V 2615 977 a(k)3047 908 y(:)291 b Fp(\(2.2\))328 1143 y(The)32 b(limit)c(exists)k(and)g(the)g(last)e (equalit)m(y)i(holds)f(b)m(y)h(the)g(subadditiv)m(e)f(ergo)s(dic)g (theo-)328 1264 y(rem)h(\(see,)i(e.g.,)f([11)o(]\).)474 1384 y(Let)i Fo(K)i Fp(=)30 b Fm(f)p Fo(k)k Fm(2)d Fh(N)45 b Fp(:)61 b(there)33 b(exists)e Fo(\022)j Fm(2)d Fp([0)p Fo(;)17 b Fp(1])30 b(with)g Fm(j)p Fo(P)2528 1399 y Fn(k)2570 1384 y Fp(\()p Fo(\022)s(;)17 b(E)6 b Fp(\))p Fm(j)30 b(\025)3021 1345 y Fi(1)p 2992 1361 95 4 v 2992 1370 a Fc(p)p 3050 1370 36 3 v 3050 1426 a Fi(2)3096 1384 y Fo(e)3141 1348 y Fn(k)r(\015)t Fi(\()p Fn(E)t Fi(\))3334 1384 y Fm(g)p Fo(:)k Fp(By)328 1517 y(\(2.1\))g(for)h(ev)m(ery)h Fo(k)f Fm(2)d Fh(N)50 b Fp(w)m(e)36 b(ha)m(v)m(e)g(that)f(at)f(least)h (one)g(of)f Fo(k)s(;)17 b(k)27 b Fp(+)c(1)p Fo(;)17 b(k)27 b Fp(+)c(2)35 b(b)s(elongs)f(to)328 1637 y Fo(K)r(:)328 1848 y Fg(De\014nition.)43 b Fp(Fix)32 b Fo(E)i Fm(2)29 b Fh(R)11 b Fo(;)44 b(m)29 b Fm(2)f Fh(R)11 b Fo(:)33 b Fp(A)g(p)s(oin)m(t)f Fo(y)f Fm(2)e Fh(Z)h Fp(will)h(b)s(e)i(called)e (\()p Fo(m;)17 b(k)s Fp(\)-regular)328 1968 y(if)31 b(there)j(exists)f (an)f(in)m(terv)-5 b(al)32 b([)p Fo(x)1506 1983 y Fi(1)1546 1968 y Fo(;)17 b(x)1645 1983 y Fi(2)1684 1968 y Fp(])p Fo(;)45 b(x)1838 1983 y Fi(2)1905 1968 y Fp(=)28 b Fo(x)2064 1983 y Fi(1)2126 1968 y Fp(+)22 b Fo(k)j Fm(\000)e Fp(1)p Fo(;)32 b Fp(con)m(taining)f Fo(y)t(;)h Fp(suc)m(h)i(that)722 2222 y Fm(j)p Fo(G)827 2237 y Fi([)p Fn(x)887 2246 y Fe(1)921 2237 y Fn(;x)981 2246 y Fe(2)1015 2237 y Fi(])1038 2222 y Fp(\()p Fo(y)t(;)17 b(x)1227 2237 y Fn(i)1255 2222 y Fp(\))p Fm(j)27 b Fo(<)h(e)1497 2181 y Fc(\000)p Fn(m)p Fc(j)p Fn(y)r Fc(\000)p Fn(x)1766 2191 y Fd(i)1792 2181 y Fc(j)1816 2222 y Fo(;)44 b Fp(and)56 b(dist)o(\()p Fo(y)t(;)17 b(x)2446 2237 y Fn(i)2474 2222 y Fp(\))27 b Fm(\025)2654 2155 y Fp(1)p 2654 2199 49 4 v 2654 2290 a(5)2713 2222 y Fo(k)s Fp(;)45 b Fo(i)28 b Fp(=)f(1)p Fo(;)17 b Fp(2)p Fo(:)328 2463 y Fp(Otherwise,)33 b Fo(y)j Fp(will)30 b(b)s(e)j(called)e(\()p Fo(m;)17 b(k)s Fp(\)-singular.)474 2583 y(This)31 b(de\014nition)e(will)f(b)s(e)i(useful)g(only)g(for)f(p) s(ositiv)m(e)h Fo(m;)h Fp(ho)m(w)m(ev)m(er,)i(it)c(can)h(b)s(e)h(used) 328 2703 y(formally)24 b(with)j(negativ)m(e)h Fo(m)f Fp(as)g(w)m(ell.)41 b(The)28 b(n)m(um)m(b)s(er)2319 2664 y Fi(1)p 2319 2680 36 4 v 2319 2738 a(5)2391 2703 y Fp(here)g(can)g(b)s (e)f(replaced,)i(equiv-)328 2824 y(alen)m(tly)-8 b(,)39 b(b)m(y)g(an)m(y)g(other)g(n)m(um)m(b)s(er)g(smaller)d(than)2213 2785 y Fi(1)p 2213 2801 V 2213 2858 a(4)2258 2824 y Fo(:)i Fp(The)i(main)d(di\013erence)i(b)s(et)m(w)m(een)328 2944 y(this)c(de\014nition)f(and)i(that)f(of)g(regularit)m(y)g(t)m(ypically) f(used)i(in)f(m)m(ultiscale)e(analysis)i(is)328 3065 y(the)i(\015exibilit)m(y)e(allo)m(w)m(ed)h(in)g(the)h(c)m(hoice)g(of)g (the)g(in)m(terv)-5 b(al)35 b([)p Fo(x)2596 3080 y Fi(1)2636 3065 y Fo(;)17 b(x)2735 3080 y Fi(2)2775 3065 y Fp(])p Fo(:)37 b Fp(It)g(is)f(going)f(to)i(b)s(e)328 3185 y(essen)m(tial)c (for)f(the)h(pro)s(of.)474 3395 y(It)c(is)g(w)m(ell)f(kno)m(wn)i(and)f (can)g(b)s(e)g(c)m(hec)m(k)m(ed)j(easily)d(that)f(v)-5 b(alues)29 b(of)g(an)m(y)g(formal)e(solu-)328 3516 y(tion)i(\011)g(of)h (the)g(equation)g Fo(H)8 b Fp(\011)27 b(=)g Fo(E)6 b Fp(\011)30 b(at)g(a)f(p)s(oin)m(t)g Fo(x)g Fm(2)f Fo(I)35 b Fm(\032)28 b Fh(Z)g Fp(can)i(b)s(e)g(reconstructed)328 3636 y(from)h(the)i(b)s(oundary)g(v)-5 b(alues)33 b(via)901 3877 y(\011\()p Fo(x)p Fp(\))28 b(=)f Fo(G)1316 3892 y Fn(I)1356 3877 y Fp(\()p Fo(x;)17 b(x)1548 3892 y Fi(1)1588 3877 y Fp(\)\011\()p Fo(x)1795 3892 y Fi(1)1857 3877 y Fm(\000)23 b Fp(1\))e(+)h Fo(G)2240 3892 y Fn(I)2280 3877 y Fp(\()p Fo(x;)17 b(x)2472 3892 y Fi(2)2512 3877 y Fp(\)\011\()p Fo(x)2719 3892 y Fi(2)2781 3877 y Fp(+)22 b(1\))p Fo(;)372 b Fp(\(2.3\))328 4047 y(where)43 b Fo(I)52 b Fp(=)44 b([)p Fo(x)916 4062 y Fi(1)956 4047 y Fo(;)17 b(x)1055 4062 y Fi(2)1094 4047 y Fp(])p Fo(:)42 b Fp(This)h(implies)c (that)j(if)f(\011)2159 4062 y Fn(E)2260 4047 y Fp(is)h(a)g(generalized) f(eigenfunction,)328 4167 y(then)d(ev)m(ery)i(p)s(oin)m(t)c Fo(y)k Fm(2)c Fh(Z)f Fp(with)j(\011)1675 4182 y Fn(E)1734 4167 y Fp(\()p Fo(y)t Fp(\))d Fm(6)p Fp(=)h(0)i(is)f(\()p Fo(m;)17 b(k)s Fp(\)-singular)35 b(for)i Fo(k)k Fp(su\016cien)m(tly)328 4288 y(large:)h Fo(k)31 b(>)d(k)841 4303 y Fi(1)880 4288 y Fp(\()p Fo(E)6 b(;)17 b(m;)g(\022)s(;)g(y)t Fp(\))p Fo(:)474 4408 y Fp(Theorem)33 b(2)f(will)f(no)m(w)i(follo)m(w)e(from) 328 4602 y Fg(LEMMA)37 b(4)49 b Ff(Supp)-5 b(ose)41 b Fo(\022)54 b(=)-61 b Fm(2)40 b Fp(\002)p Fo(;)17 b(!)44 b Ff(satis\014es)c(\(1.2\).)63 b(Then)41 b(for)f(every)i Fo(y)g Fm(2)e Fh(Z)m Fo(;)56 b(\017)40 b(>)328 4723 y Fp(0)p Fo(;)17 b(\013)60 b(<)g Fp(2)p Fo(;)53 b Ff(ther)-5 b(e)52 b(exists)g Fo(k)1403 4738 y Fi(2)1442 4723 y Fp(\()p Fo(\022)s(;)17 b(!)t(;)g(y)t(;)g(\017;)g(\013)q(;)g(E)6 b Fp(\))p Fo(;)50 b Ff(such)i(that)h(for)f(al)5 b(l)52 b Fo(k)63 b Fm(2)e Fo(K)r(;)77 b(k)63 b(>)328 4843 y(k)379 4858 y Fi(2)418 4843 y Fp(\()p Fo(\022)s(;)17 b(!)t(;)g(y)t(;)g(\017;)g (\013)q(;)g(E)6 b Fp(\))p Fo(;)32 b Ff(if)j Fo(x)g Ff(and)f Fo(y)k Ff(ar)-5 b(e)35 b(b)-5 b(oth)328 4964 y Fp(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))22 b Fm(\000)h Fo(\017;)17 b(k)s Fp(\))p Fm(\000)p Ff(singular)35 b(and)f Fm(j)p Fo(x)22 b Fm(\000)h Fo(y)t Fm(j)j Fo(>)1937 4924 y Fn(k)r Fi(+1)p 1937 4940 129 4 v 1983 4998 a(2)2075 4964 y Fo(;)35 b Ff(then)g Fm(j)p Fo(x)22 b Fm(\000)h Fo(y)t Fm(j)j Fo(>)i(k)2823 4927 y Fn(\013)2872 4964 y Fo(:)1922 5214 y Fp(5)p eop %%Page: 6 7 6 6 bop 328 631 a Fg(Remarks:)447 834 y Fp(1.)49 b(If)29 b(one)g(somewhat)g(relaxes)h(the)f(nonresonan)m(t)i(condition)c(on)i (the)h(excluded)g(set)572 955 y(\002)e(\(with)g(\002)h(still)d(ha)m (ving)i(measure)h(0)p Fo(;)g Fp(namely)e(replacing)h(the)h(sub)s(exp)s (onen)m(tial)572 1075 y(function)g(b)m(y)h(p)s(olynomial)c(deca)m(y)k (in)f(\(1.3\)\))g(one)g(can)h(obtain)e(exp)s(onen)m(tial)h(in)g Fo(k)572 1196 y Fp(repulsion)j(of)g(singular)f(clusters)i(in)f(Lemma)f (4.)447 1399 y(2.)49 b(This)e(Lemma)e(is)h(close)h(in)f(spirit)g(to)g (the)h(cen)m(tral)g(Lemmas)f(of)g(m)m(ulti-scale)572 1519 y(analysis)40 b(\([18];)46 b(see)c(also)e([19,)h(20,)g(14)o(]\))g (ho)m(w)m(ev)m(er)j(our)d(pro)s(of)f(of)h(it)f(do)s(es)i(not)572 1640 y(in)m(v)m(olv)m(e)33 b(m)m(ultiple)d(scales.)328 1843 y(T)-8 b(o)29 b(complete)g(the)g(pro)s(of)g(of)g(Theorem)g(2,)h(w) m(e)g(let)f Fo(E)6 b Fp(\()p Fo(\022)s Fp(\))29 b(b)s(e)h(a)f (generalized)f(eigen)m(v)-5 b(alue)328 1963 y(of)39 b Fo(H)527 1978 y Fn(!)r(;\025;\022)692 1963 y Fp(,)i(and)e(\011)1032 1978 y Fn(E)1092 1963 y Fp(\()p Fo(x)p Fp(\))g(the)h(corresp)s(onding)f (generalized)f(eigenfunction.)63 b(Assume)328 2084 y(\011\(0\))27 b Fm(6)p Fp(=)h(0)p Fo(:)k Fp(Then,)i(b)m(y)f(Lemma)f(4,)g(if)328 2204 y Fm(j)p Fo(x)p Fm(j)27 b Fo(>)h Fp(max)o(\()p Fo(k)840 2219 y Fi(1)880 2204 y Fp(\()p Fo(E)6 b(;)17 b(\015)5 b Fp(\()p Fo(E)h Fp(\))g Fm(\000)g Fo(\017;)17 b(\022)s(;)g Fp(0\))p Fo(;)g(k)1696 2219 y Fi(2)1734 2204 y Fp(\()p Fo(\022)s(;)g(!)t(;)g Fp(0)p Fo(;)g(\017;)g Fp(1)p Fo(:)p Fp(5)p Fo(;)g(E)6 b Fp(\)\))g(+)g(1)p Fo(;)22 b Fp(the)j(p)s(oin)m(t)e Fo(x)i Fp(is)g(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))g Fm(\000)328 2325 y Fo(\017;)17 b(k)s Fp(\))p Fm(\000)p Fp(regular)33 b(for)g(some)h Fo(k)e Fm(2)e(fj)p Fo(x)p Fm(j)22 b(\000)i Fp(1)p Fo(;)17 b Fm(j)p Fo(x)p Fm(j)p Fo(;)g Fm(j)p Fo(x)p Fm(j)k Fp(+)i(1)p Fm(g)g(\\)g Fo(K)36 b Fm(6)p Fp(=)30 b Fm(;)p Fo(;)j Fp(since)h(0)f(is)h(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))22 b Fm(\000)328 2445 y Fo(\017;)17 b(k)s Fp(\))p Fm(\000)p Fp(singular.)79 b(W)-8 b(e,)48 b(therefore,)g(obtain)43 b(that)i(there)g(exists)g(an)g (in)m(terv)-5 b(al)43 b([)p Fo(x)3360 2460 y Fi(1)3400 2445 y Fo(;)17 b(x)3499 2460 y Fi(2)3539 2445 y Fp(])328 2565 y(con)m(taining)31 b Fo(x;)i Fp(suc)m(h)h(that)338 2757 y(1)p 338 2801 49 4 v 338 2893 a(5)397 2824 y(\()p Fm(j)p Fo(x)p Fm(j)17 b(\000)i Fp(1\))27 b Fm(\024)h(j)p Fo(x)961 2839 y Fn(i)1007 2824 y Fm(\000)19 b Fo(x)p Fm(j)27 b(\024)1328 2757 y Fp(4)p 1328 2801 V 1328 2893 a(5)1387 2824 y(\()p Fm(j)p Fo(x)p Fm(j)18 b Fp(+)f(1\))p Fo(;)72 b Fm(j)p Fo(G)1938 2840 y Fi([)p Fn(x)1998 2849 y Fe(1)2032 2840 y Fn(;x)2092 2849 y Fe(2)2126 2840 y Fi(])2149 2824 y Fp(\()p Fo(x;)17 b(x)2341 2839 y Fn(i)2370 2824 y Fp(\))p Fm(j)27 b(\024)i Fo(e)2614 2783 y Fc(\000)p Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)p Fn(\017)p Fi(\))p Fc(j)p Fn(x)p Fc(\000)p Fn(x)3112 2793 y Fd(i)3138 2783 y Fc(j)3162 2824 y Fo(;)44 b(i)28 b Fp(=)f(1)p Fo(;)17 b Fp(2)p Fo(:)3365 2976 y Fp(\(2.4\))328 3117 y(Using)k(\(2.3\),)j(w)m (e)f(obtain)d(the)i(estimate:)38 b Fm(j)p Fp(\011)1953 3132 y Fn(E)2012 3117 y Fp(\()p Fo(x)p Fp(\))p Fm(j)27 b(\024)i Fp(2)p Fo(C)7 b Fp(\(\011)2544 3132 y Fn(E)2603 3117 y Fp(\)\(2)p Fm(j)p Fo(x)p Fm(j)p Fp(+1\))p Fo(e)3047 3080 y Fc(\000)p Fi(\()3139 3046 y Fd(\015)s Fe(\()p Fd(E)s Fe(\))p Fa(\000)p Fd(\017)p 3138 3065 206 4 v 3226 3106 a Fe(5)3353 3080 y Fi(\)\()p Fc(j)p Fn(x)p Fc(j\000)p Fi(1\))3609 3117 y Fo(:)328 3237 y Fp(Since,)34 b(for)g Fo(\025)c(>)f Fp(2)p Fo(;)34 b Fp(w)m(e)h(ha)m(v)m(e)g Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))30 b Fo(>)g Fp(0)k(for)f(all)f Fo(E)40 b Fp(\(note:)46 b(this)34 b(is)f(the)h(only)g(place)g(in)328 3358 y(the)c(pap)s(er)g(where)h(w)m(e)g(use)g(the)f(condition)f Fo(\025)e(>)h Fp(2\))p Fo(;)h Fp(this)h(implies)d(exp)s(onen)m(tial)i (deca)m(y)-8 b(.)p 328 3486 42 59 v 328 3698 a Ff(Pr)j(o)g(of)35 b(of)g(The)-5 b(or)g(em)35 b(3.)45 b Fp(The)34 b(fact)f(that)g(the)h (lo)m(w)m(er)f(limit)d(of)j(the)g(expression)i(in)d(\(1.4\))328 3819 y(is)41 b(alw)m(a)m(ys)h(not)g(lo)m(w)m(er)g(than)f Fm(\000)p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))p Fo(;)42 b Fp(is)g(a)f(general)g(statemen)m(t,)k(pro)m(v)m(ed)e(b)m(y)f(Craig-) 328 3939 y(Simon)28 b([10])h(for)g(an)m(y)h(quasip)s(erio)s(dic)e (\(with)h(a.e.)43 b Fo(\022)32 b Fp(form)d(true)g(ev)m(en)j(for)c(an)m (y)j(ergo)s(dic\))328 4059 y(p)s(oten)m(tial.)42 b(Therefore,)34 b(w)m(e)f(are)g(concerned)h(here)f(only)f(with)h(the)g(upp)s(er)g(b)s (ound.)474 4180 y(It)j(is)f(one)h(of)f(the)h(main)e(tec)m(hnical)i(p)s (oin)m(ts)f(of)g(m)m(ultiscale)f(analysis)h(that)g(the)h(ex-)328 4300 y(p)s(onen)m(tial)g(deca)m(y)i(of)f(a)g(Green's)h(function)e(at)h (a)g(scale)g Fo(k)j Fp(under)e(certain)f(conditions)328 4420 y(generates)h(exp)s(onen)m(tial)f(deca)m(y)i(with)e(the)h(same)f (rate)g(at)g(a)g(scale)h Fo(k)2922 4384 y Fn(\013)2971 4420 y Fo(:)f Fp(The)i(pro)s(of)d(is)328 4541 y(usually)45 b(done)h(using)g(blo)s(c)m(k-resolv)m(en)m(t)g(expansion,)k(with)45 b(the)h(com)m(binatorial)c(fac-)328 4661 y(tor)d(b)s(eing)g(killed)e(b) m(y)k(the)f(p)s(o)m(w)m(er-la)m(w)f(gro)m(wth)h(of)f(scales.)64 b(The)41 b(pro)s(of)d(of)h(Theorem)328 4782 y(3)g(will)e(consist,)k (roughly)-8 b(,)40 b(of)e(adapting)g(this)h(t)m(yp)s(e)h(of)f(argumen)m (t)g(to)f(our)h(situation.)1922 5214 y(6)p eop %%Page: 7 8 7 7 bop 328 631 a Fp(W)-8 b(e)40 b(w)m(ould)f(lik)m(e)f(to)h(note,)i (ho)m(w)m(ev)m(er,)j(that)39 b(p)s(o)m(w)m(er-la)m(w)g(repulsion)g(of)f (singular)g(clus-)328 751 y(ters,)52 b(as)47 b(in)g(Lemma)f(4,)51 b(although)46 b(used)j(in)d(the)i(follo)m(wing)d(pro)s(of,)50 b(is)d(not)g(essen-)328 872 y(tial)38 b(here.)69 b(Namely)-8 b(,)41 b(one)g(can)g(pro)m(v)m(e)h(Theorem)f(3)f(using)g(only)g(the)h (fact)g(that)f Fo(x)h Fp(is)328 992 y(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))22 b Fm(\000)h Fo(\017;)17 b Fm(j)p Fo(x)p Fm(j)p Fp(\))p Fm(\000)p Fp(regular)31 b(for)h(all)f(su\016cien) m(tly)i(large)f Fm(j)p Fo(x)p Fm(j)p Fo(:)474 1112 y Fp(Fix)37 b Fo(\013)q(;)17 b(\013)823 1127 y Fi(1)899 1112 y Fp(so)38 b(that)f(1)f Fo(<)f(\013)i(<)f(\013)1709 1127 y Fi(1)1784 1112 y Fo(<)g Fp(2)p Fo(:)h Fp(Find)2248 1086 y(\026)2245 1112 y Fo(k)k Fp(suc)m(h)e(that)e(for)g Fo(k)i(>)3136 1086 y Fp(\026)3134 1112 y Fo(k)h Fp(w)m(e)e(ha)m(v)m(e) 328 1233 y(\()p Fo(k)29 b Fp(+)d(3\))635 1197 y Fn(\013)710 1233 y Fp(+)g(4)p Fo(k)915 1197 y Fn(\013)991 1233 y Fp(+)g(5)p Fo(k)1196 1197 y Fn(\013)p Fc(\000)p Fi(1)1373 1233 y Fo(<)37 b(k)1540 1197 y Fn(\013)1585 1206 y Fe(1)1624 1233 y Fo(:)i Fp(Assume)g(without)f(loss)g(of)g(generalit)m(y)f(that)h Fo(x)h Fp(is)328 1353 y(p)s(ositiv)m(e.)63 b(Let)39 b Fo(E)45 b Fp(b)s(e)40 b(an)f(eigen)m(v)-5 b(alue)39 b(of)f Fo(H)1998 1368 y Fn(!)r(;\025;\022)2164 1353 y Fo(:)h Fp(Since)g(\011)2567 1368 y Fn(E)2666 1353 y Fm(2)g Fo(`)2812 1317 y Fi(2)2852 1353 y Fo(;)g Fp(\(it)f(ev)m(en)j(deca)m(ys)328 1474 y(exp)s(onen)m(tially)-8 b(,)34 b(as)i(w)m(e)g(already)e(pro)m(v)m (ed\))i(w)m(e)g(can)g(normalize)c(it)i(to)h(ha)m(v)m(e)h(an)f(apriori) 328 1594 y(b)s(ound)1524 1714 y Fm(j)p Fp(\011)1628 1729 y Fn(E)1687 1714 y Fp(\()p Fo(x)p Fp(\))p Fm(j)28 b(\024)g Fp(1)p Fo(;)44 b(x)28 b Fm(2)g Fh(Z)n Fo(:)995 b Fp(\(2.5\))328 1887 y(T)-8 b(ak)m(e)34 b Fo(x)f Fp(large)e(enough:)44 b Fm(j)p Fo(x)p Fm(j)27 b Fo(>)h(k)1560 1851 y Fn(\013)1557 1912 y Fi(0)1610 1887 y Fo(;)k Fp(where)735 2103 y Fo(k)786 2118 y Fi(0)853 2103 y Fp(=)27 b(max\()p Fo(k)1227 2118 y Fi(1)1266 2103 y Fp(\()p Fo(E)6 b(;)17 b(\015)5 b Fp(\()p Fo(E)h Fp(\))22 b Fm(\000)g Fo(\017;)17 b(\022)s(;)g Fp(0\))p Fo(;)g(k)2114 2118 y Fi(2)2153 2103 y Fp(\()p Fo(\022)s(;)g(!)t(;)g Fp(0)p Fo(;)g(\017;)g(\013)2630 2118 y Fi(1)2668 2103 y Fo(;)g(E)6 b Fp(\))p Fo(;)2874 2077 y Fp(\026)2872 2103 y Fo(k)r Fp(\))22 b(+)g(3)p Fo(:)328 2320 y Fp(Let)33 b Fo(k)d Fm(2)f Fo(K)39 b Fp(b)s(e)33 b(suc)m(h)h(that)1520 2536 y Fo(k)1574 2495 y Fn(\013)1651 2536 y Fo(<)28 b Fm(j)p Fo(x)p Fm(j)f(\024)h Fp(\()p Fo(k)d Fp(+)e(3\))2298 2495 y Fn(\013)2347 2536 y Fo(:)991 b Fp(\(2.6\))328 2752 y(Then)43 b Fo(k)j(>)c(k)858 2767 y Fi(0)926 2752 y Fm(\000)28 b Fp(3)p Fo(;)42 b Fp(and)f(w)m(e)i(ha)m (v)m(e,)i(b)m(y)d(Lemma)f(4,)i(that)e(ev)m(ery)j Fo(y)i Fm(2)d Fp([)3111 2713 y Fn(k)r Fi(+3)p 3111 2729 129 4 v 3158 2787 a(2)3250 2752 y Fo(;)17 b(k)3348 2716 y Fn(\013)3393 2725 y Fe(1)3432 2752 y Fp(])41 b(is)328 2873 y(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))31 b Fm(\000)g Fo(\017;)17 b(k)s Fp(\))p Fm(\000)p Fp(regular.)81 b(F)-8 b(or)44 b(eac)m(h)j(suc)m(h)f Fo(y)j Fp(pic)m(k)c(an)h(in)m(terv)-5 b(al)43 b([)p Fo(x)2936 2888 y Fi(1)2976 2873 y Fo(;)17 b(x)3075 2888 y Fi(2)3115 2873 y Fp(])45 b(from)f(the)328 2993 y(de\014nition)35 b(of)g(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))24 b Fm(\000)g Fo(\017;)17 b(k)s Fp(\))p Fm(\000)p Fp(regularit)m(y)35 b(and)h(denote)g(it)f(b)m(y)h Fo(I)8 b Fp(\()p Fo(y)t Fp(\))p Fo(:)35 b Fp(W)-8 b(e)36 b(denote)g(the)328 3113 y(b)s(oundary)h(of)f(the)h(in)m(terv)-5 b(al)35 b Fo(I)8 b Fp(\()p Fo(y)t Fp(\),)37 b(the)g(set)g Fm(f)p Fo(x)2090 3128 y Fi(1)2130 3113 y Fo(;)17 b(x)2229 3128 y Fi(2)2268 3113 y Fm(g)p Fo(;)37 b Fp(b)m(y)g Fo(@)5 b(I)j Fp(\()p Fo(y)t Fp(\))p Fo(:)37 b Fp(F)-8 b(or)35 b Fo(z)40 b Fm(2)34 b Fo(@)5 b(I)j Fp(\()p Fo(y)t Fp(\))37 b(w)m(e)328 3234 y(let)32 b Fo(z)518 3198 y Fc(0)574 3234 y Fp(b)s(e)h(the)g(neigh)m(b)s(or)f(of)g Fo(z)t(;)i Fp(\(i.e.,)e Fm(j)p Fo(z)26 b Fm(\000)d Fo(z)1967 3198 y Fc(0)1991 3234 y Fm(j)k Fp(=)h(1\))k(not)h(b)s(elonging)d(to)j Fo(I)8 b Fp(\()p Fo(y)t Fp(\))p Fo(:)474 3354 y Fp(W)-8 b(e)42 b(no)m(w)g(expand)h(\011\()p Fo(x)1380 3369 y Fi(1)1447 3354 y Fm(\000)29 b Fp(1\))41 b(in)g(\(2.3\))f(iterating)g (\(2.3\))h(with)g Fo(I)50 b Fp(=)43 b Fo(I)8 b Fp(\()p Fo(x)3279 3369 y Fi(1)3346 3354 y Fm(\000)29 b Fp(1\))p Fo(;)328 3474 y Fp(and)35 b(\011\()p Fo(x)689 3489 y Fi(2)752 3474 y Fp(+)23 b(1\))34 b(using)g(\(2.3\))g(with)g Fo(I)39 b Fp(=)31 b Fo(I)8 b Fp(\()p Fo(x)2021 3489 y Fi(2)2084 3474 y Fp(+)23 b(1\))p Fo(:)35 b Fp(Note)f(that)h Fo(I)8 b Fp(\()p Fo(x)2927 3489 y Fn(i)2978 3474 y Fm(\006)24 b Fp(1\))p Fo(;)33 b(i)f Fp(=)e(1)p Fo(;)17 b Fp(2)p Fo(;)328 3595 y Fp(are)38 b(w)m(ell-de\014ned,)h(since,)g(b)m(y)g (construction,)2078 3556 y Fn(k)r Fi(+3)p 2078 3572 V 2125 3629 a(2)2253 3595 y Fo(<)d(x)2420 3610 y Fi(1)2486 3595 y Fm(\000)26 b Fp(1)36 b Fo(<)g(x)2841 3610 y Fi(2)2907 3595 y Fp(+)25 b(1)37 b Fo(<)f(k)3260 3559 y Fn(\013)3305 3568 y Fe(1)3344 3595 y Fo(;)i Fp(and)328 3715 y(therefore)f(p)s(oin)m (ts)e(of)h(the)h(form)e Fo(x)1613 3730 y Fn(i)1666 3715 y Fm(\006)25 b Fp(1)p Fo(;)33 b(i)h Fp(=)f(1)p Fo(;)17 b Fp(2)p Fo(;)36 b Fp(are)g(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))24 b Fm(\000)i Fo(\017;)17 b(k)s Fp(\))p Fm(\000)p Fp(regular.)53 b(W)-8 b(e)328 3836 y(con)m(tin)m(ue)33 b(to)e(expand)i(eac)m(h)g(term)f(of)f(the)i(form)e(\011\()p Fo(z)t Fp(\))h(in)f(the)i(same)f(fashion)f(un)m(til)g(w)m(e)328 3956 y(arriv)m(e)h(to)g Fo(z)37 b Fp(suc)m(h)d(that)e(either)g Fo(z)g(<)c(k)35 b Fp(or)d(the)h(n)m(um)m(b)s(er)f(of)g Fo(G)2608 3971 y Fn(I)2680 3956 y Fp(terms)g(in)g(the)g(pro)s(duct)328 4076 y(b)s(ecomes)j(5)p Fo(k)821 4040 y Fn(\013)p Fc(\000)p Fi(1)960 4076 y Fo(;)g Fp(whic)m(hev)m(er)h(comes)e(\014rst.)50 b(W)-8 b(e)34 b(then)h(obtain)f(an)g(expression)h(of)f(the)328 4197 y(form)400 4413 y(\011\()p Fo(x)p Fp(\))28 b(=)892 4330 y Fb(X)738 4519 y Fn(s)p Fi(;)p Fn(z)824 4529 y Fd(i)p Fe(+1)927 4519 y Fc(2)p Fn(@)t(I)5 b Fi(\()p Fn(z)1114 4496 y Fa(0)1111 4542 y Fd(i)1137 4519 y Fi(\))1181 4413 y Fo(G)1258 4428 y Fn(I)g Fi(\()p Fn(x)p Fi(\))1393 4413 y Fp(\()p Fo(x;)17 b(z)1575 4428 y Fi(1)1615 4413 y Fp(\))p Fo(G)1730 4429 y Fn(I)5 b Fi(\()p Fn(z)1829 4406 y Fa(0)1826 4452 y Fe(1)1860 4429 y Fi(\))1892 4413 y Fp(\()p Fo(z)1979 4372 y Fc(0)1975 4438 y Fi(1)2015 4413 y Fo(;)17 b(z)2104 4428 y Fi(2)2143 4413 y Fp(\))g Fm(\001)g(\001)g(\001)d Fo(G)2407 4428 y Fn(I)5 b Fi(\()p Fn(z)2506 4410 y Fa(0)2503 4445 y Fd(s)2537 4428 y Fi(\))2568 4413 y Fp(\()p Fo(z)2655 4372 y Fc(0)2651 4438 y Fn(s)2688 4413 y Fo(;)17 b(z)2777 4428 y Fn(s)p Fi(+1)2904 4413 y Fp(\)\011\()p Fo(z)3105 4372 y Fc(0)3101 4438 y Fn(s)p Fi(+1)3228 4413 y Fp(\))p Fo(:)72 b Fp(\(2.7\))328 4724 y(where)28 b(in)e(eac)m(h)i(term)e(of)g (the)i(summation)c(w)m(e)k(ha)m(v)m(e)g Fo(z)2321 4739 y Fn(i)2377 4724 y Fo(>)g(k)s(;)44 b(i)28 b Fp(=)g(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(s;)26 b Fp(and)h(either)328 4844 y(0)32 b Fo(<)g(z)566 4808 y Fc(0)562 4869 y Fn(s)p Fi(+1)722 4844 y Fo(<)g(k)s(;)68 b(s)32 b Fm(\024)h Fp(5)p Fo(k)1270 4808 y Fn(\013)p Fc(\000)p Fi(1)1410 4844 y Fo(;)i Fp(or)g Fo(s)24 b Fp(+)g(1)32 b(=)g(5)p Fo(k)2056 4808 y Fn(\013)p Fc(\000)p Fi(1)2196 4844 y Fo(:)j Fp(Note,)i(that)e (since)g(w)m(e)i(b)s(ound)e(the)328 4965 y(n)m(um)m(b)s(er)i(of)f (expansions)h(b)m(y)h(5)p Fo(k)1540 4929 y Fn(\013)p Fc(\000)p Fi(1)1715 4965 y Fp(and)f(since,)h(b)m(y)f(construction,)h Fo(k)f(>)3111 4938 y Fp(\026)3108 4965 y Fo(k)s(;)g Fp(w)m(e)g(ha)m(v)m (e)1922 5214 y(7)p eop %%Page: 8 9 8 8 bop 338 592 a Fn(k)r Fi(+3)p 338 608 129 4 v 385 665 a(2)504 631 y Fo(<)28 b(k)j(<)c(z)842 595 y Fc(0)838 656 y Fn(i)894 631 y Fo(<)h(x)8 b Fp(+)g(4)p Fo(k)1248 595 y Fn(\013)1305 631 y Fp(+)g(5)p Fo(k)1492 595 y Fn(\013)p Fc(\000)p Fi(1)1659 631 y Fo(<)28 b(k)1817 595 y Fn(\013)1862 604 y Fe(1)1926 631 y Fp(for)d(eac)m(h)i Fo(i)h Fm(\024)g Fo(s:)e Fp(Therefore)g(eac)m(h)h Fo(z)3242 595 y Fc(0)3238 656 y Fn(i)3266 631 y Fo(;)34 b(i)28 b Fm(\024)g Fo(s;)328 751 y Fp(is)k(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))21 b Fm(\000)h Fo(\017;)17 b(k)s Fp(\))p Fm(\000)p Fp(regular,)32 b(and)h Fo(I)8 b Fp(\()p Fo(z)1732 715 y Fc(0)1728 776 y Fn(i)1756 751 y Fp(\))32 b(is)g(w)m(ell-de\014ned.)44 b(W)-8 b(e)33 b(no)m(w)g(consider)f(the)h(t)m(w)m(o)328 872 y(cases,)h(0)27 b Fo(<)h(z)829 836 y Fc(0)825 896 y Fn(s)p Fi(+1)980 872 y Fo(<)f(k)s(;)33 b Fp(and)g Fo(s)22 b Fp(+)g(1)27 b(=)h(5)p Fo(k)1836 836 y Fn(\013)p Fc(\000)p Fi(1)1975 872 y Fo(;)33 b Fp(separately)-8 b(.)474 992 y(If)33 b(0)27 b Fo(<)h(z)801 956 y Fc(0)797 1017 y Fn(s)p Fi(+1)952 992 y Fo(<)f(k)s(;)33 b Fp(w)m(e)h(ha)m(v)m(e,)g(b)m(y)f(the) g(de\014nition)f(of)g(regularit)m(y)f(and)i(\(2.5\),)328 1227 y Fm(j)p Fo(G)433 1242 y Fn(I)5 b Fi(\()p Fn(x)p Fi(\))567 1227 y Fp(\()p Fo(x;)17 b(z)749 1242 y Fi(1)789 1227 y Fp(\))p Fo(G)904 1243 y Fn(I)5 b Fi(\()p Fn(z)1003 1220 y Fa(0)1000 1266 y Fe(1)1034 1243 y Fi(\))1066 1227 y Fp(\()p Fo(z)1153 1186 y Fc(0)1149 1252 y Fi(1)1189 1227 y Fo(;)17 b(z)1278 1242 y Fi(2)1317 1227 y Fp(\))g Fm(\001)g(\001)g(\001)e Fo(G)1582 1242 y Fn(I)5 b Fi(\()p Fn(z)1681 1223 y Fa(0)1678 1259 y Fd(s)1711 1242 y Fi(\))1742 1227 y Fp(\()p Fo(z)1829 1186 y Fc(0)1825 1252 y Fn(s)1862 1227 y Fo(;)17 b(z)1951 1242 y Fn(s)p Fi(+1)2078 1227 y Fp(\)\011\()p Fo(z)2279 1186 y Fc(0)2275 1252 y Fn(s)p Fi(+1)2402 1227 y Fp(\))p Fm(j)28 b(\024)g Fo(e)2646 1186 y Fc(\000)p Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)p Fn(\017)p Fi(\)\()p Fc(j)p Fn(x)p Fc(\000)p Fn(z)3164 1195 y Fe(1)3199 1186 y Fc(j)p Fi(+)3274 1128 y Fb(P)3361 1148 y Fd(s)3361 1215 y(i)p Fe(=1)3476 1186 y Fc(j)p Fn(z)3532 1162 y Fa(0)3529 1209 y Fd(i)3555 1186 y Fc(\000)p Fn(z)3643 1196 y Fd(i)p Fe(+1)3746 1186 y Fc(j)p Fi(\))3825 1227 y Fm(\024)972 1465 y Fo(e)1017 1424 y Fc(\000)p Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)p Fn(\017)p Fi(\)\()p Fc(j)p Fn(x)p Fc(\000)p Fn(z)1535 1433 y Fd(s)p Fe(+1)1646 1424 y Fc(j\000)p Fi(\()p Fn(s)p Fi(+1\)\))1957 1465 y Fm(\024)g Fo(e)2107 1424 y Fc(\000)p Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)p Fn(\017)p Fi(\)\()p Fn(x)p Fc(\000)p Fn(k)r Fc(\000)p Fi(5)p Fn(k)2740 1401 y Fd(\013)p Fa(\000)p Fe(1)2863 1424 y Fi(\))2895 1465 y Fo(:)474 1638 y Fp(If)37 b Fo(s)25 b Fp(+)h(1)35 b(=)g(5)p Fo(k)1047 1601 y Fn(\013)p Fc(\000)p Fi(1)1186 1638 y Fo(;)j Fp(using)e(again)g(\(2.5\),)i(the)f (de\014nition)f(of)h(regularit)m(y)-8 b(,)37 b(and)g(the)328 1758 y(fact)32 b(that)h Fm(j)p Fo(x)54 b Fm(\000)i Fo(z)1047 1773 y Fi(1)1086 1758 y Fm(j)k(\025)1322 1719 y Fn(k)p 1322 1735 39 4 v 1324 1792 a Fi(5)1370 1758 y Fo(;)h Fm(j)p Fo(z)1531 1773 y Fn(i)1581 1758 y Fm(\000)23 b Fo(z)1726 1773 y Fn(i)p Fi(+1)1844 1758 y Fm(j)28 b(\025)2015 1719 y Fn(k)p 2015 1735 V 2017 1792 a Fi(5)2063 1758 y Fo(;)45 b(i)27 b Fp(=)h(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(s;)32 b Fp(w)m(e)h(can)g(estimate)475 1998 y Fm(j)p Fo(G)580 2014 y Fn(I)5 b Fi(\()p Fn(x)p Fi(\))714 1998 y Fp(\()p Fo(x;)17 b(z)896 2013 y Fi(1)936 1998 y Fp(\))p Fo(G)1051 2015 y Fn(I)5 b Fi(\()p Fn(z)1150 1992 y Fa(0)1147 2037 y Fe(1)1182 2015 y Fi(\))1213 1998 y Fp(\()p Fo(z)1300 1957 y Fc(0)1296 2023 y Fi(1)1336 1998 y Fo(;)17 b(z)1425 2013 y Fi(2)1464 1998 y Fp(\))g Fm(\001)g(\001)g(\001)e Fo(G)1729 2014 y Fn(I)5 b Fi(\()p Fn(z)1828 1995 y Fa(0)1825 2030 y Fd(s)1858 2014 y Fi(\))1890 1998 y Fp(\()p Fo(z)1977 1957 y Fc(0)1973 2023 y Fn(s)2010 1998 y Fo(;)17 b(z)2099 2013 y Fn(s)p Fi(+1)2226 1998 y Fp(\)\011\()p Fo(z)2427 1957 y Fc(0)2423 2023 y Fn(s)p Fi(+1)2550 1998 y Fp(\))p Fm(j)27 b(\024)h Fo(e)2793 1957 y Fc(\000)p Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)p Fn(\017)p Fi(\))3146 1930 y Fd(k)p 3147 1942 34 4 v 3149 1983 a Fe(5)3191 1957 y Fi(5)p Fn(k)3265 1934 y Fd(\013)p Fa(\000)p Fe(1)3392 1998 y Fo(:)328 2213 y Fp(Using)k(\(2.6\))g(w)m(e)i(obtain)d(that)i(in) e(either)i(case)449 2429 y Fm(j)p Fo(G)554 2444 y Fn(I)5 b Fi(\()p Fn(x)p Fi(\))689 2429 y Fp(\()p Fo(x;)17 b(z)871 2444 y Fi(1)910 2429 y Fp(\))p Fo(G)1025 2445 y Fn(I)5 b Fi(\()p Fn(z)1124 2422 y Fa(0)1121 2467 y Fe(1)1156 2445 y Fi(\))1187 2429 y Fp(\()p Fo(z)1274 2387 y Fc(0)1270 2453 y Fi(1)1310 2429 y Fo(;)17 b(z)1399 2444 y Fi(2)1439 2429 y Fp(\))g Fm(\001)g(\001)g(\001)d Fo(G)1703 2444 y Fn(I)5 b Fi(\()p Fn(z)1802 2425 y Fa(0)1799 2461 y Fd(s)1832 2444 y Fi(\))1864 2429 y Fp(\()p Fo(z)1951 2387 y Fc(0)1947 2453 y Fn(s)1984 2429 y Fo(;)17 b(z)2073 2444 y Fn(s)p Fi(+1)2200 2429 y Fp(\)\011\()p Fo(z)2401 2387 y Fc(0)2397 2453 y Fn(s)p Fi(+1)2524 2429 y Fp(\))p Fm(j)27 b(\024)h Fo(e)2767 2387 y Fc(\000)p Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)p Fn(\017)p Fc(\000)p Fn(\016)r Fi(\))p Fn(x)3365 2429 y Fp(\(2.8\))328 2644 y(for)42 b(an)m(y)i Fo(\016)49 b(>)c Fp(0)d(and)h Fo(x)g Fp(su\016cien)m(tly)h(large.)73 b(Finally)-8 b(,)43 b(w)m(e)h(observ)m (e)g(that)f(the)g(total)328 2764 y(n)m(um)m(b)s(er)j(of)f(terms)g(in)f (\(2.7\))h(is)g(b)s(ounded)h(ab)s(o)m(v)m(e)g(b)m(y)g(2)2484 2728 y Fi(5)p Fn(k)2558 2705 y Fd(\013)p Fa(\000)p Fe(1)2685 2764 y Fo(:)f Fp(Com)m(bining)f(it)g(with)328 2884 y(\(2.7\),)32 b(\(2.8\),\(2.6\),)g(w)m(e)h(obtain)1340 3139 y Fm(j)p Fp(\011\()p Fo(x)p Fp(\))p Fm(j)27 b(\024)h Fp(2)1784 3097 y Fi(5)p Fn(x)1869 3045 y Fd(\013)p Fa(\000)p Fe(1)p 1869 3059 119 4 v 1908 3090 a Fd(\013)2006 3139 y Fo(e)2051 3097 y Fc(\000)p Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)p Fn(\017)p Fc(\000)p Fn(\016)r Fi(\))p Fn(x)2527 3139 y Fo(:)328 3354 y Fp(Since)44 b Fo(\016)k Fp(and)43 b Fo(\017)i Fp(can)f(b)s(e)g(c)m(hosen)h(arbitrarily)c(small)g(for)j (su\016cien)m(tly)g(large)f Fo(x;)h Fp(this)328 3474 y(implies)30 b(the)j(needed)h(upp)s(er)f(b)s(ound.)p 1777 3482 42 59 v 328 3903 a Fj(3)161 b(Pro)t(of)54 b(of)g(Lemma)f(4.) 328 4122 y Fp(In)33 b(this)f(section)h(w)m(e)g(assume)g Fo(E)39 b Fp(\014xed)34 b(and)e(drop)h(it)f(from)f(most)h(of)g(the)h (notations.)474 4242 y(W)-8 b(e)24 b(start)f(with)g(outlining)d(the)j (idea)g(of)g(the)g(pro)s(of.)40 b(Ev)m(ery)25 b(\()p Fo(m;)17 b(k)s Fp(\))p Fm(\000)p Fp(singular)22 b(p)s(oin)m(t)328 4363 y Fo(x)39 b Fp(can)g(b)s(e)f(understo)s(o)s(d,)j(v)m(ery)f (roughly)-8 b(,)39 b(as)g(a)f(lo)s(cal)e(cen)m(ter)k(of)e(lo)s (calization,)e(at)i(the)328 4483 y(rate)25 b Fo(m)h Fp(and)f(on)g(the)h (scale)f Fo(k)s(:)h Fp(If)f(so,)i(for)e(an)m(y)h Fo(x)2041 4498 y Fi(1)2108 4483 y Fo(<)i(x)g(<)g(x)2454 4498 y Fi(1)2501 4483 y Fp(+)7 b Fo(k)k Fm(\000)c Fp(1)p Fo(;)25 b Fp(suc)m(h)i(that)e(b)s(oth)g Fo(x)3526 4498 y Fi(1)328 4604 y Fp(and)j Fo(x)568 4619 y Fi(1)620 4604 y Fp(+)12 b Fo(k)k Fm(\000)c Fp(1)27 b(are)h(su\016cien)m(tly)h(far)e(from)f Fo(x;)i Fp(the)h(op)s(erator)e Fo(H)2694 4619 y Fn(!)r(;\025;\022)2887 4604 y Fp(restricted)h(to)f(the)328 4724 y(in)m(terv)-5 b(al)32 b([)p Fo(x)763 4739 y Fi(1)803 4724 y Fo(;)17 b(x)902 4739 y Fi(1)964 4724 y Fp(+)23 b Fo(k)j Fm(\000)d Fp(1])33 b(should)g(ha)m(v)m(e)i(an)e(eigen)m(v)-5 b(alue)33 b(close)h(to)f Fo(E)6 b(:)33 b Fp(This)h(could)f(b)s(e)328 4844 y(manifested)25 b(b)m(y)h Fm(j)p Fo(P)1031 4859 y Fn(k)1073 4844 y Fp(\()p Fo(\022)10 b Fp(+)d Fo(x)1304 4859 y Fi(1)1344 4844 y Fo(!)t(;)17 b(E)6 b Fp(\))p Fm(j)24 b Fp(b)s(eing)h(abnormally)e(small)g(\(exp)s(onen)m(tially)h(smaller) 328 4965 y(than)36 b Fo(e)604 4929 y Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fn(k)798 4965 y Fp(\))p Fo(:)g Fp(Then,)i(for)e(eac)m(h) h(singular)e(p)s(oin)m(t)g Fo(x)i Fp(there)g(exists)g(a)f(su\016cien)m (tly)h(long)1922 5214 y(8)p eop %%Page: 9 10 9 9 bop 328 631 a Fp(\(longer)27 b(than)888 592 y Fn(k)r Fi(+1)p 888 608 129 4 v 934 665 a(2)1026 631 y Fp(\))i(in)m(terv)-5 b(al)27 b Fo(I)8 b Fp(\()p Fo(x)p Fp(\))28 b Fm(\032)g Fh(Z)d Fp(of)j(v)-5 b(alues)29 b(of)f Fo(x)2406 646 y Fi(1)2474 631 y Fp(suc)m(h)i(that)e Fm(j)p Fo(P)2988 646 y Fn(k)3030 631 y Fp(\()p Fo(\022)16 b Fp(+)e Fo(x)3274 646 y Fi(1)3314 631 y Fo(!)t(;)j(E)6 b Fp(\))p Fm(j)328 751 y Fp(is)33 b(small.)46 b(This)34 b(is)f(made)h(precise)g(in)f (Lemma)g(6.)47 b(The)35 b(main)d(tec)m(hnical)i(di\016cult)m(y)g(in)328 872 y(the)40 b(pro)s(of)e(of)h(this)g(Lemma)e(lies)i(in)f(establishing) g(a)h(uniform)e(\(in)i Fo(\022)s Fp(\))g(upp)s(er)h(b)s(ound)328 992 y(on)e Fm(j)p Fo(P)560 1007 y Fn(k)603 992 y Fp(\()p Fo(\022)s(;)17 b(E)6 b Fp(\))p Fm(j)p Fo(;)38 b Fp(that)g(is)g(a)g(sub) 5 b(ject)41 b(of)d(Lemma)f(5,)j(the)f(pro)s(of)f(of)g(whic)m(h)h(is)f (giv)m(en)h(in)328 1112 y(Section)32 b(4.)474 1233 y(The)41 b(rest)f(of)f(the)g(pro)s(of)g(consists)h(mainly)d(of)i(establishing)f (the)i(follo)m(wing)d(fact:)328 1353 y(if)43 b(the)j(p)s(oin)m(ts)e Fo(\022)960 1368 y Fi(1)1000 1353 y Fo(;)17 b(:)g(:)g(:)f(;)h(\022)1264 1368 y Fn(k)r Fi(+1)1441 1353 y Fp(are,)48 b(in)c(a)g(certain)h(sense,) k(uniformly)43 b(distributed)h(on)328 1474 y([0)p Fo(;)17 b Fp(1\))p Fo(;)31 b Fp(and)g(do)h(not)f(come)g(abnormally)e(close)j (to)f(one)h(another,)g(then)g Fo(P)3044 1489 y Fn(k)3086 1474 y Fp(\()p Fo(\022)s Fp(\))g(will)d(not)328 1594 y(b)s(e)42 b(small)e(\(in)h(the)i(sense)h(ab)s(o)m(v)m(e\))f(at)e(at)h (least)g(one)g(of)g(those)h(p)s(oin)m(ts.)71 b(Finally)-8 b(,)42 b(for)328 1714 y(Diophan)m(tine)37 b Fo(!)t(;)h Fp(an)m(y)i(p)s(oin)m(ts)e(of)h(the)g(form)e Fo(\022)30 b Fp(+)c Fo(x)2268 1729 y Fi(1)2308 1714 y Fo(!)42 b Fp(with)c Fo(x)2694 1729 y Fi(1)2773 1714 y Fp(running)g(through)h(a) 328 1835 y(union)30 b(of)h(t)m(w)m(o)g(long)f(enough)h(in)m(terv)-5 b(als,)30 b Fo(I)8 b Fp(\()p Fo(x)p Fp(\))31 b(and)g Fo(I)8 b Fp(\()p Fo(y)t Fp(\))p Fo(;)30 b Fp(are)h(going)e(to)i(b)s(e)g (uniformly)328 1955 y(distributed)44 b(\(Lemma)f(12\).)78 b(If)44 b Fo(x)h Fp(and)g Fo(y)i Fp(are)d(not)h(to)s(o)e(close)h(to)g (eac)m(h)i(other,)h(the)328 2076 y(in)m(terv)-5 b(als)38 b Fo(I)8 b Fp(\()p Fo(x)p Fp(\))39 b(and)h Fo(I)8 b Fp(\()p Fo(y)t Fp(\))37 b(will)g(not)i(in)m(tersect)h(and)f(their)g(union)f (will)f(consist)i(of)g(at)328 2196 y(least)31 b Fo(k)23 b Fp(+)d(1)32 b(p)s(oin)m(ts.)42 b(Also,)32 b(the)g(Diophan)m(tine)e (condition)g(on)h Fo(!)k Fp(\(1.2\))c(together)h(with)328 2316 y(the)43 b(nonresonan)m(t)i(condition)c(on)i Fo(\022)j Fp(\(1.3\))d(imply)e(that)i(if)f Fo(x)h Fp(is)g(not)g(v)m(ery)h(far)f (from)328 2437 y Fo(y)t(;)36 b Fp(the)i(p)s(oin)m(ts)e Fo(\022)29 b Fp(+)c Fo(x)1143 2452 y Fi(1)1183 2437 y Fo(!)t(;)36 b Fp(with)h Fo(x)1593 2452 y Fi(1)1668 2437 y Fm(2)f Fo(I)8 b Fp(\()p Fo(x)p Fp(\))25 b Fm([)h Fo(I)8 b Fp(\()p Fo(y)t Fp(\))p Fo(;)36 b Fp(do)h(not)g(come)f(to)s(o)h(close) g(to)f(one)328 2557 y(another)h(\(Lemma)f(13\).)57 b(Summing)35 b(it)h(all)f(up)j(yields)f(that)g(t)m(w)m(o)h(singular)d(p)s(oin)m(ts,) j Fo(x)328 2677 y Fp(and)33 b Fo(y)t(;)e Fp(\(if)h(they)h(are)g(not)f (v)m(ery)i(close\))f(m)m(ust)g(b)s(e)f(su\016cien)m(tly)i(far)e(from)f (eac)m(h)i(other.)474 2798 y(W)-8 b(e)33 b(will)d(no)m(w)k(pro)s(ceed)f (with)f(a)h(formal)d(pro)s(of.)474 2918 y(It)46 b(is)f(a)h(standard)g (linear)e(algebra)g(\(Cramer's)i(rule\))f(that)h(for)f(an)m(y)h Fo(x)3211 2933 y Fi(1)3251 2918 y Fo(;)67 b(x)3400 2933 y Fi(2)3490 2918 y Fp(=)328 3039 y Fo(x)383 3054 y Fi(1)445 3039 y Fp(+)22 b Fo(k)j Fm(\000)e Fp(1)p Fo(;)44 b(x)894 3054 y Fi(1)961 3039 y Fm(\024)29 b Fo(y)h Fm(\024)f Fo(x)1306 3054 y Fi(2)1378 3039 y Fp(w)m(e)34 b(ha)m(v)m(e)1034 3315 y Fm(j)p Fo(G)1139 3331 y Fi([)p Fn(x)1199 3340 y Fe(1)1233 3331 y Fn(;x)1293 3340 y Fe(2)1327 3331 y Fi(])1351 3315 y Fp(\()p Fo(x)1444 3330 y Fi(1)1483 3315 y Fo(;)17 b(y)t Fp(\))p Fm(j)82 b Fp(=)1886 3166 y Fb(\014)1886 3216 y(\014)1886 3266 y(\014)1886 3315 y(\014)1886 3365 y(\014)1923 3248 y Fo(P)1986 3263 y Fn(x)2026 3272 y Fe(2)2061 3263 y Fc(\000)p Fn(y)2157 3248 y Fp(\()p Fo(\022)25 b Fp(+)d(\()p Fo(y)j Fp(+)d(1\))p Fo(!)t Fp(\))p 1923 3292 838 4 v 2088 3384 a Fo(P)2151 3399 y Fn(k)2193 3384 y Fp(\()p Fo(\022)k Fp(+)c Fo(x)2455 3399 y Fi(1)2495 3384 y Fo(!)t Fp(\))2771 3166 y Fb(\014)2771 3216 y(\014)2771 3266 y(\014)2771 3315 y(\014)2771 3365 y(\014)2816 3315 y Fo(;)1034 3748 y Fm(j)p Fo(G)1139 3763 y Fi([)p Fn(x)1199 3772 y Fe(1)1233 3763 y Fn(;x)1293 3772 y Fe(2)1327 3763 y Fi(])1351 3748 y Fp(\()p Fo(y)t(;)17 b(x)1540 3763 y Fi(2)1578 3748 y Fp(\))p Fm(j)83 b Fp(=)1886 3598 y Fb(\014)1886 3648 y(\014)1886 3698 y(\014)1886 3748 y(\014)1886 3797 y(\014)1923 3680 y Fo(P)1986 3695 y Fn(y)r Fc(\000)p Fn(x)2118 3704 y Fe(1)2157 3680 y Fp(\()p Fo(\022)25 b Fp(+)d Fo(x)2418 3695 y Fi(1)2458 3680 y Fo(!)t Fp(\))p 1923 3724 637 4 v 1987 3816 a Fo(P)2050 3831 y Fn(k)2093 3816 y Fp(\()p Fo(\022)j Fp(+)d Fo(x)2354 3831 y Fi(1)2394 3816 y Fo(!)t Fp(\))2570 3598 y Fb(\014)2570 3648 y(\014)2570 3698 y(\014)2570 3748 y(\014)2570 3797 y(\014)2615 3748 y Fo(:)723 b Fp(\(3.1\))328 4088 y(W)-8 b(e)33 b(will)d(need)k(the)f (follo)m(wing)d(b)s(ound,)j(uniform)d(in)i Fo(\022)s(;)h Fp(on)f(the)h(n)m(umerator)g(in)e(\(3.1\):)328 4291 y Fg(LEMMA)37 b(5)49 b Ff(Supp)-5 b(ose)46 b Fo(H)1372 4306 y Fn(!)r(;\022)1524 4291 y Ff(is)g(given)h(by)g(\(1.1\))f(with)g Fo(f)11 b Fp(\()p Fo(\022)s Fp(\))50 b(=)2884 4225 y Fb(P)2971 4244 y Fn(p)2971 4317 y(k)r Fi(=0)3121 4291 y Fo(a)3172 4306 y Fn(k)3231 4291 y Fp(cos)3362 4255 y Fn(k)3421 4291 y Fo(\022)g Ff(a)328 4412 y(trigonometric)40 b(p)-5 b(olynomial,)41 b Fo(!)j Ff(any)c(irr)-5 b(ational.)62 b(Then)40 b(for)g(every)h Fo(E)j Fm(2)39 b Fh(R)11 b Fo(;)33 b(\017)39 b(>)f Fp(0)p Fo(;)328 4532 y Ff(ther)-5 b(e)38 b(exists)f Fo(k)894 4547 y Fn(f)939 4532 y Fp(\()p Fo(\017;)17 b(E)6 b(;)17 b(!)t Fp(\))37 b Ff(such)g(that)h Fm(j)p Fo(P)1839 4547 y Fn(n)1885 4532 y Fp(\()p Fo(\022)s Fp(\))p Fm(j)33 b Fo(<)g(e)2224 4496 y Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\)+)p Fn(\017)p Fi(\))p Fn(n)2597 4532 y Ff(for)38 b(al)5 b(l)37 b Fo(n)c(>)g(k)3149 4547 y Fn(f)3194 4532 y Fp(\()p Fo(\017;)17 b(E)6 b(;)17 b(!)t Fp(\))p Fo(;)328 4653 y Ff(al)5 b(l)34 b Fo(\022)s(:)328 4856 y Fp(This)f(Lemma)e(will)f(b)s(e)j(pro)m(v)m(ed)h(in)e(Section)g (4)1922 5214 y(9)p eop %%Page: 10 11 10 10 bop 474 631 a Fp(Let)29 b Fo(A)718 584 y Fn(z)s(;\022)718 656 y(k)840 631 y Fp(=)e Fm(f)p Fo(x)h Fm(2)g Fh(Z)d Fp(:)j Fm(j)p Fo(P)1410 646 y Fn(k)1452 631 y Fp(\()p Fo(\022)16 b Fp(+)d Fo(x!)t Fp(\))p Fm(j)28 b(\024)g Fo(e)2004 595 y Fn(k)r(z)2083 631 y Fm(g)p Fo(:)g Fp(W)-8 b(e)29 b(will)d(sho)m(w)j(that)f(ev)m(ery)i(singular)328 751 y(p)s(oin)m(t)23 b(\\pro)s(duces")i(a)f(cluster)h(of)f(p)s(oin)m (ts)g(b)s(elonging)e(to)i(an)h(appropriate)e Fo(A)3103 715 y Fn(z)3103 776 y(k)3146 751 y Fp(;)h(ho)m(w)m(ev)m(er,)328 872 y(for)32 b Fo(z)g(<)c(\015)5 b Fp(\()p Fo(E)h Fp(\))p Fo(;)33 b Fp(suc)m(h)h(clusters)f(in)f(the)h(set)g Fo(A)2008 836 y Fn(z)2008 896 y(k)2083 872 y Fp(should)g(b)s(e)g(far)f(apart.)328 1065 y Fg(LEMMA)37 b(6)49 b Ff(Supp)-5 b(ose)49 b Fo(y)113 b Fm(2)d Fh(Z)46 b Ff(is)k Fp(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))32 b Fm(\000)i Fo(\017;)17 b(k)s Fp(\))p Ff(-singular,)52 b Fo(\017)k(<)3122 1018 y Fn(\015)t Fi(\()p Fn(E)t Fi(\))p 3122 1041 151 4 v 3180 1099 a(3)3283 1065 y Fo(;)71 b(k)58 b(>)328 1185 y Fp(4)p Fo(k)428 1200 y Fn(f)473 1185 y Fp(\()524 1146 y Fn(\017)p 521 1162 36 4 v 521 1219 a Fi(6)566 1185 y Fo(;)17 b(E)6 b(;)17 b(!)t Fp(\))23 b(+)g(1)p Fo(:)38 b Ff(Then)e(for)h(any)g Fo(x)h Ff(such)f(that)h Fo(y)27 b Fm(\000)d Fp([)2406 1146 y Fi(3)p 2406 1162 V 2406 1219 a(4)2452 1185 y Fp(])p Fo(k)35 b Fm(\024)e Fo(x)f Fm(\024)h Fo(y)27 b Fm(\000)e Fp([)3086 1146 y Fi(3)p 3086 1162 V 3086 1219 a(4)3131 1185 y Fp(])p Fo(k)i Fp(+)d([)3373 1146 y Fn(k)r Fi(+1)p 3373 1162 129 4 v 3420 1219 a(2)3512 1185 y Fp(])p Fo(;)328 1338 y Ff(we)34 b(have)h(that)g Fo(x)g Ff(b)-5 b(elongs)34 b(to)h Fo(A)1510 1278 y Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)1728 1250 y Fd(\017)p 1726 1262 31 4 v 1726 1304 a Fe(8)1766 1278 y Fn(;\022)1510 1363 y(k)1825 1338 y Ff(.)328 1522 y(Pr)-5 b(o)g(of)p Fp(:)84 b(F)-8 b(ollo)m(ws)52 b(immediately)e(from)i(the)h(de\014nition)f(of)h(regularit)m(y)-8 b(,)57 b(\(3.1\),)h(and)328 1642 y(Lemma)31 b(5.)p 802 1650 42 59 v 474 1840 a(Assume)45 b(no)m(w)f(that)f Fo(y)1331 1855 y Fi(1)1414 1840 y Fp(and)g Fo(y)1662 1855 y Fi(2)1745 1840 y Fp(are)g(b)s(oth)h(\()p Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))29 b Fm(\000)h Fo(\017;)17 b(k)s Fp(\))p Fm(\000)p Fp(singular.)76 b(Assume)328 1960 y(without)45 b(loss)h(of)f(generalit) m(y)h(that)f Fo(y)1763 1975 y Fi(2)1853 1960 y Fo(>)50 b(y)2027 1975 y Fi(1)2066 1960 y Fo(:)c Fp(W)-8 b(e)46 b(set)h Fo(d)j Fp(=)g Fo(y)2761 1975 y Fi(2)2831 1960 y Fm(\000)32 b Fo(y)2988 1975 y Fi(1)3027 1960 y Fo(;)67 b(x)3176 1975 y Fn(i)3255 1960 y Fp(=)50 b Fo(y)3429 1975 y Fn(i)3488 1960 y Fm(\000)328 2081 y Fp([3)p Fo(=)p Fp(4)p Fo(k)s Fp(])p Fo(;)44 b(i)29 b Fp(=)f(1)p Fo(;)17 b Fp(2)p Fo(:)32 b Fp(Note)h(that)g Fo(P)1532 2096 y Fn(k)1574 2081 y Fp(\()p Fo(\022)s Fp(\))g(is)g(an)g(ev)m(en)h (function)e(of)h Fo(\022)25 b Fp(+)2861 2042 y Fn(k)r Fc(\000)p Fi(1)p 2861 2058 129 4 v 2907 2115 a(2)2999 2081 y Fo(!)36 b Fp(and)d(can)h(b)s(e)328 2201 y(written)e(as)h(a)f(p)s (olynomial)d(of)k(degree)g Fo(k)j Fp(in)31 b(cos)18 b(2)p Fo(\031)t Fp(\()p Fo(\022)25 b Fp(+)2461 2162 y Fn(k)r Fc(\000)p Fi(1)p 2461 2178 V 2507 2235 a(2)2599 2201 y Fo(!)t Fp(\))i(:)328 2351 y Fo(P)391 2366 y Fn(k)433 2351 y Fp(\()p Fo(\022)s Fp(\))h(=)689 2284 y Fb(P)776 2311 y Fn(k)776 2375 y(j)t Fi(=0)920 2351 y Fo(c)962 2366 y Fn(j)1015 2351 y Fp(cos)1145 2315 y Fn(j)1199 2351 y Fp(2)p Fo(\031)t Fp(\()p Fo(\022)c Fp(+)1522 2312 y Fn(k)r Fc(\000)p Fi(1)p 1522 2328 V 1569 2385 a(2)1661 2351 y Fo(!)t Fp(\))1791 2298 y Fi(def)1802 2351 y Fp(=)38 b Fo(Q)1993 2366 y Fn(k)2036 2351 y Fp(\(cos)17 b(2)p Fo(\031)t Fp(\()p Fo(\022)25 b Fp(+)2545 2312 y Fn(k)r Fc(\000)p Fi(1)p 2545 2328 V 2592 2385 a(2)2684 2351 y Fo(!)t Fp(\)\))p Fo(:)31 b Fp(W)-8 b(e)33 b(no)m(w)h(set)762 2607 y Fo(\022)807 2622 y Fn(j)871 2607 y Fp(=)975 2461 y Fb(\()1252 2546 y Fo(\022)26 b Fp(+)c(\()p Fo(x)1514 2561 y Fi(1)1575 2546 y Fp(+)1683 2507 y Fn(k)r Fc(\000)p Fi(1)p 1683 2523 V 1730 2580 a(2)1844 2546 y Fp(+)g Fo(j)6 b Fp(\))p Fo(!)t(;)44 b(j)33 b Fp(=)28 b(0)p Fo(;)17 b Fp(1)p Fo(;)g(:::;)g Fp([)2687 2507 y Fn(k)r Fi(+1)p 2687 2523 V 2733 2580 a(2)2825 2546 y Fp(])22 b Fm(\000)h Fp(1)1194 2667 y Fo(\022)i Fp(+)d(\()p Fo(x)1455 2682 y Fi(2)1517 2667 y Fp(+)1625 2628 y Fn(k)r Fc(\000)p Fi(1)p 1625 2644 V 1672 2702 a(2)1786 2667 y Fp(+)g Fo(j)28 b Fm(\000)23 b Fp([)2089 2628 y Fn(k)r Fi(+1)p 2089 2644 V 2135 2702 a(2)2227 2667 y Fp(]\))p Fo(!)t(;)44 b(j)34 b Fp(=)27 b([)2642 2628 y Fn(k)r Fi(+1)p 2642 2644 V 2689 2702 a(2)2781 2667 y Fp(])p Fo(;)17 b(:)g(:)g(:)f(;)h(k)328 2861 y Fp(By)47 b(the)f(assumption,)j Fo(d)i(>)1474 2822 y Fn(k)r Fi(+1)p 1474 2838 V 1521 2895 a(2)1613 2861 y Fo(:)46 b Fp(This)h(implies)c(that)j(all)e Fo(\022)2686 2876 y Fn(j)2723 2861 y Fo(;)68 b(j)56 b Fp(=)51 b(0)p Fo(;)17 b(:)g(:)g(:)f(;)h(k)s(;)46 b Fp(are)328 2981 y(di\013eren)m(t.)81 b(W)-8 b(e)46 b(ha)m(v)m(e,)j(b)m(y)d(Lemma)e(6,)k (that)d Fm(j)p Fo(Q)2198 2996 y Fn(k)2241 2981 y Fp(\(cos)17 b(2)p Fo(\031)t(\022)2579 2996 y Fn(j)2615 2981 y Fp(\))p Fm(j)49 b Fo(<)g(e)2900 2945 y Fn(k)r Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)p Fn(\017=)p Fi(8\))3302 2981 y Fo(;)66 b(j)55 b Fp(=)328 3102 y(0)p Fo(;)17 b(:)g(:)g(:)f(;)h (k)s(:)29 b Fp(W)-8 b(e)30 b(no)m(w)g(write)g(the)g(p)s(olynomial)c Fo(Q)2061 3117 y Fn(k)2104 3102 y Fp(\()p Fo(z)t Fp(\))k(in)f(Lagrange) g(in)m(terp)s(olation)e(form)328 3222 y(using)32 b(cos)17 b(2)p Fo(\031)t(\022)883 3237 y Fi(0)923 3222 y Fo(;)g(:)g(:)g(:)f(;)h Fp(cos)g(2)p Fo(\031)t(\022)1442 3237 y Fn(k)1512 3222 y Fp(:)811 3473 y Fm(j)p Fo(Q)916 3488 y Fn(k)958 3473 y Fp(\()p Fo(z)t Fp(\))p Fm(j)28 b Fp(=)g Fm(j)1329 3365 y Fn(k)1289 3390 y Fb(X)1287 3572 y Fn(j)t Fi(=0)1426 3473 y Fo(Q)1503 3488 y Fn(k)1546 3473 y Fp(\(cos)17 b(2)p Fo(\031)t(\022)1884 3488 y Fn(j)1921 3473 y Fp(\))2117 3336 y Fb(Q)2196 3423 y Fn(`)p Fc(6)p Fi(=)p Fn(j)2316 3403 y Fp(\()p Fo(z)27 b Fm(\000)c Fp(cos)17 b(2)p Fo(\031)t(\022)2826 3418 y Fn(`)2859 3403 y Fp(\))p 1969 3450 1077 4 v 1969 3475 a Fb(Q)2047 3562 y Fn(k)r Fc(6)p Fi(=)p Fn(j)2177 3541 y Fp(\(cos)h(2)p Fo(\031)t(\022)2516 3556 y Fn(j)2574 3541 y Fm(\000)23 b Fp(cos)17 b(2)p Fo(\031)t(\022)2974 3556 y Fn(`)3007 3541 y Fp(\))3055 3473 y Fm(j)282 b Fp(\(3.2\))474 3731 y(The)29 b(p)s(oin)m(ts)e Fo(\022)1003 3746 y Fn(j)1040 3731 y Fo(;)h Fp(represen)m(ting)g(t)m(w)m(o)h(long)d (pieces)i(of)g(the)g(tra)5 b(jectory)28 b(of)f(an)h(ergo)s(dic)328 3852 y(rotation,)46 b(are)f(uniformly)e(distributed)i(on)g(the)g (circle,)i(and,)i(as)c(w)m(e)h(will)c(sho)m(w)47 b(in)328 3972 y(Lemma)38 b(13,)i(cos)17 b(2)p Fo(\031)t(\022)1148 3987 y Fn(j)1224 3972 y Fp(do)39 b(not)g(come)g(to)s(o)f(close)h(to)g (one)g(another.)64 b(W)-8 b(e)39 b(claim)e(that)328 4093 y(this)26 b(implies)d(that)j(the)g(ratio)f(of)g(pro)s(ducts)i(in)e (\(3.2\))g(do)s(es)i(not)f(deviate)g(m)m(uc)m(h)g(from)f Fo(e)3499 4056 y Fi(0)3539 4093 y Fo(:)328 4213 y Fp(W)-8 b(e)31 b(mak)m(e)h(it)e(precise)h(in)g(the)g(follo)m(wing)e(Lemma)g (that)i(will)e(b)s(e)i(pro)m(v)m(ed)i(in)d(Section)h(5.)328 4383 y Fg(LEMMA)37 b(7)49 b Ff(Supp)-5 b(ose)50 b Fo(d)57 b(<)h(k)1591 4347 y Fn(\013)1691 4383 y Ff(for)51 b(some)f Fo(\013)58 b(<)f Fp(2)p Fo(;)51 b Ff(and)f Fo(\022)s(;)17 b(!)54 b Ff(as)c(in)h(The)-5 b(or)g(em)328 4503 y(2.)100 b(Then)52 b(for)h(any)g Fo(\017)63 b(>)e Fp(0)53 b Ff(ther)-5 b(e)54 b(exists)f Fo(k)2098 4518 y Fi(3)2137 4503 y Fp(\()p Fo(\017;)17 b(!)t(;)g(\013)q(;)g(\022)s(;)g(y)2614 4518 y Fi(1)2651 4503 y Fp(\))53 b Ff(such)g(that)h(for)f Fo(k)65 b(>)328 4624 y(k)379 4639 y Fi(3)418 4624 y Fp(\()p Fo(\017;)17 b(!)t(;)g(\013)q(;)g(\022)s(;)g(y)895 4639 y Fi(1)932 4624 y Fp(\))35 b Ff(we)g(have)f(that)h(for)g(any)g Fo(z)d Fm(2)c Fp([)p Fm(\000)p Fp(1)p Fo(;)17 b Fp(1])p Fo(;)1241 4793 y Fm(j)1286 4726 y Fb(Q)1364 4813 y Fn(`)p Fc(6)p Fi(=)p Fn(j)1484 4793 y Fp(\()p Fo(z)27 b Fm(\000)c Fp(cos)17 b(2)p Fo(\031)t(\022)1994 4808 y Fn(`)2027 4793 y Fp(\))p Fm(j)p 1093 4839 1149 4 v 1093 4931 a(j)1138 4865 y Fb(Q)1216 4952 y Fn(k)r Fc(6)p Fi(=)p Fn(j)1346 4931 y Fp(\(cos)g(2)p Fo(\031)t(\022)1684 4946 y Fn(j)1743 4931 y Fm(\000)22 b Fp(cos)17 b(2)p Fo(\031)t(\022)2142 4946 y Fn(`)2175 4931 y Fp(\))p Fm(j)2279 4810 y Fi(def)2289 4863 y Fp(=)2414 4795 y Fm(j)p Fo(I)2485 4810 y Fi(1)2524 4795 y Fm(j)p 2414 4839 138 4 v 2414 4931 a(j)p Fo(I)2485 4946 y Fi(2)2524 4931 y Fm(j)2589 4863 y(\024)28 b Fo(e)2739 4822 y Fn(k)r(\017)1898 5214 y Fp(10)p eop %%Page: 11 12 11 11 bop 328 631 a Fp(W)-8 b(e)35 b(can)g(no)m(w)g(complete)f(the)h (pro)s(of)f(of)g(Lemma)f(4.)50 b(Set)2486 605 y(^)2484 631 y Fo(k)34 b Fp(=)d(max)p Fm(f)p Fp(4)p Fo(k)3008 646 y Fi(cos)3106 631 y Fp(\()3157 592 y Fn(\017)p 3154 608 36 4 v 3154 665 a Fi(6)3199 631 y Fo(;)17 b(E)6 b(;)17 b(!)t Fp(\))22 b(+)328 769 y(1)p Fo(;)17 b(k)472 784 y Fi(3)511 769 y Fp(\()580 730 y Fn(\017)p 559 746 71 4 v 559 803 a Fi(16)639 769 y Fo(;)g(!)t(;)g(\013)q(;)g(\022)s(;)g(y) 1039 784 y Fi(1)1076 769 y Fp(\))p Fm(g)p Fo(:)46 b Fp(T)-8 b(ak)m(e)48 b Fo(\013)k(<)f Fp(2)p Fo(;)c Fp(and)f Fo(k)55 b Fm(2)d Fo(K)r(;)68 b(k)54 b(>)2693 743 y Fp(^)2690 769 y Fo(k)s(;)47 b Fp(and)g Fo(\022)3067 784 y Fi(0)3153 769 y Fp(suc)m(h)h(that)328 889 y Fm(j)p Fo(P)419 904 y Fn(k)461 889 y Fp(\()p Fo(\022)544 904 y Fi(0)584 889 y Fp(\))p Fm(j)32 b(\025)830 850 y Fi(1)p 801 866 95 4 v 801 876 a Fc(p)p 860 876 36 3 v 55 x Fi(2)922 889 y Fp(exp)q(\()p Fo(k)s(\015)5 b Fp(\()p Fo(E)h Fp(\)\))p Fo(:)35 b Fp(Let)g Fo(z)1695 904 y Fi(0)1767 889 y Fp(=)d(cos)17 b(2)p Fo(\031)t Fp(\()p Fo(\022)2213 904 y Fi(0)2276 889 y Fp(+)2386 850 y Fn(k)r Fc(\000)p Fi(1)p 2386 866 129 4 v 2433 924 a(2)2525 889 y Fo(!)t Fp(\))p Fo(:)34 b Fp(Supp)s(ose)i Fo(d)c(<)g(k)3319 853 y Fn(\013)3368 889 y Fo(:)j Fp(W)-8 b(e)328 1022 y(ev)j(aluate)35 b Fo(Q)790 1037 y Fn(k)833 1022 y Fp(\()p Fo(z)916 1037 y Fi(0)956 1022 y Fp(\))h(using)g(\(3.2\))f(and)h(Lemma)f(7)h(to)f (estimate)h(the)g(ratio)e(of)i(pro)s(ducts)328 1142 y(in)c(\(3.2\).)43 b(W)-8 b(e)33 b(obtain)1089 1311 y(1)p 1047 1356 132 4 v 1047 1373 a Fm(p)p 1130 1373 49 4 v 82 x Fp(2)1189 1379 y Fo(e)1234 1338 y Fn(k)r(\015)t Fi(\()p Fn(E)t Fi(\))1455 1379 y Fm(\024)28 b(j)p Fo(Q)1665 1394 y Fn(k)1708 1379 y Fp(\()p Fo(z)1791 1394 y Fi(0)1830 1379 y Fp(\))p Fm(j)g(\024)g Fo(e)2074 1338 y Fn(k)r Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\))p Fc(\000)2358 1311 y Fd(\017)p 2356 1323 31 4 v 2356 1364 a Fe(8)2396 1338 y Fi(\))2428 1379 y Fp(\()p Fo(k)d Fp(+)d(1\))p Fo(e)2782 1311 y Fd(k)q(\017)p 2782 1323 61 4 v 2782 1364 a Fe(16)328 1645 y Fp(and)33 b(the)g(con)m(tradiction)e(pro)m(v)m(es)j(Lemma)e(4)g(with)g Fo(k)2283 1660 y Fi(2)2350 1645 y Fp(=)27 b(max\()2676 1618 y(^)2673 1645 y Fo(k)s(;)17 b(k)2822 1660 y Fi(4)2861 1645 y Fp(\()p Fo(\017)p Fp(\)\))p Fo(:)p 3123 1653 42 59 v 328 2057 a Fj(4)161 b(P)l(olynomials)37 b(and)f(Measure.)64 b(Pro)t(of)37 b(of)e(Lemma)g(5.)328 2288 y Fp(W)-8 b(e)41 b(in)m(tro)s(duce)g(the)g(upp)s(er)h(Ly)m(apuno)m(v)g(exp)s(onen)m(t,) 48 b(\026)-53 b Fo(\015)5 b Fp(\()p Fo(E)h(;)17 b(\022)s Fp(\))42 b(=)p 2788 2209 136 4 v 42 w(lim)2923 2303 y Fn(k)r Fc(!1)3117 2241 y Fi(ln)11 b Fc(k)p Fn(M)3290 2253 y Fd(k)3329 2241 y Fi(\()p Fn(\022)r(;E)t Fi(\))p Fc(k)p 3117 2265 412 4 v 3304 2322 a Fn(k)3539 2288 y Fo(:)328 2408 y Fp(Craig-Simon)35 b([10)o(])j(sho)m(w)m(ed)i(that)e (for)f(an)m(y)h(ergo)s(dic)f(p)s(oten)m(tial)j(\026)-53 b Fo(\015)5 b Fp(\()p Fo(E)h(;)17 b(\022)s Fp(\))37 b Fm(\024)g Fo(\015)5 b Fp(\()p Fo(E)h Fp(\))37 b(for)328 2529 y(a.e.)51 b Fo(\022)38 b Fp(\(for)d(all)e Fo(\022)38 b Fp(in)c(the)i(quasip)s(erio)s(dic)d(case)j(\(1.1\)\).)51 b(Th)m(us,)37 b(b)m(y)f(\(2.1\),)f(for)g(ev)m(ery)i Fo(\022)328 2649 y Fp(and)f(su\016cien)m(tly)g(large)f Fo(n;)h Fp(w)m(e)h(ha)m(v)m (e)g Fm(j)p Fo(P)1841 2664 y Fn(n)1887 2649 y Fp(\()p Fo(\022)s Fp(\))p Fm(j)c Fo(<)g(e)2226 2613 y Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\)+)p Fn(\017)p Fi(\))p Fn(n)2562 2649 y Fo(:)j Fp(As)g(b)s(efore,)h Fo(P)3159 2664 y Fn(k)3202 2649 y Fp(\()p Fo(\022)s Fp(\))f(is)f(an)328 2769 y(ev)m(en)i(function) e(of)g Fo(\022)28 b Fp(+)1235 2730 y Fn(k)r Fc(\000)p Fi(1)p 1235 2746 129 4 v 1281 2804 a(2)1373 2769 y Fo(!)39 b Fp(and)d(can)g(b)s(e)g(written)f(as)h(a)f(p)s(olynomial)d(of)j (degree)i Fo(k)s(p)328 2890 y Fp(in)32 b(cos)18 b(2)p Fo(\031)t Fp(\()p Fo(\022)25 b Fp(+)914 2851 y Fn(k)r Fc(\000)p Fi(1)p 914 2867 V 961 2924 a(2)1053 2890 y Fo(!)t Fp(\))p Fo(:)32 b Fp(W)-8 b(e)34 b(will)d(sho)m(w)j(that)f(this) g(implies)d(that)j(for)g(large)f Fo(n;)46 b Fm(j)p Fo(P)3368 2905 y Fn(n)3414 2890 y Fp(\()p Fo(\022)s Fp(\))p Fm(j)328 3010 y Fp(can)33 b(b)s(e)g(b)s(ounded)g(ab)s(o)m(v)m(e)g(uniformly)e (in)g Fo(\022)s Fp(.)44 b(The)34 b(k)m(ey)g(statemen)m(t)f(is)328 3190 y Fg(THEOREM)j(8)49 b Ff(L)-5 b(et)28 b Fo(Q)p Fp(\()p Fo(x)p Fp(\))h(=)1558 3123 y Fb(P)1646 3150 y Fn(n)1646 3214 y(j)t Fi(=0)1789 3190 y Fo(c)1831 3205 y Fn(j)1868 3190 y Fo(x)1923 3154 y Fn(j)1988 3190 y Ff(b)-5 b(e)27 b(an)h(arbitr)-5 b(ary)28 b Fo(n)2691 3154 y Ff(t)q Fn(h)2797 3190 y Ff(de)-5 b(gr)g(e)g(e)27 b(p)-5 b(olynomial.)328 3310 y(Supp)g(ose)34 b Fm(j)p Fo(Q)p Fp(\()p Fo(z)887 3325 y Fi(0)927 3310 y Fp(\))p Fm(j)27 b Fp(=)g Fo(a)1174 3274 y Fn(n)1222 3310 y Fo(;)34 b Ff(some)g Fo(a)28 b(>)g Fp(0)p Fo(;)44 b(z)1880 3325 y Fi(0)1947 3310 y Fm(2)29 b Fp([)p Fm(\000)p Fp(1)p Fo(;)17 b Fp(1])p Fo(:)34 b Ff(Then)g(for)h(any)g Fp(0)27 b Fo(<)h(b)g(<)f(a)35 b Ff(and)328 3430 y(su\016ciently)j(lar)-5 b(ge)37 b Fo(n)h Ff(\()p Fo(n)33 b(>)g(n)1455 3445 y Fi(0)1495 3430 y Fp(\()p Fo(a=b)p Fp(\))p Ff(\))38 b(we)f(have)g Fm(jf)p Fo(\022)f Fm(2)e Fp(\(0)p Fo(;)17 b(\031)t Fp(\))32 b(:)h Fm(j)p Fo(Q)p Fp(\(cos)17 b Fo(\022)s Fp(\))p Fm(j)33 b Fo(<)g(b)3330 3394 y Fn(n)3378 3430 y Fm(gj)f(\024)328 3551 y Fo(c)p Fp(\()p Fo(a;)17 b(b)p Fp(\))28 b Fo(<)f(\031)t(:)328 3730 y Fp(Here)43 b(and)g(in)f(the)h(future)g(w)m(e)g(use)h Fm(j)73 b(\001)h(j)42 b Fp(for)g(Leb)s(esgue)i(measure.)74 b(The)43 b(non)m(triv-)328 3851 y(ial)d(statemen)m(t)i(here)g(is)g (that)f(the)h(measure)h(can)f(b)s(e)g(b)s(ounded)g(b)m(y)h(a)e(constan) m(t)i(not)328 3971 y(dep)s(enden)m(t)34 b(on)f Fo(n)g Fp(or)f(the)h(p)s(olynomial)c Fo(Q:)328 4091 y Fg(Remark:)63 b Fp(The)43 b(theorem)f(is)g(certainly)g(true)h(for)e Fm(jf)p Fo(z)49 b Fm(2)c Fp([)p Fm(\000)p Fp(1)p Fo(;)17 b Fp(1])44 b(:)h Fm(j)p Fo(Q)p Fp(\()p Fo(z)t Fp(\))p Fm(j)g Fo(<)f(b)3441 4055 y Fn(n)3488 4091 y Fm(gj)328 4212 y Fp(as)g(w)m(ell.)77 b(W)-8 b(e)44 b(form)m(ulate)f(it)g(with)g (the)i(cosine)f(b)s(ecause)h(it)e(is)h(con)m(v)m(enien)m(t)h(for)f(our) 328 4332 y(application)30 b(and)j(also)e(simpli\014es)g(the)i(pro)s (of.)474 4453 y(T)-8 b(o)35 b(complete)g(the)h(pro)s(of)e(of)h(Lemma)e (5)i(w)m(e)h(set)g Fo(A)2385 4468 y Fn(n)2465 4453 y Fp(=)31 b Fm(f)p Fo(\022)36 b Fm(2)c Fp(\(0)p Fo(;)17 b(\031)t Fp(\))31 b(:)i(for)f(all)e Fo(k)35 b(>)328 4573 y(n;)17 b Fm(j)p Fo(P)521 4588 y Fn(k)563 4573 y Fp(\(cos)g Fo(\022)s Fp(\))p Fm(j)51 b(\024)g Fo(e)1086 4537 y Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\)+)p Fn(\017=)p Fi(2\))p Fn(k)1488 4573 y Fm(g)p Fo(:)46 b Fp(Then)h Fm(j)p Fo(A)1980 4537 y Fn(c)1980 4598 y(n)2027 4573 y Fm(j)f Fp(go)s(es)g(to)g(0)f(as)i Fo(n)j Fm(!)h(1)p Fo(;)45 b Fp(ho)m(w)m(ev)m(er)k(if)328 4693 y(there)42 b(exists)h Fo(\022)h Fp(suc)m(h)f(that)e Fm(j)p Fo(P)1494 4708 y Fn(k)1537 4693 y Fp(\(cos)17 b Fo(\022)s Fp(\))p Fm(j)43 b Fo(>)f(e)2042 4657 y Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\)+)p Fn(\017)p Fi(\))p Fn(k)2374 4693 y Fo(;)g Fp(some)f Fo(k)46 b(>)d(n;)f Fp(then)g Fm(j)p Fo(A)3371 4657 y Fn(c)3371 4718 y(n)3418 4693 y Fm(j)g(\025)328 4844 y(jf)p Fo(\022)37 b Fm(2)e Fp(\(0)p Fo(;)17 b(\031)t Fp(\))34 b(:)h Fm(j)p Fo(P)1004 4859 y Fn(k)1046 4844 y Fp(\(cos)17 b Fo(\022)s Fp(\))p Fm(j)35 b Fo(>)f(e)1535 4808 y Fi(\()p Fn(\015)t Fi(\()p Fn(E)t Fi(\)+)p Fn(\017=)p Fi(2\))p Fn(k)1937 4844 y Fm(gj)h(\025)g Fo(\031)29 b Fm(\000)c Fo(c)p Fp(\()p Fo(e)2483 4764 y Fd(\015)s Fe(\()p Fd(E)s Fe(\)+)p Fd(\017)p 2483 4783 204 4 v 2569 4824 a(p)2701 4844 y Fo(;)17 b(e)2800 4764 y Fd(\015)s Fe(\()p Fd(E)s Fe(\)+)p Fd(\017=)p Fe(2)p 2800 4783 267 4 v 2917 4824 a Fd(p)3080 4844 y Fp(\))35 b Fo(>)f Fp(0)p Fo(:)j Fp(This)328 4965 y(con)m(tradiction)31 b(pro)m(v)m(es)k(Lemma)c(5.)p 1740 4973 42 59 v 1898 5214 a(11)p eop %%Page: 12 13 12 12 bop 328 631 a Ff(Pr)-5 b(o)g(of)32 b(of)g(The)-5 b(or)g(em)31 b(8.)42 b Fp(W)-8 b(e)30 b(start)g(with)f(the)h(follo)m (wing)e(elemen)m(tary)h(Lemma,)g(pro)m(v)m(en)328 751 y(b)s(elo)m(w:)328 941 y Fg(LEMMA)37 b(9)49 b Ff(F)-7 b(or)37 b(any)g(me)-5 b(asur)g(able)36 b(set)i Fo(B)f Fm(2)c Fh(R)11 b Fo(;)37 b Fm(j)p Fo(B)5 b Fm(j)32 b Fp(=)g Fo(m;)38 b Ff(any)f Fo(n)c Fm(2)g Fh(N)15 b Fo(;)37 b Ff(and)g(any)328 1061 y Fo(\016)47 b(<)42 b(m=n)h Ff(one)g(c)-5 b(an)42 b(\014nd)h Fo(x)1409 1076 y Fi(1)1448 1061 y Fo(;)17 b(:)g(:)g(:)f(;)h(x)1722 1076 y Fn(n)p Fi(+1)1902 1061 y Fm(2)43 b Fo(B)48 b Ff(such)43 b(that)g Fm(j)p Fo(x)2653 1076 y Fn(i)2710 1061 y Fm(\000)28 b Fo(x)2870 1076 y Fn(j)2907 1061 y Fm(j)p Fo(=\016)46 b Fm(2)e Fh(N)57 b Ff(for)43 b(al)5 b(l)328 1182 y Fo(i)28 b Fm(6)p Fp(=)f Fo(j:)328 1371 y Fp(In)45 b(other)f(w)m(ords,)k(one)d(can)f(\014nd)h Fo(n)30 b Fp(+)g(1)44 b(elemen)m(ts)h(of)e(an)i(arithmetic)d (progression)328 1491 y(with)37 b(step)i Fo(\016)j Fp(b)s(elonging)35 b(to)j Fo(B)5 b(:)38 b Fp(W)-8 b(e)38 b(will)d(no)m(w)k(pro)m(v)m(e)f (Theorem)h(8.)58 b(Let)38 b Fo(B)k Fp(=)36 b Fm(f)p Fo(\022)j Fm(2)328 1612 y Fp(\(0)p Fo(;)17 b(\031)t Fp(\))29 b(:)i Fm(j)p Fo(Q)p Fp(\(cos)17 b Fo(\022)s Fp(\))p Fm(j)30 b Fo(<)g(b)1224 1576 y Fn(n)1272 1612 y Fm(g)p Fo(:)k Fp(Let)g Fo(m)c Fp(=)h Fm(j)p Fo(B)5 b Fm(j)p Fo(:)33 b Fp(W)-8 b(e)35 b(\014x)g Fo(\016)f(<)2479 1573 y Fn(m)p 2479 1589 63 4 v 2489 1646 a(n)2551 1612 y Fo(;)g Fp(and)h(pic)m(k,)g (according)e(to)328 1732 y(Lemma)d(9,)i Fo(x)838 1747 y Fi(1)878 1732 y Fo(;)17 b(:)g(:)g(:)e(;)i(x)1151 1747 y Fn(n)p Fi(+1)1316 1732 y Fm(2)28 b Fo(B)5 b(;)32 b Fp(elemen)m(ts)g(of)f(an)h(arithmetic)d(progression)j(with)f(step)328 1853 y Fo(\016)n(:)f Fp(W)-8 b(e)29 b(write)g Fo(Q)p Fp(\()p Fo(x)p Fp(\))h(in)e(a)h(Lagrange)g(in)m(terp)s(olation)d(form)i (using)h(cos)18 b Fo(x)2941 1868 y Fi(1)2980 1853 y Fo(;)f(:)g(:)g(:)f (;)h Fp(cos)g Fo(x)3401 1868 y Fn(n)p Fi(+1)3539 1853 y Fo(:)328 1973 y Fp(Then)328 2239 y Fo(a)379 2198 y Fn(n)454 2239 y Fp(=)27 b Fm(j)p Fo(Q)p Fp(\()p Fo(z)745 2254 y Fi(0)785 2239 y Fp(\))p Fm(j)g Fp(=)982 2065 y Fb(\014)982 2114 y(\014)982 2164 y(\014)982 2214 y(\014)982 2264 y(\014)982 2314 y(\014)1009 2131 y Fn(n)p Fi(+1)1016 2156 y Fb(X)1015 2338 y Fn(j)t Fi(=1)1159 2239 y Fo(Q)p Fp(\(cos)17 b Fo(x)1476 2254 y Fn(j)1513 2239 y Fp(\))1639 2102 y Fb(Q)1717 2189 y Fn(k)r Fc(6)p Fi(=)p Fn(j)1847 2169 y Fp(\()p Fo(z)1930 2184 y Fi(0)1992 2169 y Fm(\000)22 b Fp(cos)c Fo(x)2294 2184 y Fn(k)2337 2169 y Fp(\))p 1561 2216 891 4 v 1561 2241 a Fb(Q)1640 2328 y Fn(k)r Fc(6)p Fi(=)p Fn(j)1770 2307 y Fp(\(cos)f Fo(x)2010 2322 y Fn(j)2069 2307 y Fm(\000)23 b Fp(cos)17 b Fo(x)2371 2322 y Fn(k)2414 2307 y Fp(\))2462 2065 y Fb(\014)2462 2114 y(\014)2462 2164 y(\014)2462 2214 y(\014)2462 2264 y(\014)2462 2314 y(\014)2517 2239 y Fo(<)28 b Fp(\()p Fo(n)p Fp(+1\))p Fo(b)2921 2198 y Fn(n)2984 2239 y Fp(max)3059 2297 y Fn(j)3183 2089 y Fb(\014)3183 2139 y(\014)3183 2189 y(\014)3183 2239 y(\014)3183 2289 y(\014)3297 2102 y(Q)3376 2189 y Fn(k)r Fc(6)p Fi(=)p Fn(j)3506 2169 y Fp(\()p Fo(z)3589 2184 y Fi(0)3651 2169 y Fm(\000)22 b Fp(cos)17 b Fo(x)3952 2184 y Fn(k)3996 2169 y Fp(\))p 3220 2216 V 3220 2241 a Fb(Q)3299 2328 y Fn(k)r Fc(6)p Fi(=)p Fn(j)3429 2307 y Fp(\(cos)g Fo(x)3669 2322 y Fn(j)3728 2307 y Fm(\000)22 b Fp(cos)c Fo(x)4030 2322 y Fn(k)4073 2307 y Fp(\))4121 2089 y Fb(\014)4121 2139 y(\014)4121 2189 y(\014)4121 2239 y(\014)4121 2289 y(\014)3365 2446 y Fp(\(4.1\))328 2567 y(W)-8 b(e)27 b(denote)g(the)g(last)f(ratio)f(b)m (y)j Fo(I)34 b Fp(and)27 b(will)e(estimate)g(separately)i(the)g(n)m (umerator)g(and)328 2687 y(denominator)k(in)h Fo(I)8 b(:)32 b Fp(First,)328 2943 y(exp)q(\()518 2860 y Fb(X)515 3045 y Fn(j)t Fc(6)p Fi(=)p Fn(k)657 2943 y Fp(ln)16 b Fm(j)p Fo(z)828 2958 y Fi(0)868 2943 y Fm(\000)p Fp(cos)h Fo(x)1147 2958 y Fn(j)1184 2943 y Fm(j)p Fp(\))28 b Fm(\024)g Fp(exp)q(\()1610 2835 y Fn(n)1571 2860 y Fb(X)1570 3042 y Fn(j)t Fi(=0)1709 2943 y Fp(ln)16 b Fm(j)p Fp(1)p Fm(\000)p Fp(cos\()p Fo(\031)t Fm(\000)p Fo(j)6 b(\016)t Fp(\))p Fm(j)p Fp(\))28 b Fm(\024)g Fp(exp)q(\(1)p Fo(=\016)2943 2826 y Fb(Z)3026 2852 y Fn(n\016)2989 3015 y Fi(0)3123 2943 y Fp(ln\(1+cos)16 b Fo(x)p Fp(\))p Fo(dx)p Fp(+ln)h(2\))p Fo(:)3365 3149 y Fp(\(4.2\))328 3269 y(Similarly)-8 b(,)328 3576 y(exp)q(\()518 3493 y Fb(X)515 3677 y Fn(j)t Fc(6)p Fi(=)p Fn(k)657 3576 y Fp(ln)16 b Fm(j)h Fp(cos)g Fo(x)1002 3591 y Fn(k)1045 3576 y Fm(\000)p Fp(cos)g Fo(x)1324 3591 y Fn(j)1361 3576 y Fm(j)p Fp(\))28 b Fm(\025)g Fp(exp)q(\()1932 3449 y Fi([)1962 3421 y Fd(n)p Fe(+1)p 1962 3435 116 4 v 2004 3475 a(2)2087 3449 y Fi(])1959 3493 y Fb(X)1788 3715 y Fo(j)34 b Fp(=)28 b Fm(\000)p Fp([)2080 3676 y Fn(n)p Fi(+1)p 2080 3692 133 4 v 2129 3750 a(2)2223 3715 y Fp(])1906 3836 y Fo(j)34 b Fm(6)p Fp(=)27 b(0)2308 3576 y(ln)16 b Fm(j)h Fp(cos\()p Fo(\031)t(=)p Fp(2+)p Fo(j)6 b(\016)t Fp(\))p Fm(j)p Fp(\))26 b Fm(\025)j Fp(exp)q(\(2)p Fo(=\016)3529 3459 y Fb(Z)3622 3456 y Fd(n)p Fe(+1)p 3622 3470 116 4 v 3664 3511 a(2)3747 3485 y Fn(\016)3575 3647 y Fi(0)3801 3576 y Fp(ln)16 b(sin)h Fo(xdx)p Fp(\))3365 3963 y(\(4.3\))328 4084 y(W)-8 b(e)33 b(set)g Fo(m)733 4099 y Fi(1)801 4084 y Fp(=)27 b Fo(n\016)n(;)34 b Fp(and,)e(com)m (bining)f(\(4.2\),)h(and)h(\(4.3\))f(w)m(e)i(obtain)423 4350 y Fo(I)h Fm(\024)28 b Fp(2)17 b(exp)837 4203 y Fb( )946 4282 y Fo(n)p 913 4326 125 4 v 913 4418 a(m)998 4433 y Fi(1)1064 4203 y Fb( )1130 4350 y Fm(\000)1224 4232 y Fb(Z)1307 4259 y Fn(\031)1270 4421 y(m)1332 4430 y Fe(1)1388 4350 y Fp(ln)o(\(1)22 b(+)g(cos)c Fo(x)p Fp(\))p Fo(dx)k Fp(+)g(2)2209 4232 y Fb(Z)2292 4259 y Fn(\031)r(=)p Fi(2)2255 4421 y Fn(m)2317 4430 y Fe(1)2352 4421 y Fn(=)p Fi(2)2443 4350 y Fp(ln)16 b(sin)g Fo(xdx)2838 4203 y Fb(!)2927 4350 y Fm(\000)22 b Fp(ln)16 b(sin)3271 4282 y Fo(m)3356 4297 y Fi(1)p 3271 4326 V 3309 4418 a Fp(2)3405 4203 y Fb(!)3365 4532 y Fp(\(4.4\))328 4653 y(where)34 b(w)m(e)f(used)h(the)f(elemen)m(tary)g(iden)m(tit)m(y)f(\(5.8\).)43 b(W)-8 b(e)33 b(set)702 4910 y Fo(g)t Fp(\()p Fo(m)p Fp(\))27 b(=)h(exp)q(\(1)p Fo(=m)p Fp(\()p Fm(\000)1547 4793 y Fb(Z)1630 4819 y Fn(\031)1593 4981 y(m)1693 4910 y Fp(ln\(1)21 b(+)h(cos)c Fo(x)p Fp(\))p Fo(dx)k Fp(+)g(2)2514 4793 y Fb(Z)2597 4819 y Fn(\031)r(=)p Fi(2)2560 4981 y Fn(m=)p Fi(2)2731 4910 y Fp(ln)16 b(sin)g Fo(xdx)p Fp(\))p Fo(:)1898 5214 y Fp(12)p eop %%Page: 13 14 13 13 bop 328 631 a Fp(It)39 b(is)f(easily)f(v)m(eri\014ed)j(that)e Fo(g)k Fp(is)c(a)g(monotone)g(decreasing)h(function)f(on)h(\(0)p Fo(;)17 b(\031)t Fp(])p Fo(;)38 b Fp(and)328 751 y Fo(g)t Fp(\()p Fo(\031)t Fp(\))27 b(=)g(1)p Fo(:)33 b Fp(F)-8 b(or)31 b Fo(m)1012 766 y Fi(1)1080 751 y Fm(\025)1195 712 y Fn(\031)p 1195 728 43 4 v 1199 786 a Fi(3)1280 751 y Fp(w)m(e)j(ha)m(v)m(e)g(sin)1795 712 y Fn(m)1857 721 y Fe(1)p 1795 728 98 4 v 1826 786 a Fi(2)1930 751 y Fm(\025)28 b Fp(1)p Fo(=)p Fp(2)k(and)h(w)m(e)g(can)g(rewrite)g (\(4.1\))f(as)1439 995 y Fo(a)p 1439 1039 52 4 v 1444 1131 a(b)1517 916 y Fb( )1744 995 y Fp(1)p 1593 1039 352 4 v 1593 1131 a(4\()p Fo(n)22 b Fp(+)g(1\))1954 916 y Fb(!)2034 912 y Fe(1)p 2030 924 39 4 v 2030 965 a Fd(n)2110 1062 y Fo(<)28 b(g)t Fp(\()p Fo(m)2388 1077 y Fi(1)2427 1062 y Fp(\))328 1324 y(whic)m(h)37 b(implies)e Fo(m)1032 1339 y Fi(1)1107 1324 y Fm(\024)g Fo(g)1270 1288 y Fc(\000)p Fi(1)1381 1228 y Fb(\020)1440 1285 y Fn(a)p 1440 1301 38 4 v 1444 1358 a(b)1488 1324 y Fp(\()1630 1285 y Fi(1)p 1536 1301 224 4 v 1536 1358 a(4\()p Fn(n)p Fi(+1\))1769 1324 y Fp(\))1821 1260 y Fe(1)p 1817 1272 39 4 v 1817 1313 a Fd(n)1869 1228 y Fb(\021)1956 1324 y Fp(and,)j(\014nally)-8 b(,)37 b(for)f(an)m(y)i Fo(\017)f Fp(and)g(su\016cien)m(tly)328 1462 y(large)31 b Fo(n;)i Fp(w)m(e)h(obtain)d Fo(m)d Fm(\024)h Fp(\(1)21 b(+)h Fo(\017)p Fp(\))p Fo(g)1684 1426 y Fc(\000)p Fi(1)1778 1462 y Fp(\()1826 1423 y Fn(a)p 1826 1439 38 4 v 1830 1497 a(b)1874 1462 y Fp(\))p Fo(:)p 2021 1470 42 59 v 328 1682 a Ff(Pr)-5 b(o)g(of)35 b(of)f(L)-5 b(emma)34 b(9.)43 b Fp(W)-8 b(e)33 b(will)d(need)k(an)f(elemen)m(tary) 328 1886 y Fg(LEMMA)k(10)49 b Ff(L)-5 b(et)39 b Fo(A)33 b Fp(=)h Fm([)1418 1849 y Fc(1)1418 1910 y Fn(i)p Fi(=1)1536 1886 y Fo(A)1609 1901 y Fn(i)1638 1886 y Fo(;)50 b(A)1788 1901 y Fn(i)1854 1886 y Ff(ar)-5 b(e)38 b(me)-5 b(asur)g(able)37 b(sets,)h Fo(i)c Fm(2)g Fh(N)15 b Fo(;)38 b Ff(and)f(supp)-5 b(ose)328 2006 y(every)28 b Fo(x)g Fm(2)h Fo(A)f Ff(b)-5 b(elongs)28 b(to)g(no)g(mor)-5 b(e)28 b(than)h Fo(k)i Ff(di\013er)-5 b(ent)28 b Fo(A)2420 2021 y Fn(j)2457 1964 y Fc(0)2480 2006 y Ff(s)p Fo(:)g Ff(Then)g Fm(j)p Fo(A)p Fm(j)f(\025)3096 1967 y Fi(1)p 3094 1983 39 4 v 3094 2040 a Fn(k)3159 1940 y Fb(P)3247 1966 y Fc(1)3247 2031 y Fn(i)p Fi(=1)3382 2006 y Fm(j)p Fo(A)3483 2021 y Fn(i)3511 2006 y Fm(j)p Fo(:)328 2223 y Ff(Pr)-5 b(o)g(of.)43 b Fp(W)-8 b(e)33 b(use)g(induction)f(in)g Fo(k)s(:)g Fp(Set)h Fo(B)1867 2238 y Fn(j)1932 2223 y Fp(=)27 b Fo(A)2108 2238 y Fn(j)2167 2223 y Fm(\\)c Fp(\()p Fm([)2360 2176 y Fn(j)t Fc(\000)p Fi(1)2360 2246 y Fn(i)p Fi(=1)2487 2223 y Fo(A)2560 2238 y Fn(i)2588 2223 y Fp(\))p Fo(;)44 b(j)34 b Fp(=)28 b(2)p Fo(;)17 b(:)g(:)g(:)e(;)33 b Fp(and)f(write)608 2395 y Fc(1)583 2420 y Fb(X)586 2603 y Fn(i)p Fi(=1)719 2503 y Fm(j)p Fo(A)820 2518 y Fn(i)848 2503 y Fm(j)c Fp(=)f Fm(j)p Fo(A)1108 2518 y Fi(1)1148 2503 y Fm(j)21 b Fp(+)h Fm(j)p Fo(A)1396 2518 y Fi(2)1436 2503 y Fm(n)p Fo(A)1559 2518 y Fi(1)1598 2503 y Fm(j)g Fp(+)g Fm(\001)17 b(\001)g(\001)j Fp(+)i Fm(j)p Fo(A)2083 2518 y Fn(n)2130 2503 y Fm(n)g([)2268 2462 y Fn(n)p Fc(\000)p Fi(1)2268 2528 y Fn(i)p Fi(=1)2428 2503 y Fo(A)2501 2518 y Fn(i)2529 2503 y Fm(j)g Fp(+)g Fm(\001)17 b(\001)g(\001)j Fp(+)2938 2395 y Fc(1)2913 2420 y Fb(X)2916 2603 y Fn(i)p Fi(=2)3050 2503 y Fm(j)p Fo(B)3152 2518 y Fn(i)3180 2503 y Fm(j)27 b Fp(=)1062 2864 y(=)g Fm(j)p Fo(A)p Fm(j)22 b Fp(+)1438 2756 y Fc(1)1414 2781 y Fb(X)1417 2963 y Fn(i)p Fi(=2)1550 2864 y Fm(j)p Fo(B)1652 2879 y Fn(i)1680 2864 y Fm(j)28 b(\024)g(j)p Fo(A)p Fm(j)21 b Fp(+)h(\()p Fo(k)k Fm(\000)c Fp(1\))p Fm(j)p Fo(A)p Fm(j)27 b Fp(=)h Fo(k)s Fm(j)p Fo(A)p Fm(j)328 3121 y Fp(since)38 b(if)e Fo(x)h Fm(2)f(\\)926 3080 y Fn(k)926 3145 y(j)t Fi(=1)1053 3121 y Fo(B)1127 3136 y Fn(i)1151 3146 y Fd(j)1188 3121 y Fo(;)53 b Fp(2)36 b Fm(\024)h Fo(i)1500 3136 y Fi(1)1575 3121 y Fo(<)f(i)1720 3136 y Fi(2)1796 3121 y Fo(<)g Fm(\001)17 b(\001)g(\001)34 b Fo(<)i(i)2205 3136 y Fn(k)2248 3121 y Fo(;)i Fp(then)g Fo(x)g Fp(necessarily)g(b)s(elongs)f(to)328 3254 y Fm(\\)394 3213 y Fn(k)394 3279 y(j)t Fi(=0)521 3254 y Fo(A)594 3269 y Fn(i)618 3279 y Fd(j)687 3254 y Fp(for)32 b(some)h Fo(i)1114 3269 y Fi(0)1181 3254 y Fo(<)28 b(i)1318 3269 y Fi(1)1357 3254 y Fo(;)33 b Fp(whic)m(h)g(is)f(a)h(con)m(tradiction.)p 2513 3262 42 59 v 474 3474 a(T)-8 b(o)36 b(complete)g(the)g(pro)s(of)f (of)h(Lemma)e(9)i(w)m(e)h(let)e Fo(A)f Fp(=)f([0)p Fo(;)17 b(\016)t Fp(])36 b(and)g(set)h Fo(A)3176 3489 y Fn(i)3237 3474 y Fp(=)d(\()p Fo(B)29 b Fm(\000)328 3594 y Fo(i\016)t Fp(\))d Fm(\\)g Fo(A;)53 b(i)36 b Fm(2)h Fh(Z)m Fo(:)h Fp(Assume)g(the)g(statemen)m(t)g(of)f(the)h(Lemma)e(is)h(false,)i(whic) m(h)f(means)328 3715 y(ev)m(ery)c Fo(x)28 b Fm(2)h Fo(A)j Fp(b)s(elongs)g(to)h(no)f(more)g(than)h Fo(n)f Fp(di\013eren)m(t)h Fo(A)2493 3730 y Fn(j)2530 3673 y Fc(0)2553 3715 y Fp(s)p Fo(:)g Fp(Then,)h(b)m(y)f(Lemma)f(10,)439 3928 y Fo(m)p 439 3972 86 4 v 453 4064 a(n)562 3995 y(>)c(\016)j Fp(=)d Fm(j)p Fo(A)p Fm(j)f(\025)h(j)22 b([)1221 3954 y Fc(1)1221 4020 y Fn(i)p Fi(=)p Fc(\0001)1452 3995 y Fo(A)1525 4010 y Fn(i)1554 3995 y Fm(j)27 b(\025)1729 3928 y Fp(1)p 1724 3972 59 4 v 1724 4064 a Fo(n)1876 3887 y Fc(1)1851 3912 y Fb(X)1809 4095 y Fn(i)p Fi(=)p Fc(\0001)2030 3995 y Fm(j)p Fo(A)2131 4010 y Fn(i)2159 3995 y Fm(j)g Fp(=)2332 3928 y(1)p 2328 3972 V 2328 4064 a Fo(n)2479 3887 y Fc(1)2455 3912 y Fb(X)2412 4095 y Fn(i)p Fi(=)p Fc(\0001)2633 3995 y Fm(j)p Fo(B)g Fm(\\)c Fp(\()p Fo(A)f Fp(+)g Fo(i\016)t Fp(\))p Fm(j)28 b Fp(=)3369 3928 y Fo(m)p 3369 3972 86 4 v 3383 4064 a(n)328 4287 y Fp(This)33 b(con)m(tradiction)e(pro)m(v)m (es)j(the)f(Lemma.)p 2006 4295 42 59 v 1898 5214 a(13)p eop %%Page: 14 15 14 14 bop 328 631 a Fj(5)161 b(Uniform)55 b(distribution)f(and)h(small) h(denomina-)570 814 y(tors.)71 b(Pro)t(of)54 b(of)g(Lemma)e(7.)328 1033 y Fp(Let)36 b Fo(\036)p Fp(\()p Fo(n)p Fp(\))f(b)s(e)h(a)g (monotone)f(increasing)g(function,)h Fo(\036)p Fp(\()p Fo(n)p Fp(\))d(=)g Fo(o)p Fp(\()p Fo(n)p Fp(\))p Fo(:)j Fp(W)-8 b(e)36 b(will)d(sa)m(y)k(that)328 1153 y Fo(x)383 1168 y Fi(1)423 1153 y Fo(;)17 b(:)g(:)g(:)f(;)h(x)697 1168 y Fn(n)784 1153 y Fp(are)40 b Fo(\036)p Fm(\000)p Fp(uniformly)e(distributed)h(on)h([)p Fo(a;)17 b(b)p Fp(])41 b(if)d(for)i(an)m(y)h Fo(f)51 b Fm(2)40 b Fo(C)7 b Fp(\([)p Fo(a;)17 b(b)p Fp(]\))41 b(w)m(e)328 1273 y(ha)m(v)m(e)34 b Fm(j)598 1207 y Fb(P)685 1233 y Fn(n)685 1298 y(i)p Fi(=1)820 1273 y Fo(f)11 b Fp(\()p Fo(x)972 1288 y Fn(i)1000 1273 y Fp(\))22 b Fm(\000)1210 1234 y Fn(n)p 1170 1250 123 4 v 1170 1308 a(b)p Fc(\000)p Fn(a)1319 1202 y Fb(R)1374 1229 y Fn(b)1358 1299 y(a)1425 1273 y Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(dx)p Fm(j)28 b(\024)g Fo(\036)p Fp(\()p Fo(n)p Fp(\)V)-8 b(ar)o Fo(f)5 b(:)474 1394 y Fp(Clearly)-8 b(,)23 b(if)e Fo(x)957 1409 y Fi(1)997 1394 y Fo(;)c(:)g(:)g(:)f(;)h(x)1271 1409 y Fn(n)1339 1394 y Fp(are)22 b Fo(\036)p Fm(\000)p Fp(uniformly)e (distributed)h(and)h Fo(y)2775 1409 y Fi(1)2814 1394 y Fo(;)17 b(:)g(:)g(:)f(;)h(y)3081 1409 y Fn(n)3149 1394 y Fp(are)k Fo( )t Fm(\000)p Fp(uniformly)328 1514 y(distributed,)j (then)e Fo(x)1112 1529 y Fi(1)1152 1514 y Fo(;)17 b(:)g(:)g(:)f(;)h(x) 1426 1529 y Fn(n)1473 1514 y Fo(;)g(y)1565 1529 y Fi(1)1604 1514 y Fo(;)g(:)g(:)g(:)f(;)h(y)1871 1529 y Fn(n)1939 1514 y Fp(are)22 b Fo(\036)p Fp(\()p Fo(n)p Fp(\))q(+)q Fo( )t Fp(\()p Fo(n)p Fp(\))p Fm(\000)p Fp(uniformly)d(distributed.)328 1635 y(W)-8 b(e)35 b(will)e(sa)m(y)i(that)g Fo(f)42 b Fm(2)32 b Fo(C)7 b Fp(\([)p Fo(a;)17 b(b)p Fp(]\))35 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b(:)g(:)g(:)f(;)h(n)p Fp(])33 b Ff(we)328 3075 y(have:)815 3191 y Fn(n)776 3216 y Fb(X)784 3387 y Fd(i)p Fe(=1)643 3435 y Fd(i)p Fa(6)p Fe(=)p Fd(j)737 3450 y Fe(1)772 3435 y Fd(;:::)o(;j)891 3455 y( )r Fe(\()p Fd(n)p Fe(\))1046 3299 y Fp(ln)16 b Fm(j)p Fo(z)26 b Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)1495 3314 y Fn(i)1523 3299 y Fp(\))p Fm(j)83 b(\024)1920 3231 y Fo(n)p 1842 3275 215 4 v 1842 3367 a(b)23 b Fm(\000)f Fo(a)2083 3152 y Fb( )2149 3181 y(Z)2232 3208 y Fn(b)2195 3370 y(a)2283 3299 y Fp(ln)16 b Fm(j)p Fo(z)26 b Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fm(j)p Fo(dx)22 b Fp(+)g Fo(\017)3063 3152 y Fb(!)815 3550 y Fn(n)776 3575 y Fb(X)784 3747 y Fd(i)p Fe(=1)643 3795 y Fd(i)p Fa(6)p Fe(=)p Fd(j)737 3810 y Fe(1)772 3795 y Fd(;:::)o(;j)891 3815 y( )r Fe(\()p Fd(n)p Fe(\))1046 3658 y Fp(ln)16 b Fm(j)p Fo(z)26 b Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)1495 3673 y Fn(i)1523 3658 y Fp(\))p Fm(j)83 b(\025)1920 3591 y Fo(n)p 1842 3635 V 1842 3727 a(b)23 b Fm(\000)f Fo(a)2083 3512 y Fb( )2149 3541 y(Z)2232 3568 y Fn(b)2195 3730 y(a)2283 3658 y Fp(ln)16 b Fm(j)p Fo(z)26 b Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fm(j)p Fo(dx)22 b Fm(\000)h Fo(\017)3065 3512 y Fb(!)3153 3658 y Fp(+)136 b(\(5.2\))1672 3944 y(+)84 b Fo(\036)p Fp(\()p Fo(n)p Fp(\))p Fo(p)p Fp(\()p Fo(f)11 b Fp(\))17 b(ln)127 b(min)2410 3990 y Fd(i)p Fe(=1)p Fd(;:::;n)2322 4048 y(i)p Fa(6)p Fe(=)p Fd(j)2416 4063 y Fe(1)2451 4048 y Fd(;:::;j)2571 4068 y( )r Fe(\()p Fd(n)p Fe(\))2725 3944 y Fm(j)p Fo(z)26 b Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)3076 3959 y Fn(i)3104 3944 y Fp(\))p Fm(j)328 4353 y Ff(Pr)-5 b(o)g(of.)52 b Fp(F)-8 b(or)35 b(0)e Fo(<)g(\016)k(<)c Fp(1)i(put)h Fo(g)1508 4368 y Fn(\016)1546 4353 y Fp(\()p Fo(z)t Fp(\))d(=)1813 4179 y Fb(8)1813 4254 y(<)1813 4403 y(:)2039 4247 y Fp(ln)16 b Fm(j)p Fo(z)t Fm(j)p Fo(;)45 b Fm(j)p Fo(z)t Fm(j)27 b Fo(>)h(\016)2059 4458 y Fp(ln)16 b Fo(\016)n(;)45 b Fm(j)p Fo(z)t Fm(j)28 b(\024)g Fo(\016)n(:)2684 4353 y Fp(Giv)m(en)35 b Fo(\017)f(>)f Fp(0)p Fo(;)i Fp(\014x)i Fo(\016)t 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y Fo(g)1082 683 y Fn(\016)1120 668 y Fp(\()p Fm(j)p Fo(z)k Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fm(j)p Fp(\))p Fo(dx)22 b Fp(+)g Fo(\036)p Fp(\()p Fo(n)p Fp(\)V)-8 b(ar)o Fo(g)2229 683 y Fn(\016)2267 668 y Fp(\()p Fm(j)p Fo(z)26 b Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fm(j)p Fp(\))22 b Fm(\000)g Fo( )t Fp(\()p Fo(n)p Fp(\))17 b(ln)f Fo(\016)31 b Fm(\024)328 927 y(\024)521 859 y Fo(n)p 443 903 V 443 995 a(b)23 b Fm(\000)f Fo(a)667 927 y Fp(\()705 810 y Fb(Z)788 836 y Fn(b)752 998 y(a)839 927 y Fp(ln\()p Fm(j)p Fo(z)t Fm(\000)p Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fm(j)p Fp(\))p Fo(dx)p Fp(+)p Fo(\017=)p Fp(2\)+)p Fo(\036)p Fp(\()p Fo(n)p Fp(\))p Fo(p)p Fp(\()p Fo(f)g Fp(\)\()p Fm(\000)17 b Fp(ln)e Fo(\016)t Fp(+ln)o(\(max)2720 976 y Fn(x)2847 927 y Fm(j)p Fo(z)t Fm(\000)p Fo(f)c Fp(\()p Fo(x)p Fp(\))p Fm(j)p Fp(\)\))p Fm(\000)p Fo( )t Fp(\()p Fo(n)p Fp(\))17 b(ln)g Fo(\016)32 b Fm(\024)773 1185 y(\024)966 1118 y Fo(n)p 888 1162 V 888 1253 a(b)22 b Fm(\000)h Fo(a)1112 1185 y Fp(\()1150 1068 y Fb(Z)1233 1094 y Fn(b)1196 1256 y(a)1284 1185 y Fp(ln)o(\()p Fm(j)p Fo(z)k Fm(\000)22 b Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fm(j)p Fp(\))p Fo(dx)22 b Fp(+)g Fo(\017)p Fp(\))p Fo(;)45 b Fp(for)27 b Fo(n)h(>)g(n)2625 1200 y Fi(1)2664 1185 y Fp(\()p Fo(\017;)17 b(\036;)g( )t(;)g(f)11 b Fp(\))p Fo(:)474 1381 y Fp(Let)145 b(min)737 1428 y Fd(i)p Fe(=1)p Fd(;:::;n)649 1485 y(i)p Fa(6)p Fe(=)p Fd(j)743 1500 y Fe(1)778 1485 y Fd(;:::;j)898 1506 y( )r Fe(\()p Fd(n)p Fe(\))1052 1381 y Fm(j)p Fo(z)26 b Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)1403 1396 y Fn(i)1431 1381 y Fp(\))p Fm(j)27 b Fp(=)h Fo(c)g(<)f Fp(1)p Fo(:)33 b Fp(F)-8 b(or)31 b(an)i(estimate)f(from)f(b)s(elo)m(w)h(w)m(e)i(write:)1025 1630 y Fn(n)986 1655 y Fb(X)995 1827 y Fd(i)p Fe(=1)853 1874 y Fd(i)p Fa(6)p Fe(=)p Fd(j)947 1889 y Fe(1)982 1874 y Fd(;:::;j)1102 1895 y( )r Fe(\()p Fd(n)p Fe(\))1256 1738 y Fp(ln)16 b Fm(j)p Fo(z)26 b Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)1705 1753 y Fn(i)1733 1738 y Fp(\))p Fm(j)28 b Fp(=)2102 1630 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b Fm(\000)h Fo(a)834 2378 y Fp(\()872 2261 y Fb(Z)955 2287 y Fn(b)918 2449 y(a)1006 2378 y Fp(ln)16 b Fm(j)p Fo(z)26 b Fm(\000)d Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fm(j)p Fo(dx)22 b Fm(\000)h Fo(\017)p Fp(\))f(+)g Fo(\036)p Fp(\()p Fo(n)p Fp(\))p Fo(p)p Fp(\()p Fo(f)11 b Fp(\))17 b(ln)e Fo(c;)44 b Fp(for)27 b Fo(n)h(>)g(n)2941 2393 y Fi(2)2981 2378 y Fp(\()p Fo(\017;)17 b(\036;)g( )t Fp(\))p 3358 2386 42 59 v 328 2650 a(Let)33 b Fm(f)p Fo(x)p Fm(g)f Fp(denote)i(the)f(fractional)d(part)i(of)g Fo(x:)h Fp(W)-8 b(e)33 b(will)e(use)i(the)g(follo)m(wing)328 2818 y Fg(LEMMA)k(12)49 b Ff(F)-7 b(or)40 b(any)h Fo(!)j Ff(satisfying)d(\(1.2\))f(and)g(any)h Fo(x)e Fm(2)g Fh(R)11 b Fo(;)41 b Ff(p)-5 b(oints)40 b Fm(f)p Fo(x)p Fm(g)p Fo(;)17 b Fm(f)p Fo(x)27 b Fp(+)328 2939 y Fo(!)t Fm(g)p Fo(;)17 b(:)g(:)g(:)e(;)i Fm(f)p Fo(x)28 b Fp(+)g(\()p Fo(n)h Fm(\000)g Fp(1\))p Fo(!)t Fm(g)42 b Ff(ar)-5 b(e)43 b Fo(\036)p Fm(\000)p Ff(uniformly)g(distribute)-5 b(d)44 b(on)f Fp([0)p Fo(;)17 b Fp(1])p Fo(;)43 b Ff(with)g Fo(\036)p Fp(\()p Fo(n)p Fp(\))g(=)328 3059 y Fo(c)370 3074 y Fi(1)409 3059 y Fp(\()p Fo(!)t Fp(\))p Fo(n)608 3023 y Fi(1)p Fc(\000)p Fn(r)r Fi(\()p Fn(!)r Fi(\))832 2999 y Fa(\000)p Fe(1)936 3059 y Fp(ln)16 b Fo(n:)35 b Ff(Her)-5 b(e)35 b Fo(r)s Fp(\()p Fo(!)t Fp(\))f Ff(is)h(given)f(by)h (\(1.2\).)328 3227 y Fp(This)k(is)f(a)g(sligh)m(tly)f(di\013eren)m(t)i (form)e(of)h(a)h(kno)m(wn)h(statemen)m(t)f(\(see,)i(e.g.,)f([33]\).)62 b(F)-8 b(or)328 3348 y(the)33 b(reader's)h(con)m(v)m(enience)g(w)m(e)g (pro)m(vide)f(a)f(pro)s(of)g(here:)328 3468 y Ff(Pr)-5 b(o)g(of.)57 b Fp(Let)38 b Fo(p)871 3483 y Fn(n)918 3468 y Fo(=q)1010 3483 y Fn(n)1094 3468 y Fp(b)s(e)g(the)f(sequence)j(of)d (con)m(tin)m(ued)h(fraction)f(appro)m(ximan)m(ts)g(of)f Fo(!)t(:)328 3588 y Fp(Let)i Fo(n)p Fp(\()p Fo(k)s Fp(\))g(b)s(e)g(suc) m(h)i(that)d Fo(q)1357 3604 y Fn(n)p Fi(\()p Fn(k)r Fi(\))1535 3588 y Fm(\024)g Fo(k)j(<)d(q)1896 3604 y Fn(n)p Fi(\()p Fn(k)r Fi(\)+1)2126 3588 y Fp(.)60 b(W)-8 b(e)38 b(will)e(use)j Fo(r)h Fp(for)e Fo(r)s Fp(\()p Fo(!)t Fp(\))p Fo(:)f Fp(W)-8 b(riting)328 3709 y Fo(k)31 b Fp(=)c Fo(b)554 3724 y Fn(n)602 3709 y Fo(q)645 3724 y Fn(n)704 3709 y Fp(+)12 b Fo(b)833 3724 y Fn(n)p Fc(\000)p Fi(1)971 3709 y Fo(q)1014 3724 y Fn(n)p Fc(\000)p Fi(1)1164 3709 y Fp(+)g Fm(\001)17 b(\001)g(\001)11 b Fp(+)h Fo(b)1510 3724 y Fi(1)1551 3709 y Fo(q)1594 3724 y Fi(1)1646 3709 y Fp(+)g Fo(b)1775 3724 y Fi(0)1843 3709 y Fp(and)28 b(using)f(the)h(Denjo)m(y-Koksma)g(inequalit)m(y)328 3829 y(\(see,)34 b(e.g.,)f(Lemma)e(4.1,)h(Ch.)h(3)g([9)o(]\),)g(w)m(e)h (get)328 3921 y Fb(\014)328 3971 y(\014)328 4021 y(\014)328 4071 y(\014)328 4120 y(\014)328 4170 y(\014)356 3988 y Fn(k)r Fc(\000)p Fi(1)360 4012 y Fb(X)359 4195 y Fn(j)t Fi(=0)501 4095 y Fo(f)11 b Fp(\()p Fm(f)p Fo(\022)25 b Fp(+)d Fo(j)6 b(!)t Fm(g)p Fp(\))21 b Fm(\000)h Fo(k)1206 3978 y Fb(Z)1289 4005 y Fi(1)1252 4167 y(0)1345 4095 y Fo(f)11 b Fp(\()p Fo(\022)s Fp(\))p Fo(d\022)1627 3921 y Fb(\014)1627 3971 y(\014)1627 4021 y(\014)1627 4071 y(\014)1627 4120 y(\014)1627 4170 y(\014)1682 4095 y Fm(\024)28 b Fp(\()p Fo(b)1866 4110 y Fi(0)1906 4095 y Fp(+)p Fm(\001)17 b(\001)g(\001)n Fp(+)p Fo(b)2215 4110 y Fn(n)2263 4095 y Fp(\)V)-8 b(ar)o(\()p Fo(f)11 b Fp(\))28 b Fm(\024)2720 3949 y Fb( )2824 3988 y Fn(n)2785 4012 y Fb(X)2788 4195 y Fn(i)p Fi(=0)2922 3949 y Fb(")2980 4028 y Fo(q)3023 4043 y Fn(i)p Fi(+1)p 2980 4072 162 4 v 3025 4164 a Fo(q)3068 4179 y Fn(i)3152 3949 y Fb(#!)3283 4095 y Fp(V)-8 b(ar)o(\()p Fo(f)11 b Fp(\))p Fo(:)3365 4303 y Fp(\(5.3\))328 4469 y(Since)24 b(\(1.2\))g(implies)e Fo(q)1165 4484 y Fn(i)p Fi(+1)1311 4469 y Fo(<)1424 4416 y Fn(q)1458 4393 y Fd(r)1456 4439 y(i)p 1424 4446 69 4 v 1443 4504 a Fn(c)1503 4469 y Fo(;)i Fp(where)h Fo(c)j Fp(=)2011 4423 y Fn(c)p Fi(\()p Fn(!)r Fi(\))p 2011 4446 132 4 v 2037 4504 a(2)p Fn(\031)2152 4469 y Fo(;)c Fp(w)m(e)i(ha)m(v)m (e)2565 4422 y Fn(q)2597 4432 y Fd(i)p Fe(+1)p 2565 4446 135 4 v 2604 4504 a Fn(q)2636 4514 y Fd(i)2738 4469 y Fo(<)2942 4422 y Fn(q)2974 4432 y Fd(i)p Fe(+1)p 2851 4446 318 4 v 2851 4506 a Fi(\()p Fn(cq)2941 4516 y Fd(i)p Fe(+1)3044 4506 y Fi(\))3071 4487 y Fe(1)p Fd(=r)3206 4469 y Fp(=)3320 4409 y Fn(q)3354 4374 y Fe(1)p Fa(\000)p Fe(1)p Fd(=r)3352 4432 y(i)p Fe(+1)p 3320 4446 210 4 v 3360 4506 a Fn(c)3391 4487 y Fe(1)p Fd(=r)3539 4469 y Fp(.)328 4606 y(The)33 b(righ)m(t-hand)f(side)h(of)f(\(5.3\))g(can)h (no)m(w)g(b)s(e)g(estimated)f(as)420 4873 y Fm(\024)526 4702 y Fb(0)526 4852 y(@)598 4873 y Fo(c)640 4832 y Fc(\000)p Fi(1)p Fn(=r)820 4756 y(n)p Fi(\()p Fn(k)r Fi(\))828 4790 y Fb(X)831 4973 y Fn(i)p Fi(=1)973 4873 y Fo(q)1020 4822 y Fi(1)p Fc(\000)p Fi(1)p Fn(=r)1016 4896 y(i)1241 4873 y Fp(+)1413 4806 y Fo(k)p 1349 4850 184 4 v 1349 4942 a(q)1392 4957 y Fn(n)p Fi(\()p Fn(k)r Fi(\))1542 4702 y Fb(1)1542 4852 y(A)1632 4873 y Fp(V)-8 b(ar)o(\()p Fo(f)11 b Fp(\))28 b Fo(<)2049 4727 y Fb( )2115 4873 y Fo(c)2157 4832 y Fc(\000)p Fi(1)p Fn(=r)2320 4873 y Fo(n)p Fp(\()p Fo(k)s Fp(\))p Fo(q)2555 4822 y Fi(1)p Fc(\000)p Fi(1)p Fn(=r)2551 4902 y(n)p Fi(\()p Fn(k)r Fi(\))2776 4873 y Fp(+)2949 4806 y Fo(k)p 2884 4850 V 2884 4942 a(q)2927 4957 y Fn(n)p Fi(\()p Fn(k)r Fi(\))3078 4727 y Fb(!)3160 4873 y Fp(V)-8 b(ar\()p Fo(f)11 b Fp(\))p Fo(:)1898 5214 y Fp(15)p eop %%Page: 16 17 16 16 bop 328 646 a Fp(Since)40 b Fo(k)j(<)d(q)843 661 y Fn(n)p Fi(\()p Fn(k)r Fi(\)+1)1114 646 y Fm(\024)1242 576 y Fn(q)1276 553 y Fd(r)1274 603 y(n)p Fe(\()p Fd(k)q Fe(\))p 1242 623 156 4 v 1304 680 a Fn(c)1408 646 y Fp(,)h(w)m(e)g(ha)m (v)m(e)g Fo(q)1902 661 y Fn(n)p Fi(\()p Fn(k)r Fi(\))2083 646 y Fm(\025)g Fp(\()p Fo(ck)s Fp(\))2373 610 y Fi(1)p Fn(=r)2481 646 y Fp(,)h(and)2816 607 y Fn(k)p 2757 623 V 2757 680 a(q)2789 694 y Fd(n)p Fe(\()p Fd(k)q Fe(\))2963 646 y Fm(\024)f Fo(c)3123 610 y Fc(\000)p Fi(1)p Fn(=r)3286 646 y Fo(k)3340 610 y Fi(1)p Fc(\000)p Fi(1)p Fn(=r)3539 646 y Fp(.)328 812 y(Also,)35 b(for)f(an)m(y)h Fo(!)t(;)48 b(n)p Fp(\()p Fo(k)s Fp(\))31 b Fm(\024)1393 765 y Fi(2)12 b(ln)g Fn(q)1543 773 y Fd(n)1585 765 y Fi(\()p Fn(k)r Fi(\))p 1393 788 286 4 v 1483 846 a(ln)f(2)1723 812 y Fp(\(see,)36 b(e.g.)50 b(Ch.)36 b(4)e([9]\).)50 b(Th)m(us)36 b(w)m(e)g(can)f(con)m(tin)m(ue)328 932 y(our)d(estimate)g(as)1403 1052 y Fm(\024)c Fp(\()p Fo(c)1588 1067 y Fi(1)1627 1052 y Fo(k)1681 1011 y Fi(1)p Fc(\000)p Fi(1)p Fn(=r)1896 1052 y Fp(ln)16 b Fo(k)s Fp(\)V)-8 b(ar\()p Fo(f)11 b Fp(\))p Fo(:)p 2450 1060 42 59 v 474 1322 a Fp(Besides)39 b(Lemma)e(12,)h(the)h(Diophan)m(tine)d(prop)s(erties)i(of)f Fo(!)42 b Fp(and)c(the)g(arithmetic)328 1443 y(prop)s(erties)25 b(of)f Fo(\022)k Fp(will)23 b(only)h(pla)m(y)g(a)h(role)f(in)g(the)h (follo)m(wing)d(simple)i(Lemma)f(that)i(allo)m(ws)328 1563 y(us)33 b(to)f(estimate)g(the)h(small)d(denominators)i(in)g (\(3.2\),)328 1763 y Fg(LEMMA)37 b(13)49 b Ff(Supp)-5 b(ose)36 b Fo(\022)45 b(=)-61 b Fm(2)31 b Fp(\002)p Fo(;)47 b(!)40 b Ff(satis\014es)35 b(\(1.2\),)h(then)g(for)h Fo(k)i Ff(su\016ciently)d(lar)-5 b(ge)328 1883 y(\()35 b Fo(k)30 b(>)e(k)639 1898 y Fi(5)678 1883 y Fp(\()p Fo(y)764 1898 y Fi(1)803 1883 y Fo(;)17 b(\022)s(;)g(!)t Fp(\))p Ff(\),)459 2154 y Fm(j)g Fp(cos)f(2)p Fo(\031)t(\022)803 2169 y Fn(j)862 2154 y Fm(\000)23 b Fp(cos)17 b(2)p Fo(\031)t(\022)1262 2169 y Fn(`)1295 2154 y Fm(j)27 b(\025)i Fp(exp)1621 2033 y Fb(\022)1682 2154 y Fm(\000)p Fp(\()p Fo(d)23 b Fp(+)f Fo(k)s(=)p Fp(4)f(+)h(1\))2397 2078 y Fe(1)p 2337 2090 150 4 v 2337 2131 a(2)p Fd(r)r Fe(\()p Fd(!)r Fe(\))2501 2033 y Fb(\023)2802 2087 y Fo(c)p Fp(\()p Fo(!)t Fp(\))p 2588 2131 609 4 v 2588 2223 a(\()p Fo(d)g Fp(+)g Fo(k)j Fp(+)d(1\))3058 2194 y Fn(r)r Fi(\()p Fn(!)r Fi(\))3207 2154 y Fo(:)131 b Fp(\(5.4\))328 2447 y Ff(Pr)-5 b(o)g(of.)43 b Fp(De\014ne)33 b Fo(i)963 2462 y Fn(j)1028 2447 y Fm(2)28 b Fh(Z)i Fp(b)m(y)j Fo(i)1389 2462 y Fn(j)1453 2447 y Fp(=)1567 2397 y Fn(\022)1600 2407 y Fd(j)1632 2397 y Fc(\000)p Fn(\022)p 1567 2424 156 4 v 1621 2482 a(!)1754 2447 y Fm(\000)23 b Fp(\()p Fo(x)1947 2462 y Fi(1)2009 2447 y Fp(+)2117 2408 y Fn(k)r Fc(\000)p Fi(1)p 2117 2424 129 4 v 2164 2482 a(2)2256 2447 y Fp(\))p Fo(;)44 b(j)33 b Fp(=)28 b(0)p Fo(;)17 b(:)g(:)g(:)f(;)h(k)s(:)32 b Fp(Then)422 2663 y Fo(i)455 2678 y Fn(j)514 2663 y Fm(\000)22 b Fo(x)668 2678 y Fi(1)736 2663 y Fm(2)28 b Fp([[)p Fm(\000)p Fp(3)p Fo(k)s(=)p Fp(4])p Fo(;)17 b(:)g(:)g(:)f(;)h Fp([)p Fm(\000)p Fo(k)s(=)p Fp(4])22 b(+)g(1])g Fm(\\)h Fp([)p Fo(d)e Fm(\000)i Fp([3)p Fo(k)s(=)p Fp(4])p Fo(;)17 b(:)g(:)g(:)e(;)i(d)22 b Fm(\000)h Fp([)p Fo(k)s(=)p Fp(4])f(+)g(1])p Fo(:)93 b Fp(\(5.5\))328 2879 y(W)-8 b(e)33 b(write)328 3119 y Fm(j)17 b Fp(cos)f(2)p Fo(\031)t(\022)672 3134 y Fn(j)729 3119 y Fm(\000)21 b Fp(cos)c(2)p Fo(\031)t(\022)1127 3134 y Fn(`)1160 3119 y Fm(j)27 b Fp(=)h(2)p Fm(j)17 b Fp(sin)e(2)p Fo(\031)t Fp(\()p Fo(\022)23 b Fp(+)d Fo(x)1913 3134 y Fi(1)1953 3119 y Fo(!)j Fp(+)2143 3052 y Fo(k)j Fm(\000)c Fp(1)g(+)g Fo(i)2521 3067 y Fn(j)2580 3052 y Fp(+)g Fo(i)2711 3067 y Fn(`)p 2143 3096 601 4 v 2419 3188 a Fp(2)2754 3119 y Fo(!)t Fp(\))17 b(sin)e(2)p Fo(\031)t Fp(\()p Fo(i)3188 3134 y Fn(j)3245 3119 y Fm(\000)21 b Fo(i)3376 3134 y Fn(`)3409 3119 y Fp(\))p Fo(!)t Fm(j)p Fo(:)3365 3271 y Fp(\(5.6\))328 3391 y(Since)33 b Fo(\022)42 b(=)-61 b Fm(2)29 b Fp(\002)j(w)m(e)i(ha)m(v)m(e)f(that)g(for)f(su\016cien)m (tly)h(large)f Fo(j)h Fp(:)g Fo(j)h(>)27 b(k)2678 3406 y Fi(6)2717 3391 y Fp(\()p Fo(\022)s(;)17 b(!)t Fp(\))1223 3628 y Fm(j)g Fp(sin)e(2)p Fo(\031)t Fp(\()p Fo(\022)25 b Fp(+)d Fo(j)6 b(=)p Fp(2\))p Fo(!)t Fm(j)26 b Fo(>)i Fp(exp)q(\()p Fm(\000)p Fo(j)2502 3552 y Fe(1)p 2442 3564 150 4 v 2442 3605 a(2)p Fd(r)r Fe(\()p Fd(!)r Fe(\))2606 3628 y Fp(\))p Fo(:)694 b Fp(\(5.7\))328 3844 y(Therefore)25 b(w)m(e)h(can)e(com)m(bine)g(\(1.2\),\(5.7\),\(5.5\),)g(and)h(\(5.6)e (\))i(to)e(obtain)h(the)g(statemen)m(t)328 3965 y(of)32 b(the)h(Lemma)e(with)h Fo(k)1228 3980 y Fi(5)1268 3965 y Fp(\()p Fo(y)1354 3980 y Fi(1)1393 3965 y Fo(;)17 b(\022)s(;)g(!)t Fp(\))26 b(=)i Fo(k)1813 3980 y Fi(6)1852 3965 y Fp(\()p Fo(\022)d Fp(+)d Fo(y)2106 3980 y Fi(1)2145 3965 y Fo(!)t(;)17 b(!)t Fp(\).)p 2476 3973 42 59 v 328 4182 a Ff(Pr)-5 b(o)g(of)25 b(of)g(L)-5 b(emma)25 b(7.)39 b Fp(Set)23 b Fo(\036)p Fp(\()p Fo(n)p Fp(\))28 b(=)f(2)p Fo(c)1712 4197 y Fi(1)1751 4182 y Fp(\()p Fo(!)t Fp(\)\()1940 4143 y Fn(n)p 1940 4159 43 4 v 1944 4217 a Fi(2)1992 4182 y Fp(\))2030 4146 y Fi(1)p Fc(\000)p Fn(r)r Fi(\()p Fn(!)r Fi(\))2254 4123 y Fa(\000)p Fe(1)2359 4182 y Fp(ln)2466 4143 y Fn(n)p 2466 4159 V 2470 4217 a Fi(2)2519 4182 y Fo(:)22 b Fp(According)g(to)g(Lemma)f(12)328 4303 y(and)h(a)f(remark) h(in)f(the)h(b)s(eginning)e(of)h(this)h(section)g(p)s(oin)m(ts)f Fo(\022)2518 4318 y Fi(0)2558 4303 y Fo(;)c(:)g(:)g(:)f(;)h(\022)2822 4318 y Fn(n)2890 4303 y Fp(are)22 b Fo(\036)p Fm(\000)p Fp(uniformly)328 4423 y(distributed)h(on)h([0)p Fo(;)17 b Fp(1])p Fo(:)23 b Fp(Th)m(us)i(w)m(e)f(can)g(apply)f(Lemma)g(11)g (with)g Fo(f)11 b Fp(\()p Fo(x)p Fp(\))28 b(=)f(cos)17 b Fo(x;)45 b( )t Fp(\()p Fo(n)p Fp(\))28 b(=)328 4544 y(1)p Fo(;)44 b(\036)p Fp(\()p Fo(n)p Fp(\))p Fo(:)32 b Fp(The)h(sp)s(ecial)e(prop)s(ert)m(y)i(of)f(the)g(cosine)g(that)g (leads)g(to)g(sp)s(ectral)g(uniformit)m(y)328 4664 y(in)g(energy)h (\(absence)i(of)d(mobilit)m(y)d(edge,)34 b(in)e(particular\))e(is)i (that)920 4809 y Fb(Z)1003 4836 y Fi(1)967 4998 y(0)1060 4926 y Fp(ln)15 b Fm(j)p Fo(z)27 b Fm(\000)c Fp(cos)17 b(2)p Fo(\031)t(x)p Fm(j)p Fo(dx)27 b Fp(=)h Fm(\000)17 b Fp(ln)f(2)27 b(for)32 b(an)m(y)d Fo(z)j Fm(2)c Fp([)p Fm(\000)p Fp(1)p Fo(;)17 b Fp(1])392 b(\(5.8\))1898 5214 y(16)p eop %%Page: 17 18 17 17 bop 328 631 a Fp(The)38 b(imp)s(ortan)m(t)e(part)h(is)g(that)g (the)h(in)m(tegral)e(in)g(\(5.8\))h(is)g(constan)m(t)h(o)m(v)m(er)h Fo(z)t(;)f Fp(not)f(the)328 751 y(actual)28 b(v)-5 b(alue.)42 b(In)29 b(particular,)f(it)f(implies)g(that)h Fm(j)p Fo(I)2197 766 y Fi(1)2236 751 y Fm(j)h Fp(and)g Fm(j)p Fo(I)2550 766 y Fi(2)2589 751 y Fm(j)f Fp(are)h(of)g(the)g(same)g (order.)328 872 y(Precisely)-8 b(,)33 b(w)m(e)g(obtain,)e(b)m(y)i (Lemma)e(11,)g(\(5.8\))h(and)g(\(5.4\),)g(that)g(for)f Fo(k)k Fp(large)c(enough,)328 1005 y Fo(k)g(>)c Fp(max)711 908 y Fb(\020)761 1005 y Fo(n)819 1020 y Fi(cos)918 1005 y Fp(\()969 965 y Fn(\017)p 966 981 36 4 v 966 1039 a Fi(3)1011 1005 y Fo(;)17 b(\036)p Fp(\()p Fo(n)p Fp(\))p Fo(;)g Fp(1\))p Fo(;)g(k)1473 1020 y Fi(5)1511 1005 y Fp(\()p Fo(y)1597 1020 y Fi(1)1636 1005 y Fo(;)g(\022)s(;)g(!)t Fp(\))1875 908 y Fb(\021)1940 1005 y Fo(;)32 b Fp(w)m(e)i(ha)m(v)m(e) 1251 1275 y Fm(j)p Fo(I)1322 1290 y Fi(1)1361 1275 y Fm(j)28 b(\024)g Fp(exp)1688 1154 y Fb(\024)1731 1275 y Fp(\()p Fo(k)e Fp(+)c(1\))2048 1154 y Fb(\022)2108 1275 y Fm(\000)17 b Fp(ln)f(2)22 b(+)2484 1208 y Fo(\017)p 2479 1252 49 4 v 2479 1344 a Fp(3)2538 1154 y Fb(\023\025)3365 1275 y Fp(\(5.9\))328 1516 y(and)1696 1636 y Fm(j)p Fo(I)1767 1651 y Fi(2)1806 1636 y Fm(j)27 b(\025)h Fp(exp)18 b Fo(S)1125 b Fp(\(5.10\))328 1806 y(where)328 2052 y Fo(S)33 b Fp(=)28 b(\()p Fo(k)s Fp(+1\))797 1930 y Fb(\022)857 2052 y Fm(\000)17 b Fp(ln)f(2)22 b Fm(\000)1234 1984 y Fo(\017)p 1230 2028 V 1230 2120 a Fp(3)1288 1930 y Fb(\023)1350 2052 y Fm(\000)p Fp(2)p Fo(\036)p Fp(\()p Fo(k)s Fp(+1\))1806 1906 y Fb( )1871 2052 y Fp(\()p Fo(d)f Fp(+)h Fo(k)s(=)p Fp(4)g(+)g(1\))2438 2011 y Fi(\(2)p Fn(r)r Fi(\()p Fn(!)r Fi(\)\))2661 1987 y Fa(\000)p Fe(1)2772 2052 y Fm(\000)g Fp(ln)3193 1984 y Fo(c)p Fp(\()p Fo(!)t Fp(\))p 2979 2028 609 4 v 2979 2120 a(\()p Fo(d)g Fp(+)g Fo(k)j Fp(+)d(1\))3449 2091 y Fn(r)r Fi(\()p Fn(!)r Fi(\))3598 1906 y Fb(!)3680 2052 y Fo(:)328 2344 y Fp(Supp)s(ose)32 b Fo(d)27 b Fm(\024)h Fo(k)946 2308 y Fn(\013)996 2344 y Fo(;)i Fp(some)h Fo(\013)d(<)g Fp(2)p Fo(:)i Fp(Then)i(w)m(e)g(can)f (write)f Fm(j)p Fo(I)8 b Fm(j)27 b Fp(=)2663 2298 y Fc(j)p Fn(I)2714 2307 y Fe(1)2747 2298 y Fc(j)p 2663 2321 105 4 v 2663 2378 a(j)p Fn(I)2714 2387 y Fe(2)2747 2378 y Fc(j)2805 2344 y Fm(\024)h Fp(exp)q(\(\()p Fo(k)21 b Fp(+)d(1\))3398 2305 y Fi(2)p Fn(\017)p 3398 2321 64 4 v 3412 2378 a Fi(3)3490 2344 y Fp(+)328 2478 y Fo(o)p Fp(\()p Fo(k)s Fp(\)\))32 b(where)i Fo(o)p Fp(\()p Fo(k)s Fp(\))e(dep)s(ends)j(only)d(on)g Fo(!)t(;)17 b(\013)q(:)31 b Fp(This)i(pro)m(v)m(es)i(Lemma)c(7.)p 3065 2486 42 59 v 328 2902 a Fj(References)377 3121 y Fp([1])48 b(S.)28 b(Aubry)g(and)f(G.)g(Andre,)i(Analyticit)m(y)d(breaking)h(and)g (Anderson)i(lo)s(calization)528 3242 y(in)j(incommensurate)g(lattices.) g(Ann.)h(Israel)f(Ph)m(ys.)j(So)s(c.)d Fg(3)p Fp(,)h(133{140)e (\(1980\))377 3442 y([2])48 b(S.)26 b(Aubry)-8 b(,)28 b(The)f(new)g(concept)g(of)e(transition)f(b)m(y)j(breaking)e(of)g (analyticit)m(y)-8 b(.)24 b(Solid)528 3562 y(State)33 b(Sci.)f Fg(8)p Fp(,)h(264)f(\(1978\).)377 3762 y([3])48 b(J.)31 b(Avron)g(and)g(B.)f(Simon,)g(Almost)f(p)s(erio)s(dic)g(Sc)m (hr\177)-49 b(odinger)31 b(op)s(erators.)f(I)s(I.)h(The)528 3883 y(in)m(tegrated)i(densit)m(y)g(of)f(states.)i(Duk)m(e)f(Math.)g (J.)g Fg(50)p Fp(,)g(369{391)e(\(1983\).)377 4083 y([4])48 b(J.)e(Avron)h(and)f(B.)g(Simon,)h(Singular)e(con)m(tin)m(uous)h(sp)s (ectrum)g(for)g(a)f(class)h(of)528 4203 y(almost)31 b(p)s(erio)s(dic)g (Jacobi)h(matrices.)g(Bull.)f(AMS)i Fg(6)p Fp(,)g(81{85)e(\(1982\).)377 4403 y([5])48 b(J.)24 b(Avron,)i(P)-8 b(.)24 b(v)-5 b(an)23 b(Mouc)m(he,)k(and)c(B.)h(Simon,)g(On)f(the)h(measure)g(of)f(the)g(sp)s (ectrum)528 4524 y(for)39 b(the)h(almost)e(Mathieu)h(op)s(erator.)g (Comm)m(un.)g(Math.)h(Ph)m(ys.)p Fg(132)p Fp(,)j(103{118)528 4644 y(\(1990\).)377 4844 y([6])48 b(J.)36 b(Bellisard,)f(R.)h(Lima,)e (and)i(D.)g(T)-8 b(estard,)37 b(A)f(metal-insulator)c(transition)i(for) 528 4965 y(the)f(almost)e(Mathieu)i(mo)s(del.)e(Comm)m(un.)h(Math.)h (Ph)m(ys.)h Fg(88)p Fp(,)f(207{234)e(\(1983\).)1898 5214 y(17)p eop %%Page: 18 19 18 18 bop 377 631 a Fp([7])48 b(Y.)d(Berezanskii,)k(Expansions)c(in)f (eigenfunctions)h(of)f(selfadjoin)m(t)g(op)s(erators.)528 751 y(T)-8 b(ransl.)33 b(Math.)g(Monogr.,)f(V)-8 b(ol.)32 b(17.)g(Pro)m(vidence,)i(RI:)f(Am.)f(Math.)h(So)s(c.)f(1968.)377 952 y([8])48 b(V.)40 b(Ch)m(ulaevsky)i(and)d(F.)g(Dely)m(on,)i(Purely)f (absolutely)e(con)m(tin)m(uous)i(sp)s(ectrum)528 1072 y(for)32 b(almost)f(Mathieu)i(op)s(erators.)f(J.)h(Stat.)f(Ph)m(ys.)j Fg(55)p Fp(,)e(1279{1284)d(\(1989\).)377 1273 y([9])48 b(I.)59 b(Cornfeld,)64 b(S.)58 b(F)-8 b(omin,)63 b(and)58 b(Y)-8 b(a.)57 b(Sinai,)63 b(Ergo)s(dic)57 b(Theory)-8 b(.)60 b(New)f(Y)-8 b(ork:)528 1393 y(Springer)33 b(1982)328 1594 y([10])48 b(W.)36 b(Craig)d(and)i(B.)g(Simon,)f(Subharmonicit)m(y) g(of)g(the)h(Ly)m(apuno)m(v)i(index.)e(Duk)m(e)528 1714 y(Math.)e(J.)g Fg(50)p Fp(,)g(551{560)e(\(1983\).)328 1915 y([11])48 b(H.)26 b(L.)f(Cycon,)j(R.)d(G.)g(F)-8 b(ro)s(ese,)26 b(W.)g(Kirsc)m(h,)h(B.)e(Simon,)g(Sc)m(hr\177)-49 b(odinger)26 b(Op)s(erators.)528 2035 y(Berlin,)32 b(Heidelb)s(erg,)g (New)h(Y)-8 b(ork:)44 b(Springer)32 b(1987.)328 2236 y([12])48 b(F.)f(Dely)m(on,)k(Absence)e(of)d(lo)s(calization)d(for)k (the)h(almost)d(Mathieu)i(equation.)528 2356 y(J.Ph)m(ys.)35 b Fg(A)i(20)p Fp(,)c(L21{L23)e(\(1987\).)328 2557 y([13])48 b(E.)37 b(Dinaburg)e(and)h(Y)-8 b(a.)36 b(Sinai,)f(The)i (one-dimensional)d(Sc)m(hr\177)-49 b(odinger)36 b(equation)528 2677 y(with)d(a)f(quasi-p)s(erio)s(dic)e(p)s(oten)m(tial.)h(F)-8 b(unct.)33 b(Anal.)f(Appl.)g Fg(9)p Fp(,)h(279{289)e(\(1975\).)328 2878 y([14])48 b(H.)25 b(v)m(on)g(Dreifus)f(and)h(A.)g(Klein,)f(A)h (new)g(pro)s(of)f(of)g(lo)s(calization)d(in)j(the)h(Anderson)528 2998 y(tigh)m(t)32 b(binding)f(mo)s(del.)h(Comm.)f(Math.)i(Ph)m(ys.)h Fg(124)p Fp(,)f(285{299)e(\(1989\).)328 3199 y([15])48 b(L.)30 b(H.)g(Eliasson,)g(Flo)s(quet)f(solutions)g(for)g(the)i (one-dimensional)c(quasi-p)s(erio)s(dic)528 3319 y(Sc)m(hr\177)-49 b(odinger)33 b(equation.)g(Comm)m(un.)f(Math.)h(Ph)m(ys.)h Fg(146)p Fp(,)f(447{482)e(\(1992\).)328 3520 y([16])48 b(L.)31 b(H.)f(Eliasson,)g(Discrete)h(one-dimensional)c(quasi-p)s(erio) s(dic)i(Sc)m(hr\177)-49 b(odinger)30 b(op-)528 3641 y(erators)j(with)f (pure)i(p)s(oin)m(t)d(sp)s(ectrum.)i(Acta)g(Math.)g Fg(179)g Fp(153{196)e(\(1997\).)328 3841 y([17])48 b(A.)f(Figotin)d(and)i(L.)g (P)m(astur,)51 b(The)c(p)s(ositivit)m(y)d(of)i(Ly)m(apuno)m(v)i(exp)s (onen)m(t)g(and)528 3962 y(absence)28 b(of)d(absolutely)g(con)m(tin)m (uous)i(sp)s(ectrum)f(for)f(almost)f(Mathieu)i(equation.)528 4082 y(J.)33 b(Math.)g(Ph)m(ys.)i Fg(25)p Fp(,)d(774{777)f(\(1984\).) 328 4283 y([18])48 b(J.)33 b(F)-8 b(r\177)-49 b(ohlic)m(h)30 b(and)j(T.)f(Sp)s(encer,)i(Absence)g(of)e(di\013usion)f(in)g(the)i (Anderson)g(tigh)m(t-)528 4403 y(binding)i(mo)s(del)g(for)g(large)g (disorder)h(or)g(lo)m(w)f(energy)-8 b(,)38 b(Comm)m(un.)e(Math.)g(Ph)m (ys.)528 4523 y Fg(88)p Fp(,)d(151{184)e(\(1983\).)328 4724 y([19])48 b(J.)43 b(F)-8 b(r\177)-49 b(ohlic)m(h,)43 b(F.)f(Martinelli,)g(E.)g(Scopp)s(ola,)i(and)f(T.)f(Sp)s(encer,)k (Constructiv)m(e)528 4844 y(pro)s(of)f(of)g(lo)s(calization)c(in)k(the) h(Anderson)h(tigh)m(t)d(binding)g(mo)s(del.)g(Comm)m(un.)528 4965 y(Math.)33 b(Ph)m(ys.)i Fg(101)p Fp(,)e(21{46)e(\(1985\).)1898 5214 y(18)p eop %%Page: 19 20 19 19 bop 328 631 a Fp([20])48 b(J.)23 b(F)-8 b(r\177)-49 b(ohlic)m(h,)22 b(T.)h(Sp)s(encer,)i(and)d(P)-8 b(.)22 b(Witt)m(w)m(er,)j(Lo)s(calization)19 b(for)i(a)h(class)g(of)g(one)g (di-)528 751 y(mensional)30 b(quasi-p)s(erio)s(dic)f(Sc)m(hr\177)-49 b(odinger)31 b(op)s(erators.)g(Comm)m(un.)f(Math.)i(Ph)m(ys.)528 872 y Fg(132)p Fp(,)h(5{25)f(\(1990\).)328 1075 y([21])48 b(F.)g(Gesztesy)i(and)d(B.)h(Simon,)i(The)f(xi)e(function,)k(Acta)d (Math.)g Fg(176)g Fp(49{71)528 1196 y(\(1996\).)328 1399 y([22])g(M.)30 b(Goldsh)m(tein,)f(Laplace)g(transform)f(metho)s(d)g(in) h(p)s(erturbation)f(theory)i(of)e(the)528 1519 y(sp)s(ectrum)49 b(of)e(Sc)m(hr\177)-49 b(odinger)48 b(op)s(erator)f Fo(I)8 b(I)56 b Fp(\(one-dimensional)45 b(quasi-p)s(erio)s(dic)528 1640 y(p)s(oten)m(tials\).)32 b(Preprin)m(t)h(\(1992\).)328 1843 y([23])48 b(F.)38 b(Germinet,)h(Dynamical)d(lo)s(calization)e(I)s (I)k(with)g(an)h(application)c(to)j(the)h(Al-)528 1963 y(most)32 b(Mathieu)h(op)s(erator,)f(to)g(app)s(ear)h(in)e(J.)i(Stat.)g (Ph)m(ys.)328 2167 y([24])48 b(A.)35 b(Gordon,)f(On)h(the)g(p)s(oin)m (t)e(sp)s(ectrum)i(of)f(the)h(one-dimensional)c(Sc)m(hr\177)-49 b(odinger)528 2287 y(op)s(erator.)32 b(Usp.)i(Math.)f(Nauk.)g Fg(31)p Fp(,)g(257)f(\(1976\).)328 2491 y([25])48 b(A.)25 b(Gordon,)g(S.)f(Jitomirsk)-5 b(a)m(y)m(a,)25 b(Y.)f(Last,)i(and)e(B.)g (Simon,)h(Dualit)m(y)d(and)i(singular)528 2611 y(con)m(tin)m(uous)30 b(sp)s(ectrum)f(in)f(the)h(almost)e(Mathieu)h(equation.)h(Acta)g (Math.,)h Fg(178)p Fp(,)528 2731 y(169{183)h(\(1997\).)328 2935 y([26])48 b(B.)27 b(Hel\013er)f(and)g(J.)h(Sj\177)-49 b(ostrand,)28 b(Semiclassical)c(analysis)h(for)h(Harp)s(er's)h (equation.)528 3055 y(I)s(I)s(I.)e(Can)m(tor)h(structure)g(of)e(the)h (sp)s(ectrum.)h(Mm.)e(So)s(c.)h(Math.)g(F)-8 b(rance)25 b(\(N.S.\))h Fg(39)p Fp(,)528 3176 y(1{124)32 b(\(1989\).)328 3379 y([27])48 b(M.)38 b(Herman,)f(Une)h(metho)s(de)e(p)s(our)h (minorer)e(les)i(exp)s(osan)m(ts)h(de)f(Ly)m(apuno)m(v)i(et)528 3499 y(quelques)25 b(exemples)e(mon)m(tran)m(t)g(le)f(caractere)h(lo)s (cal)e(d'un)i(theoreme)g(d'Arnold)g(et)528 3620 y(de)30 b(moser)e(sur)h(le)f(tore)h(en)g(dimension)e(2.)i(Comm)m(un.)f(Math.)h (Helv.)g Fg(58)p Fp(,)g(453{502)528 3740 y(\(1983\).)328 3944 y([28])48 b(S.)28 b(Jitomirsk)-5 b(a)m(y)m(a,)27 b(Anderson)h(lo)s(calization)c(for)i(the)i(almost)e(Mathieu)h (equation;)528 4064 y(A)33 b(nonp)s(erturbativ)m(e)g(pro)s(of.)f(Comm)m (un.)g(Math.)h(Ph)m(ys.)h Fg(165)p Fp(,)f(49{58)f(\(1994\).)328 4267 y([29])48 b(S.)g(Jitomirsk)-5 b(a)m(y)m(a,)49 b(Anderson)g(lo)s (calization)43 b(for)k(the)h(almost)d(Mathieu)i(equa-)528 4388 y(tion,)32 b(I)s(I:)h(P)m(oin)m(t)g(sp)s(ectrum)g(for)f Fo(\025)c(>)g Fp(2)p Fo(;)33 b Fp(Comm)m(un.)f(Math.)h(Ph)m(ys.)p Fg(168)p Fp(,)i(563{570)528 4508 y(\(1995\).)328 4711 y([30])48 b(S.)33 b(Jitomirsk)-5 b(a)m(y)m(a,)31 b(Almost)f(ev)m (erything)k(ab)s(out)d(the)i(almost)d(Mathieu)i(op)s(erator)528 4832 y(I)s(I,)38 b(Pro)s(ceedings)g(of)f(XI)g(In)m(ternational)f (Congress)j(of)e(Mathematical)e(Ph)m(ysics,)528 4952 y(P)m(aris)e(1994,)f(In)m(t.)h(Press,)h(373{382)d(\(1995\).)1898 5214 y(19)p eop %%Page: 20 21 20 20 bop 328 631 a Fp([31])48 b(S.)42 b(Jitomirsk)-5 b(a)m(y)m(a,)42 b(Almost)f(Mathieu)g(equations)h(with)f(w)m(eakly)i (Liouville)38 b(fre-)528 751 y(quencies:)45 b(second)34 b(critical)c(constan)m(ts.)k(In)f(preparation.)328 955 y([32])48 b(S.)40 b(Jitomirsk)-5 b(a)m(y)m(a)37 b(and)i(Y.)h(Last,)h (Dimensional)35 b(Hausdor\013)40 b(prop)s(erties)f(of)g(sin-)528 1075 y(gular)32 b(con)m(tin)m(uous)h(sp)s(ectra.)g(Ph)m(ys.)i(Rev.)e (Lett.)g Fg(76)p Fp(,)g(1765{1769)d(\(1996\).)328 1279 y([33])48 b(S.)k(Jitomirsk)-5 b(a)m(y)m(a)49 b(and)j(Y.)f(Last,)56 b(Anderson)d(Lo)s(calization)48 b(for)i(the)i(Almost)528 1399 y(Mathieu)37 b(Equation,)g(I)s(I)s(I.)g(Uniform)d(Lo)s (calization,)h(Con)m(tin)m(uit)m(y)h(of)g(Gaps,)i(and)528 1519 y(Measure)c(of)f(the)g(Sp)s(ectrum,)f(Comm)m(un.)g(Math.)h(Ph)m (ys.)i Fg(195)p Fp(,1{14)d(\(1998\).)328 1723 y([34])48 b(S.)29 b(Jitomirsk)-5 b(a)m(y)m(a)27 b(and)h(Y.)g(Last,)i(P)m(o)m(w)m (er-La)m(w)g(sub)s(ordinacy)e(and)g(singular)f(sp)s(ec-)528 1843 y(tra,)33 b(I)s(I.)f(Line)h(op)s(erators.)f(Preprin)m(t.)328 2046 y([35])48 b(S.)28 b(Jitomirsk)-5 b(a)m(y)m(a)26 b(and)i(B.)g(Simon,)f(Op)s(erators)g(with)h(singular)e(con)m(tin)m (uous)i(sp)s(ec-)528 2167 y(trum,)c(I)s(I)s(I.)e(Almost)e(p)s(erio)s (dic)g(Sc)m(hr\177)-49 b(odinger)22 b(op)s(erators,)i(Comm)m(un.)e (Math.)g(Ph)m(ys.)528 2287 y Fg(165)p Fp(,)33 b(201{205)e(\(1994\).)328 2491 y([36])48 b(Y.)36 b(Last,)g(A)f(relation)e(b)s(et)m(w)m(een)k (absolutely)e(con)m(tin)m(uous)g(sp)s(ectrum)h(of)f(ergo)s(dic)528 2611 y(Jacobi)46 b(matrices)f(and)h(the)h(sp)s(ectra)f(of)g(p)s(erio)s (dic)e(appro)m(ximan)m(ts.)i(Comm)m(un.)528 2731 y(Math.)33 b(Ph)m(ys.)i Fg(151)p Fp(,)e(183{192)d(\(1993\).)328 2935 y([37])48 b(Y.)35 b(Last,)g(Almost)e(ev)m(erything)i(ab)s(out)f (the)g(almost)f(Mathieu)h(op)s(erator)g(I.)g(Pro-)528 3055 y(ceedings)43 b(of)f(XI)g(In)m(ternational)f(Congress)i(of)f (Mathematical)e(Ph)m(ysics,)46 b(P)m(aris)528 3176 y(1994,)32 b(In)m(t.)h(Press,)i(366{372)c(\(1995\).)328 3379 y([38])48 b(Y.)38 b(Last)f(and)h(B.)f(Simon,)g(Eigenfuctions,)i (transfermatrices,)f(and)f(absolutely)528 3499 y(con)m(tin)m(uous)e(sp) s(ectrum)f(of)g(one-dimensional)d(Sc)m(hr\177)-49 b(odinger)34 b(op)s(erators,)g(In)m(v)m(en)m(t.)528 3620 y(Math.)f Fg(135)p Fp(,)g(329{367)e(\(1999\).)328 3823 y([39])48 b(R.)31 b(P)m(eierls,)g(Zur)f(Theorie)g(des)i(Diamagnetism)m(us)c(v)m (on)j(Leitungselektronen.)g(Z.)528 3944 y(Ph)m(ys.,)k Fg(80)p Fp(,)e(763-791)d(\(1933\).)328 4147 y([40])48 b(M.)40 b(Sh)m(ubin,)g(Discrete)f(Magnetic)g(Laplacian,)f(Comm)m(un.)h (Math.)g(Ph)m(ys.)i Fg(164,)528 4267 y Fp(259{275)31 b(\(1994\).)328 4471 y([41])48 b(B.)33 b(Simon,)e(Sc)m(hr\177)-49 b(odinger)33 b(semigroups,)f(Bull.)f(AMS)j Fg(7,)e Fp(447{526)f (\(1982\).)328 4674 y([42])48 b(Y)-8 b(a.)92 b(Sinai,)105 b(Anderson)93 b(lo)s(calization)88 b(for)j(one-dimensional)e (di\013erence)528 4795 y(Sc)m(hr\177)-49 b(odinger)42 b(op)s(erator)e(with)h(quasi-p)s(erio)s(dic)e(p)s(oten)m(tial.)g(J.)j (Stat.)f(Ph)m(ys.)i Fg(46)p Fp(,)528 4915 y(861{909)31 b(\(1987\).)1898 5214 y(20)p eop %%Page: 21 22 21 21 bop 328 631 a Fp([43])48 b(P)-8 b(.B.)26 b(Wiegmann,)g(On)g(the)g (singular)e(sp)s(ectrum)h(of)g(the)h(almost)e(Mathieu)i(op)s(era-)528 751 y(tor)j(-)f(Arithmetics)f(and)i(Can)m(tor)g(sp)s(ectra)h(of)e(in)m (tegrable)g(mo)s(dels.)f(Prog.)i(Theor.)528 872 y(Ph)m(ys.)35 b(Suppl.)d Fg(134)p Fp(,)h(171-181)e(\(1999\).)1898 5214 y(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9910011429919--