Content-Type: multipart/mixed; boundary="-------------0109211030876" This is a multi-part message in MIME format. ---------------0109211030876 Content-Type: text/plain; name="01-333.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-333.keywords" Random operators, localization, weak disorder, Lifshitz tails ---------------0109211030876 Content-Type: application/postscript; name="weak-dis.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="weak-dis.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: /home1/klopp/PAPTEX/SOUMIS/WEAK-DISORDER/weak-dis.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips %+ /home1/klopp/PAPTEX/SOUMIS/WEAK-DISORDER/weak-dis.dvi -o weak-dis.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.09.21:1713 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict 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y Fn(!)1980 5552 y Fq(';)17 b(')p Fm(i)27 b(\024)h Fq(\015)5 b Fs(\()p 2417 5497 V Fq(!)25 b Fm(\000)e Fq(\015)2659 5511 y Fn(\021)2701 5552 y Fs(\))p Fq(:)1989 5690 y Fg(3)p eop %%Page: 4 4 4 3 bop -118 241 a Fs(As)33 b(b)s(oth)f Fm(\000)p Fs(\001)i(and)e Fq(V)694 256 y Fn(!)777 241 y Fs(are)h(non-negativ)m(e,)f(this)g (implies)1649 427 y Fm(h\000)p Fs(\001)p Fq(';)17 b(')p Fm(i)28 b(\024)g Fq(C)7 b(\015)e(;)-2468 b Fs(\(0.7\))1608 578 y Fm(h)p Fq(V)1704 593 y Fn(!)1754 578 y Fq(';)17 b(')p Fm(i)27 b(\024)p 2097 524 65 4 v 28 w Fq(!)e Fm(\000)e Fq(\015)2339 537 y Fn(\021)2381 578 y Fq(:)-2526 b Fs(\(0.8\))-118 775 y(If)37 b(w)m(e)g(cut)h Fq(')e Fs(o\013)h(in)f(momen)m(tum)f(space) j(outside)f(of)f(a)g(ball)f(of)i(radius)f Fq(L)2660 739 y Fi(\000)p Fp(1)2755 775 y Fs(,)i(b)m(y)h(\(0.7\))o(,)f(the)f(part)g (of)52 b(^)-65 b Fq(')37 b Fs(outside)-118 894 y(of)h(that)g(ball)e(is) i(of)g Fq(L)698 858 y Fp(2)738 894 y Fs(-norm)f Fd(.)h Fs(\()p Fq(\015)5 b(L)1306 858 y Fp(2)1345 894 y Fs(\))1383 858 y Fp(1)p Fn(=)p Fp(2)1531 894 y Fs(\(here)39 b(and)g(in)e(the)i (rest)g(of)f(the)h(pap)s(er,)h Fq(a)d Fd(.)h Fq(b)h Fs(means)f Fq(a)g Fm(\024)g Fq(C)7 b(b)38 b Fs(for)-118 1013 y(some)d(constan)m(t) h Fq(C)7 b Fs(\).)51 b(W)-8 b(e)36 b(pic)m(k)f Fq(L)h Fs(so)f(that)g Fq(\015)1591 977 y Fp(1)p Fn(=)p Fp(2)1701 1013 y Fq(L)d Fm(\034)g Fq(\015)1987 977 y Fn(\021)2064 1013 y Fs(\(here)k(and)f(in)g(the)g(rest)h(of)f(the)g(pap)s(er,)h Fq(a)c Fm(\034)g Fq(b)k Fs(means)-118 1130 y Fq(a=b)i Fm(!)e Fs(0\).)61 b(So,)39 b(in)f(\(0.8\))o(,)i(w)m(e)f(can)g(assume)f (that)g(the)h(\\F)-8 b(ourier)37 b(transform")g(of)g Fq(')h Fs(has)h(supp)s(ort)g(in)e(a)h(ball)e(of)-118 1246 y(radius)d Fq(L)242 1210 y Fi(\000)p Fp(1)337 1246 y Fs(.)47 b(The)35 b(uncertain)m(t)m(y)g(principle)d(then)i(states)h (that,)f(mo)s(dulo)e(a)i(small)d(error,)j Fq(')g Fs(can)g(b)s(e)g (replaced)g(b)m(y)-118 1362 y(a)29 b(function)f(that)h(is)f(constan)m (t)i(o)m(v)m(er)g(cub)s(e)g(of)f(size)g Fq(L)1817 1326 y Fi(0)1868 1362 y Fm(\034)e Fq(L)p Fs(;)k(the)f(error)e(\(estimated)h (b)m(y)h(means)f(of)f(Lemma)g(6.2\))g(is)-118 1478 y Fq(L)-52 1442 y Fi(0)-28 1478 y Fq(=L)p Fs(.)41 b(W)-8 b(e)26 b(pic)m(k)g Fq(L)581 1442 y Fi(0)631 1478 y Fs(so)f(that)h(the)g (error)g(term)f Fq(L)1633 1442 y Fi(0)1657 1478 y Fq(=L)j Fm(\034)f Fq(\015)1983 1442 y Fn(\021)2025 1478 y Fs(.)41 b(This)26 b(no)m(w)g(enables)g(us)h(to)e(transform)g(condition)f (\(0.8\))-118 1595 y(in)m(to)634 1783 y Fl(X)690 1993 y Fn(j)794 1677 y Fl(0)794 1857 y(@)970 1810 y Fs(1)p 892 1855 206 4 v 892 1946 a(\()p Fq(L)996 1917 y Fi(0)1019 1946 y Fs(\))1057 1917 y Fn(d)1216 1783 y Fl(X)1124 1999 y Fi(j)p Fn(k)r Fi(\000)p Fn(k)1275 2009 y Fj(j)1306 1999 y Fi(j\024)p Fn(L)1429 1980 y Fe(0)1451 1878 y Fs(\()p Fq(!)1550 1893 y Fn(k)1615 1878 y Fm(\000)p 1714 1823 65 4 v 22 w Fq(!)t Fs(\))1817 1677 y Fl(1)1817 1857 y(A)1920 1878 y Fq(a)1971 1893 y Fn(j)2036 1878 y Fm(\024)k(\000)p Fq(\015)2274 1836 y Fn(\021)2349 1878 y Fs(and)2555 1783 y Fl(X)2611 1993 y Fn(j)2716 1878 y Fq(a)2767 1893 y Fn(j)2831 1878 y Fs(=)g(1)22 b(+)g Fq(O)s Fs(\()p Fq(\015)3276 1836 y Fn(\021)3317 1878 y Fs(\))p Fq(:)-118 2180 y Fs(The)41 b(probabilit)m(y)e(of)h(this)g(ev)m(en)m(t)j(is)d(then)h(estimated)f (using)g(large)g(deviations)g(tec)m(hniques;)46 b(the)41 b(exp)s(onen)m(tial)-118 2302 y(estimate)g(one)g(gets)h(is)f Fq(P)57 b Fd(.)43 b Fq(e)1071 2265 y Fi(\000)p Fn(\015)1166 2242 y Fc(2)p Fj(\021)1235 2265 y Fp(\()p Fn(L)1310 2242 y Fe(0)1333 2265 y Fp(\))1360 2242 y Fj(d)1400 2302 y Fs(.)71 b(Saturating)40 b(the)i(t)m(w)m(o)g(error)g(estimates)f(giv)m (en)g(ab)s(o)m(v)m(e,)k(w)m(e)e(get)e Fq(L)i Fm(\020)-118 2419 y Fq(\015)-62 2383 y Fi(\000)p Fp(1)p Fn(=)p Fp(2+)p Fn(\021)229 2419 y Fs(and)33 b Fq(L)485 2383 y Fi(0)538 2419 y Fm(\020)d Fq(\015)701 2383 y Fi(\000)p Fp(1)p Fn(=)p Fp(2+2)p Fn(\021)1027 2419 y Fs(\(here)k(and)f(in)g(the)h(rest)g (of)f(the)g(pap)s(er,)h Fq(a)c Fm(\020)f Fq(b)34 b Fs(means)f Fq(C)3162 2383 y Fi(\000)p Fp(1)3257 2419 y Fq(b)c Fm(\024)h Fq(a)f Fm(\024)g Fq(C)7 b(b)34 b Fs(for)f(some)-118 2538 y(constan)m(t)41 b Fq(C)7 b Fs(\).)65 b(Hence,)43 b Fq(P)14 b Fs(,)41 b(the)f(probabilit)m(y)e(w)m(e)j(w)m(an)m(t)g(to)f(estimate,) h(is)e(small)e(if)i Fq(\015)3113 2502 y Fp(2)p Fn(\021)r Fp(+2)p Fn(d\021)r Fi(\000)p Fn(d=)p Fp(2)3557 2538 y Fm(!)g Fs(+)p Fm(1)h Fs(when)-118 2654 y Fq(\015)e Fm(!)33 b Fs(0.)52 b(This)36 b(is)f(the)i(case)f(if)f Fq(\021)i(<)32 b(d=)p Fs(\(4)p Fq(d)24 b Fs(+)g(4\).)52 b(In)36 b(Theorem)g(0.2,)h (the)f(limit)c(v)-5 b(alue)35 b(for)g Fq(\021)40 b Fs(is)35 b Fq(d=)p Fs(\(4)p Fq(d)23 b Fs(+)h(6\);)37 b(the)-118 2770 y(di\013erence)28 b(comes)f(from)f(the)i(fact)f(that)g(w)m(e)h (only)f(assumed)g(that)g(the)h(random)e(v)-5 b(ariables)26 b(are)h(square)i(in)m(tegrable;)-118 2887 y(hence,)37 b(one)f(has)g(to)f(tak)m(e)h(their)f(p)s(ossible)g(un)m(b)s(oundedness) j(in)m(to)c(accoun)m(t)j(in)d(the)i(error)f(estimates)g(and)h(in)e(the) -118 3003 y(large)d(deviation)h(estimation.)-118 3166 y(W)-8 b(e)33 b(no)m(w)g(describ)s(e)g(our)g(results)g(in)f(their)g (full)e(generalit)m(y)-8 b(.)1640 3449 y(1.)55 b Fr(The)38 b(resul)-7 b(ts)-118 3623 y Fs(Let)39 b Fq(H)46 b Fs(b)s(e)39 b(a)f(translational)e(in)m(v)-5 b(arian)m(t)37 b(Jacobi)h(matrix)f (with)h(exp)s(onen)m(tial)g(o\013-diagonal)e(deca)m(y)k(that)e(is)g Fq(H)46 b Fs(=)-118 3739 y(\(\()p Fq(h)14 3754 y Fn(k)r Fi(\000)p Fn(k)147 3735 y Fe(0)172 3739 y Fs(\)\))248 3759 y Fn(k)r(;k)346 3740 y Fe(0)367 3759 y Fi(2)p Fk(Z)465 3740 y Fj(d)531 3739 y Fs(suc)m(h)34 b(that,)-1 3904 y Fy(H0:)117 b Fm(\017)42 b Fq(h)439 3919 y Fi(\000)p Fn(k)564 3904 y Fs(=)p 668 3823 99 4 v 28 w Fq(h)724 3919 y Fn(k)799 3904 y Fs(for)32 b Fq(k)f Fm(2)d Fo(Z)1193 3868 y Fn(d)1263 3904 y Fs(and)33 b(for)f(some)g Fq(k)f Fm(6)p Fs(=)d(0,)k Fq(h)2196 3919 y Fn(k)2267 3904 y Fm(6)p Fs(=)27 b(0.)175 4020 y Fm(\017)41 b Fs(there)33 b(exists)h Fq(c)27 b(>)h Fs(0)k(suc)m(h)i(that,)f(for)f Fq(k)f Fm(2)d Fo(Z)1892 3984 y Fn(d)1929 4020 y Fs(,)1710 4250 y Fm(j)p Fq(h)1794 4265 y Fn(k)1837 4250 y Fm(j)f(\024)2007 4183 y Fs(1)p 2007 4227 49 4 v 2010 4318 a Fq(c)2066 4250 y(e)2111 4209 y Fi(\000)p Fn(c)p Fi(j)p Fn(k)r Fi(j)2278 4250 y Fq(:)-2423 b Fs(\(1.1\))-118 4479 y(The)26 b(in\014nite)e (matrix)f Fq(H)32 b Fs(de\014nes)27 b(a)d(b)s(ounded)i(self-adjoin)m(t) d(op)s(erator)h(on)h Fq(`)2645 4443 y Fp(2)2685 4479 y Fs(\()p Fo(Z)2792 4443 y Fn(d)2829 4479 y Fs(\).)41 b(Using)25 b(the)g(F)-8 b(ourier)24 b(transform,)-118 4595 y(it)29 b(is)h(easily)g(seen)i(that)e Fq(H)38 b Fs(is)30 b(unitarily)e(equiv)-5 b(alen)m(t)30 b(to)g(the)h(m)m (ultiplication)26 b(b)m(y)31 b(the)g(function)f Fq(\022)h Fm(7!)c Fq(h)p Fs(\()p Fq(\022)s Fs(\))j(de\014ned)-118 4712 y(b)m(y)1148 4914 y Fq(h)p Fs(\()p Fq(\022)s Fs(\))e(=)1459 4820 y Fl(X)1466 5032 y Fn(k)r Fi(2)p Fk(Z)1620 4914 y Fq(h)1676 4929 y Fn(k)1719 4914 y Fq(e)1764 4873 y Fn(ik)r(\022)1898 4914 y Fs(where)34 b Fq(\022)d Fs(=)c(\()p Fq(\022)2442 4929 y Fp(1)2482 4914 y Fq(;)17 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y Fl(X)1922 380 y Fp(1)p Fi(\024)p Fn(j)t Fi(\024)p Fn(d)2135 262 y Fs(\(1)25 b(+)h(cos\()p Fq(\022)2562 277 y Fn(i)2591 262 y Fs(\)\).)57 b(In)38 b(this)f(case,)j(assumption)c(\(H1\))h(is)-118 480 y(satis\014ed.)94 644 y(Let)43 b Fq(V)336 659 y Fn(!)428 644 y Fs(b)s(e)f(the)h(diagonal) c(matrix)i(de\014ned)i(b)m(y)h(\(0.2\))e(with)f(en)m(tries)i(the)f (indep)s(enden)m(t)i(iden)m(tically)c(dis-)-118 760 y(tributed)32 b(real)g(v)-5 b(alued)32 b(random)g(v)-5 b(ariables)31 b(\()p Fq(!)1618 775 y Fn(k)1660 760 y Fs(\))1698 779 y Fn(k)r Fi(2)p Fk(Z)1834 760 y Fj(d)k Fs(that)d(satisfy)-1 898 y Fy(H2:)42 b Fs(the)k(random)g(v)-5 b(ariables)45 b(\()p Fq(!)1286 913 y Fn(k)1329 898 y Fs(\))1367 917 y Fn(k)r Fi(2)p Fk(Z)1502 898 y Fj(d)k Fs(are)d(non)h(trivial,)g(are)g (lo)m(w)m(er)f(semib)s(ounded)h(and)f(for)g(some)g Fq(r)54 b Fm(2)99 1014 y Fs(\(1)p Fq(;)17 b Fs(+)p Fm(1)p Fs(],)32 b(they)h(satisfy)1377 1182 y Fm(8)p Fq(r)1479 1141 y Fi(0)1530 1182 y Fq(<)27 b(r)m(;)115 b Fo(E)12 b Fs(\()p 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2733 2798 a Fi(j)p Fn(E)t Fi(\000)p Fn(\015)p 2904 2760 47 3 v 4 w(!)q Fi(jj)p Fn(m)p Fi(\000)p Fn(n)p Fi(j)3172 2839 y Fq(:)-118 3069 y Fs(One)41 b(can)h(also)e(relax)g(assumption)h (\(H1\))g(on)g(the)g(bac)m(kground)h(op)s(erator)f Fq(h)p Fs(.)69 b(F)-8 b(rom)39 b(the)j(pro)s(of)e(giv)m(en)h(in)f(the)-118 3185 y(presen)m(t)31 b(pap)s(er,)f(it)f(is)g(clear)g(that,)h(at)f(the)h (exp)s(ense)h(of)e(c)m(hanging)g Fq(\021)t Fs(\()p Fq(r)s Fs(\),)h(Theorems)g(1.1)f(and)g(1.3)g(still)f(hold)g(if)h(one)-118 3301 y(assumes)35 b(that)f Fq(h)g Fs(assumes)h(its)e(minim)m(um)e(at)j (\014nitely)f(man)m(y)h(p)s(oin)m(ts.)48 b(In)34 b(dimension)f(2,)h (using)f(the)i(tec)m(hniques)-118 3417 y(dev)m(elop)s(ed)45 b(in)e([12],)k(one)d(can)g(pro)m(v)m(e)i(that)e(Theorems)g(1.1)g(and)g (1.3)g(hold)f(for)g(an)m(y)i(non)f(constan)m(t,)k(analytic)-118 3534 y(function)32 b Fq(h)p Fs(.)-118 3708 y(1.1.)56 b Fy(The)26 b(la)m(y)m(out)e(of)i(the)g(pap)s(er.)50 b Fs(T)-8 b(o)22 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y(certain)23 b(ev)m(en)m(ts)k (\(expressed)f(only)e(in)f(terms)g(of)h(the)g(random)f(v)-5 b(ariables\))23 b(in)g(sections)h(4.3)g(and)g(4.4.)40 b(In)24 b(section)g(4.5,)-118 4521 y(w)m(e)35 b(carry)f(out)g(this)f (estimation)f(follo)m(wing)f(the)k(ideas)e(outlined)g(at)g(the)i(end)f (of)g(the)g(in)m(tro)s(duction.)46 b(The)35 b(pap)s(er)-118 4638 y(is)d(completed)g(b)m(y)i(an)e(app)s(endix,)h(section)g(6,)f (where)i(w)m(e)f(gather)g(some)f(useful)h(tec)m(hnical)f(results.)1270 4835 y(2.)54 b Fr(The)38 b(pr)n(oof)g(of)g(Theorem)g(1.2)94 5010 y Fs(T)-8 b(o)38 b(deduce)h(Theorem)f(1.2)f(from)g(Theorem)h(1.1,) g(w)m(e)h(use)g(the)f(\014nite)f(v)m(olume)g(lo)s(calization)d (criterion)i(de-)-118 5126 y(v)m(elop)s(ed)d(in)e(Theorem)i(3.1)f(of)g ([4].)44 b(T)-8 b(o)32 b(b)s(e)h(able)f(to)g(use)h(this)g(criterion,)e (w)m(e)i(\014rst)g(need)h(to)e(c)m(hec)m(k)j(that,)d(for)g(some)-118 5242 y(0)27 b Fq(<)h(s)g(<)f(\034)11 b Fs(,)29 b(our)e(random)g(v)-5 b(ariables)26 b(are)h Fq(s)p Fs(-regular)f(in)h(the)h(sense)h(of)e (de\014nition)f(3.1)h(in)g([4])g(i.e.)41 b(the)28 b Fq(s)p Fs(-decoupling)-118 5358 y(prop)s(ert)m(y)k(\(1.7\))e(in)g([4];)h(this) g(is)f(done)i(in)e(section)h(6.3.)42 b(Let)32 b(us)f(note)g(that,)g(if) f Fq(h)h Fs(is)f(a)h(trigonometric)d(p)s(olynomial,)-118 5475 y(the)k Fq(s)p Fs(-regularit)m(y)e(is)h(not)g(necessary)j(as)e(w)m (e)g(can)g(use)g(Theorem)g(1.1)f(of)g([4];)h(this)g(applies)e(in)h (particular)f(when)i Fq(H)-118 5591 y Fs(is)g(the)h(free)g(Laplace)f (op)s(erator.)1989 5690 y Fg(6)p eop %%Page: 7 7 7 6 bop 94 241 a Fs(Let)40 b(us)f(no)m(w)h(describ)s(e)g(the)f (\014nite)g(v)m(olume)g(criterion.)61 b(Therefore,)42 b(w)m(e)e(consider)f(the)h(resolv)m(en)m(t)g(of)f Fq(H)4024 256 y Fn(!)r(;\015)-118 359 y Fs(restricted)i(to)g(a)g(cub)s(e)g(with)g (Diric)m(hlet)e(b)s(oundary)i(conditions.)68 b(More)41 b(precisely)-8 b(,)43 b(pic)m(k)f(\003)3370 374 y Fn(L)3463 359 y Fm(\032)h Fo(Z)3652 323 y Fn(d)3690 359 y Fs(,)g(the)e(cub)s(e) -118 475 y(cen)m(tered)29 b(at)f(0)f(of)g(sidelength)h(2)p Fq(L)12 b Fs(+)g(1.)42 b(Let)28 b Fq(H)1610 439 y Fn(D)1602 505 y(!)r(;\015)t Fi(j)p Fp(\003)1777 516 y Fj(L)1854 475 y Fs(b)s(e)g(the)g(random)f(Hamiltonian)d Fq(H)3135 490 y Fn(!)r(;\015)3273 475 y Fs(restricted)k(to)f(the)h(b)s(o)m(x)-118 611 y(\003)-50 626 y Fn(L)34 611 y Fs(with)j(Diric)m(hlet)f(b)s (oundary)i(conditions,)f(i.e.,)h Fq(H)1861 575 y Fn(D)1853 641 y(!)r(;\015)t Fi(j)p Fp(\003)2028 652 y Fj(L)2106 611 y Fs(=)27 b(\005)2282 626 y Fp(\003)2331 637 y Fj(L)2382 611 y Fq(H)2463 626 y Fn(!)r(;\015)2573 611 y Fs(\005)2646 626 y Fp(\003)2695 637 y Fj(L)2777 611 y Fs(where)33 b(\005)3131 626 y Fp(\003)3180 637 y Fj(L)3263 611 y Fs(is)e(the)i(pro)5 b(jector)32 b(on)m(to)-118 741 y(the)37 b(sites)h(in)e(\003)463 756 y Fn(L)515 741 y Fs(.)57 b(Rephrased)38 b(with)f(our)g(notations,)g(to)g(c)m(hec)m(k)i(the)f(lo) s(calization)33 b(criterion)i(dev)m(elop)s(ed)j(in)f([4],)-118 857 y(w)m(e)d(need)f(to)f(c)m(hec)m(k)j(that,)e(for)f Fq(\015)37 b Fs(su\016cien)m(tly)d(small,)c(for)i(some)g(cub)s(e)i (\003)2574 872 y Fn(L)2625 857 y Fs(,)f(one)g(has)457 1073 y Fq(C)527 1088 y Fp(0)567 1073 y Fq(L)633 1032 y Fp(2)p Fn(d)709 1073 y Fq(\015)765 1032 y Fi(\000)p Fp(2)p Fn(s)987 979 y Fl(X)956 1190 y Fn(m)p Fi(2)p Fp(\003)1114 1201 y Fj(L)908 1268 y Fn(n)p Fi(2)p Fk(Z)1048 1245 y Fj(d)1079 1268 y Fi(n)p Fp(\003)1163 1279 y Fj(L)1226 1073 y Fq(e)1271 1032 y Fi(\000)p Fn(c)p Fi(j)p Fn(m)p Fi(\000)p Fn(n)p Fi(j)1560 1073 y Fo(E)12 b Fs(\()p Fm(jh)p Fq(\016)1768 1088 y Fp(0)1813 1073 y Fq(;)17 b Fs(\()p Fq(E)28 b Fs(+)22 b Fq(i")g Fm(\000)h Fq(H)2375 1089 y Fn(!)r(;\015)t Fi(j)p Fp(\003)2550 1100 y Fj(L)2600 1073 y Fs(\))2638 1032 y Fi(\000)p Fp(1)2732 1073 y Fq(\016)2775 1088 y Fn(m)2842 1073 y Fm(ij)2909 1032 y Fn(s)2945 1073 y Fs(\))p Fq(e)3028 1032 y Fn(\016)r Fp(\()p Fn(E)t Fp(\))p Fi(j)p Fn(n)p Fi(j)p Fn(=C)3339 1041 y Fc(0)3406 1073 y Fq(<)28 b Fs(1)-3677 b(\(2.1\))-118 1462 y(where)34 b Fq(C)234 1477 y Fp(0)305 1462 y Fs(is)f(a)f(constan)m(t)h(dep)s (ending)g(only)f(on)h Fq(h)f Fs(and)h(on)f(the)h(random)f(v)-5 b(ariables)31 b(\()p Fq(!)3139 1477 y Fn(\015)3184 1462 y Fs(\))3222 1481 y Fn(\015)t Fi(2)p Fk(Z)3359 1462 y Fj(d)r Fs(.)-118 1578 y(Let)i(us)g(pro)m(v)m(e)h(the)f(b)s(ound)f (\(2.1\);)h(therefore,)g(w)m(e)g(use)-118 1777 y Fy(Theorem)k(2.1)h Fs(\([12)o(]\))p Fy(.)49 b Ff(Ther)-5 b(e)34 b(exists)h Fq(C)f(>)28 b Fs(0)35 b Ff(such)f(that,)h(for)g Fq(L)28 b Fm(\025)g Fs(1)p Ff(,)35 b Fq(\015)d Fm(2)d Fs([0)p Fq(;)17 b Fs(1])34 b Ff(and)g Fq(E)g Fm(2)28 b Fo(R)5 b Ff(,)41 b(one)34 b(has)864 1982 y Fo(P)p Fs(\()p Fm(f)p Fq(H)1100 1941 y Fn(D)1092 2007 y(!)r(;\015)t Fi(j)p Fp(\003)1267 2018 y Fj(L)1353 1982 y Ff(admits)g(an)h(eigenvalue)61 b Fm(\024)28 b Fq(E)6 b Fs(\))28 b Fm(\024)g Fq(C)7 b(L)2804 1941 y Fn(d)2845 1982 y Fm(N)2927 1997 y Fn(\015)2971 1982 y Fs(\()p Fq(E)f Fs(\))p Fq(:)-118 2189 y Fs(In)26 b([12],)h(Theorem)f(2.1)f(w)m(as)i(pro)m(v)m(ed)g(in)f(dimension)e(2;)k (the)e(pro)s(of)f(carries)h(o)m(v)m(er)h(to)e(arbitrary)g(dimension)f (without)-118 2305 y(a)32 b(c)m(hange.)45 b(W)-8 b(e)33 b(do)f(not)g(rep)s(eat)h(it)f(here.)94 2422 y(Fix)j Fq(\021)k Fs(as)c(in)g(Theorem)h(1.2)f(and)g(\014x)h Fq(\021)g(<)c(\021)1750 2386 y Fi(0)1806 2422 y Fq(<)g(\021)t Fs(\()p Fq(r)s Fs(\).)51 b(De\014ne)36 b(\012)2542 2437 y Fn(\015)t(;\021)2639 2418 y Fe(0)2662 2437 y Fn(;L)2766 2422 y Fs(=)c Fm(f)p Fs(there)k(exists)g(an)f(eigen)m(v)-5 b(alue)35 b(of)-118 2545 y Fq(H)-37 2561 y Fn(!)r(;\015)t Fi(j)p Fp(\003)138 2572 y Fj(L)222 2545 y Fm(\024)g Fq(\015)5 b Fs(\()p 428 2490 65 4 v Fq(!)28 b Fm(\000)e Fq(\015)676 2509 y Fn(\021)713 2486 y Fe(0)740 2545 y Fs(\))p Fm(g)p Fs(.)55 b(Using)36 b(Theorem)h(1.1)f(in)g(conjunction)g(with)h(Theorem)f(2.1)h (w)m(e)g(obtain)f(that,)h(there)-118 2682 y(exists)c Fq(\015)202 2697 y Fn(\021)271 2682 y Fq(>)28 b Fs(0)k(and)h Fq(")27 b(>)h Fs(0)k(suc)m(h)i(that,)f(for)f(0)27 b Fq(<)h(\015)k(<)c (\015)1930 2697 y Fn(\021)1971 2682 y Fs(,)33 b(1)27 b Fm(\024)i Fq(L)f Fm(\024)g Fq(e)2457 2646 y Fn(\015)2497 2623 y Fe(\000)p Fj("=)p Fc(2)2645 2682 y Fs(,)33 b(one)g(has)1238 2905 y Fo(P)p Fs(\(\012)1405 2920 y Fn(\015)t(;\021)1502 2901 y Fe(0)1527 2920 y Fn(;L)1599 2905 y Fs(\))27 b Fm(\024)i Fq(C)7 b(e)1892 2864 y Fn(d\015)1968 2840 y Fe(\000)p Fj("=)p Fc(2)2116 2905 y Fq(e)2161 2864 y Fi(\000)p Fn(\015)2256 2840 y Fe(\000)p Fj(")2370 2905 y Fm(\024)28 b Fq(e)2520 2864 y Fi(\000)2585 2836 y Fc(1)p 2585 2848 31 3 v 2585 2890 a(2)2625 2864 y Fn(\015)2665 2840 y Fe(\000)p Fj(")2751 2905 y Fq(:)-2896 b Fs(\(2.2\))-118 3111 y(By)34 b(a)e(Com)m(b)s(es-Thomas)i(estimate,)e(Lemma)g(6.1,)h(w)m (e)h(get)f(that,)h(for)e Fq(!)g Fm(62)e Fs(\012)2769 3126 y Fn(\015)t(;\021)r(;L)2972 3111 y Fs(and)j Fq(E)i Fm(2)29 b Fs([0)p Fq(;)17 b(\015)5 b Fs(\()p 3578 3056 65 4 v Fq(!)25 b Fm(\000)e Fq(\015)3820 3075 y Fn(\021)3862 3111 y Fs(\)],)34 b(one)-118 3227 y(has)667 3356 y Fl(\014)667 3416 y(\014)700 3441 y Fm(h)p Fq(\016)782 3456 y Fn(n)829 3441 y Fq(;)17 b Fs(\()p Fq(E)28 b Fs(+)22 b Fq(i")g Fm(\000)g Fq(H)1390 3456 y Fn(!)r(;\015)t Fi(j)p Fp(\003)1565 3467 y Fj(L)1615 3441 y Fs(\))1653 3399 y Fi(\000)p Fp(1)1748 3441 y Fq(\016)1791 3456 y Fn(m)1857 3441 y Fm(i)1896 3356 y Fl(\014)1896 3416 y(\014)1957 3441 y Fm(\024)28 b Fq(C)7 b(\015)2195 3399 y Fi(\000)p Fp(1)p Fi(\000)p Fn(\021)2382 3441 y Fq(e)2427 3399 y Fi(\000)2482 3328 y Fm(p)p 2565 3328 447 4 v 71 x Fi(j)p Fn(E)t Fi(\000)p Fn(\015)t Fp(\()p 2763 3362 47 3 v Fn(!)r Fi(\000)p Fn(\015)2904 3380 y Fj(\021)2938 3366 y Fe(0)2965 3399 y Fp(\))p Fi(jj)p Fn(m)p Fi(\000)p Fn(n)p Fi(j)p Fn(=C)3305 3441 y Fq(;)-3450 b Fs(\(2.3\))-118 3665 y(for)33 b Fq(C)k Fm(\025)268 3640 y Fs(~)246 3665 y Fq(C)316 3680 y Fp(0)389 3665 y Fs(where)694 3640 y(~)672 3665 y Fq(C)742 3680 y Fp(0)811 3665 y Fq(>)29 b Fs(0)34 b(is)f(a)g(constan)m(t)i(indep)s(enden)m(t)g (of)e Fq(\015)39 b Fs(and)34 b Fq(\021)t Fs(.)46 b(De\014ne)35 b Fq(\016)t Fs(\()p Fq(E)6 b Fs(\))29 b(:=)3314 3580 y 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Fm(2)g Fo(Z)962 5438 y Fn(d)1000 5475 y Fs(,)g(consider)f(the)g (random)f(v)-5 b(ariables)25 b Fq(!)2402 5490 y Fn(n)2449 5475 y Fs(\()p Fq(\025)p Fs(\))i(=)h(inf)6 b(\()p Fq(!)2930 5490 y Fn(n)2977 5475 y Fq(;)17 b(\025)p Fs(\).)41 b(They)28 b(are)e(indep)s(enden)m(t,)-118 5591 y(iden)m(tically)37 b(distributed)i(and)h(b)s(ounded.)64 b(Let)40 b(us)g(estimate)e(the)i (di\013erence)g Fo(E)13 b Fs(\()p Fq(!)3008 5606 y Fp(0)3079 5591 y Fm(\000)28 b Fq(!)3245 5606 y Fp(0)3284 5591 y Fs(\()p Fq(\025)p Fs(\)\).)63 b(Therefore,)43 b(w)m(e)1989 5690 y Fg(8)p eop %%Page: 9 9 9 8 bop -118 241 a Fs(write)244 404 y(0)27 b Fm(\024)h Fo(E)13 b Fs(\()p Fq(!)585 419 y Fp(0)652 404 y Fm(\000)23 b Fq(!)813 419 y Fp(0)852 404 y Fs(\()p Fq(\025)p Fs(\)\))k(=)h Fo(E)1231 323 y Fl(\000)1283 404 y Fs(\()p Fq(!)1382 419 y Fp(0)1443 404 y Fm(\000)22 b Fq(\025)p Fs(\))p Fy(1)1693 419 y Fi(f)p Fn(!)1772 428 y Fc(0)1807 419 y Fi(\025)p Fn(\025)p Fi(g)1943 323 y Fl(\001)2016 404 y Fm(\024)2126 309 y Fl(X)2121 521 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Fn(!)r(;\015)3027 4962 y Fs(,)h(etc.)94 5126 y(Let)f(us)f(no)m (w)h(brie\015y)f(explain)f(the)h(pro)s(of)f(of)h(Theorem)g(3.1.)42 b(First,)31 b(w)m(e)h(in)m(tro)s(duce)f(a)f(p)s(erio)s(dic)g(appro)m (xima-)-118 5242 y(tion)e(sc)m(heme)j(\(that)e(has)g(pro)m(v)m(ed)i (useful)e(for)g(related)g(issues)h(in)f([11,)g(14)o(]\).)43 b(This)29 b(sc)m(heme)i(deliv)m(ers)e(a)g(sequence)j(of)-118 5358 y(v)m(ery)j(quic)m(kly)f(con)m(v)m(erging)g(\014nite)f(v)m(olume)g (appro)m(ximations)f(to)h(the)h(densit)m(y)g(of)f(states.)47 b(T)-8 b(o)34 b(analyze)f(the)h(\014nite)-118 5475 y(v)m(olume)c (densit)m(y)i(of)e(states,)j(w)m(e)e(recall)f(the)h(Flo)s(quet)f (theory)i(of)e(p)s(erio)s(dic)f(op)s(erators.)43 b(W)-8 b(e)31 b(use)h(it)e(to)g(reduce)i(the)-118 5591 y(problem)f(of)h (estimating)f(the)i(densit)m(y)g(of)f(states)i(to)e(the)h(probabilit)m (y)e(of)h(estimating)e(a)j(w)m(ell)e(c)m(hosen)k(ev)m(en)m(t.)1989 5690 y Fg(9)p eop %%Page: 10 10 10 9 bop -118 241 a Fs(4.1.)56 b 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y(\(see)k(e.g.)g([22)o(]\);)j(it)c(is)f(a)h(non-decreasing)h (function.)65 b(Its)41 b(deriv)-5 b(ativ)m(e,)42 b Fq(d)p Fm(N)2728 1215 y Fn(N)2713 1276 y(!)r(;\015)2823 1251 y Fs(,)g(is)d(a)h(p)s(ositiv)m(e)g(measure)h(and,)h(it)-118 1372 y(satis\014es,)33 b(for)f Fq(')c Fm(2)g(C)661 1336 y Fi(1)655 1396 y Fp(0)736 1372 y Fs(\()p Fo(R)5 b Fs(\),)673 1603 y(\()p Fq(';)17 b(d)p Fm(N)967 1562 y Fn(N)952 1628 y(!)r(;\015)1061 1603 y Fs(\))27 b(=)1230 1467 y Fl(Z)1285 1693 y Fk(R)1354 1603 y Fq(')p Fs(\()p Fq(x)p Fs(\))p Fq(d)p Fm(N)1697 1562 y Fn(N)1682 1628 y(!)r(;\015)1792 1603 y Fs(\()p Fq(x)p Fs(\))h(=)2193 1536 y(1)p 2064 1580 307 4 v 2064 1671 a(2)p Fq(N)33 b Fs(+)22 b(1)2473 1508 y Fl(X)2397 1732 y Fn(k)r Fi(2)p Fk(Z)2533 1709 y Fj(d)2533 1755 y Fc(2)p Fj(N)5 b Fc(+1)2692 1603 y Fm(h)p Fq(\016)2774 1618 y Fn(k)2817 1603 y Fq(;)17 b(')p Fs(\()p Fq(H)3052 1562 y Fn(N)3044 1628 y(!)r(;\015)3154 1603 y Fs(\))p Fq(\016)3235 1618 y Fn(k)3277 1603 y Fm(i)p Fq(:)-3461 b Fs(\(4.1\))-118 1912 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b(W)-8 b(e)33 b(kno)m(w)h(\(see)f([21,)g(6)o(]\))g(that)1084 3909 y(\()p Fq(';)17 b(d)p Fm(N)e Fs(\))26 b(=)1546 3773 y Fl(Z)1601 3999 y Fk(R)1670 3909 y Fq(')p Fs(\()p Fq(\025)p Fs(\))p Fq(d)p Fm(N)41 b Fs(=)28 b Fo(E)12 b Fs(\()p Fm(h)p Fq(\016)2325 3924 y Fp(0)2371 3909 y Fq(;)17 b(')p Fs(\()p Fq(H)2598 3924 y Fn(!)r(;\015)2707 3909 y Fs(\))p Fq(\016)2788 3924 y Fp(0)2828 3909 y Fm(i)p Fs(\))p Fq(:)-118 4144 y Fs(Av)m(eraging)26 b(\(4.1\))e(in)g Fq(!)k Fs(and)d(using)f(the) h(fact)g(that)f(the)h(random)f(v)-5 b(ariables)23 b(\()p Fq(!)2741 4159 y Fn(n)2788 4144 y Fs(\))2826 4159 y Fn(n)p Fi(2)p Fk(Z)2989 4144 y Fs(are)i(i.i.d)e(and)i(the)g(translation)-118 4260 y(in)m(v)-5 b(ariance)32 b(of)g Fq(H)8 b Fs(,)32 b(w)m(e)h(get)1265 4439 y Fo(E)12 b Fs(\(\()p Fq(';)17 b(d)p Fm(N)1657 4398 y Fn(N)1642 4463 y(!)r(;\015)1757 4439 y Fs(\)\))28 b(=)f Fo(E)13 b Fs(\()p Fm(h)p Fq(\016)2145 4454 y Fp(0)2190 4439 y Fq(;)k(')p Fs(\()p Fq(H)2425 4398 y Fn(n)2417 4463 y(!)r(;\015)2527 4439 y Fs(\))p Fq(\016)2608 4454 y Fp(0)2647 4439 y Fm(i)p Fs(\))p Fq(:)-2869 b Fs(\(4.4\))-118 4623 y(So)36 b(that)g(w)m(e)h(w)m(an)m(t)g(to)f (estimate)f Fm(h)p Fq(\016)1224 4638 y Fp(0)1264 4623 y Fq(;)17 b(')p Fs(\()p Fq(H)1499 4587 y Fn(n)1491 4648 y(!)r(;\015)1600 4623 y Fs(\))p Fq(\016)1681 4638 y Fp(0)1721 4623 y Fm(i)24 b(\000)h(h)p Fq(\016)1968 4638 y Fp(0)2008 4623 y Fq(;)17 b(')p Fs(\()p Fq(H)2235 4638 y Fn(!)r(;\015)2344 4623 y Fs(\))p Fq(\016)2425 4638 y Fp(0)2465 4623 y Fm(i)p Fs(.)54 b(Therefore)37 b(w)m(e)h(use)f(Hel\013er-Sj\177)-49 b(ostrand's)-118 4744 y(form)m(ula)31 b(\([8)o(]\))i(that)f(reads)1098 4976 y Fq(')p Fs(\()p Fq(H)1281 4991 y Fn(!)1331 4976 y Fs(\))27 b(=)1547 4908 y Fq(i)p 1510 4953 108 4 v 1510 5044 a Fs(2)p Fq(\031)1644 4840 y Fl(Z)1699 5066 y Fk(C)1778 4908 y Fq(@)22 b Fs(~)-66 b Fq(')p 1778 4953 121 4 v 1785 5044 a(@)p 1841 4989 50 4 v 5 w(z)1909 4976 y Fs(\()p Fq(z)t Fs(\))23 b Fm(\001)e Fs(\()p Fq(z)27 b Fm(\000)c Fq(H)2397 4991 y Fn(!)2447 4976 y Fs(\))2485 4935 y Fi(\000)p Fp(1)2579 4976 y Fq(dz)k Fm(^)c Fq(d)p 2842 4921 V(z)t(:)-3036 b Fs(\(4.5\))-118 5211 y(where)49 b(~)-64 b Fq(')32 b Fs(is)g(an)h(almost)e(analytic)g(extension)i(of)f Fq(')h Fs(\(see)g([19]\))g(i.e.)43 b(a)32 b(function)g(satisfying)-18 5358 y(1.)41 b(for)32 b Fq(z)g Fm(2)c Fo(R)5 b Fs(,)55 b(~)-65 b Fq(')p Fs(\()p Fq(z)t Fs(\))28 b(=)g Fq(')p Fs(\()p Fq(z)t Fs(\);)-18 5475 y(2.)41 b(supp\()16 b(~)-65 b Fq(')q Fs(\))27 b Fm(\032)i(f)p Fq(z)j Fm(2)c Fo(C)20 b Fs(;)55 b Fm(j)p Fs(Im)o(\()p Fq(z)t Fs(\))p Fm(j)28 b Fq(<)g Fs(1)p Fm(g)p Fs(;)-18 5591 y(3.)57 b(~)-65 b Fq(')27 b Fm(2)i(S)7 b Fs(\()p Fm(f)p Fq(z)33 b Fm(2)28 b Fo(C)20 b Fs(;)55 b Fm(j)p Fs(Im)o(\()p Fq(z)t Fs(\))p Fm(j)28 b Fq(<)f Fs(1)p Fm(g)p Fs(\);)1969 5690 y Fg(10)p eop %%Page: 11 11 11 10 bop -18 294 a Fs(4.)41 b(the)e(family)d(of)j(functions)f Fq(x)h Fm(7!)1363 227 y Fq(@)21 b Fs(~)-65 b Fq(')p 1363 271 121 4 v 1370 363 a(@)p 1426 308 50 4 v 5 w(z)1494 294 y Fs(\()p Fq(x)26 b Fs(+)h Fq(iy)t Fs(\))e Fm(\001)h(j)p Fq(y)t Fm(j)2026 253 y Fi(\000)p Fn(n)2165 294 y Fs(\(for)38 b(0)g Fq(<)h Fm(j)p Fq(y)t Fm(j)d Fq(<)i Fs(1\))h(is)f(b)s(ounded)i(in) e Fm(S)7 b Fs(\()p Fo(R)e Fs(\))45 b(for)38 b(an)m(y)99 440 y Fq(n)g Fm(2)g Fo(N)9 b Fs(;)48 b(more)37 b(precisely)-8 b(,)41 b(there)e(exists)g Fq(C)45 b(>)38 b Fs(1)g(suc)m(h)i(that,)g (for)e(all)e Fq(p;)17 b(q)t(;)g(r)40 b Fm(2)e Fo(N)9 b Fs(,)46 b(there)39 b(exists)h Fq(C)3841 455 y Fn(p;q)3971 440 y Fq(>)e Fs(0)99 556 y(suc)m(h)c(that)477 776 y(sup)422 862 y Fp(0)p Fn(<)p Fi(j)p Fn(y)r Fi(j\024)p Fp(1)696 776 y Fs(sup)702 858 y Fn(x)p Fi(2)p Fk(R)859 632 y Fl(\014)859 692 y(\014)859 751 y(\014)859 811 y(\014)892 776 y Fq(x)947 735 y Fn(p)1025 709 y Fq(@)1081 673 y Fn(q)p 997 753 151 4 v 997 845 a Fq(@)5 b(x)1108 816 y Fn(q)1174 636 y Fl(\022)1248 776 y Fm(j)p Fq(y)t Fm(j)1356 735 y Fi(\000)p Fn(r)1469 776 y Fm(\001)1529 709 y Fq(@)22 b Fs(~)-66 b Fq(')p 1529 753 121 4 v 1536 845 a(@)p 1592 790 50 4 v 5 w(z)1660 776 y Fs(\()p Fq(x)22 b Fs(+)g Fq(iy)t Fs(\))1996 636 y Fl(\023)2069 632 y(\014)2069 692 y(\014)2069 751 y(\014)2069 811 y(\014)2130 776 y Fm(\024)28 b Fq(C)2312 735 y Fn(r)2350 776 y Fq(C)2420 791 y Fn(p;q)2618 776 y Fs(sup)2529 863 y Fn(q)2563 840 y Fe(0)2586 863 y Fi(\024)p Fn(r)r Fp(+)p Fn(q)r Fp(+2)2617 936 y Fn(p)2653 913 y Fe(0)2675 936 y Fi(\024)p Fn(p)2870 776 y Fs(sup)2876 858 y Fn(x)p Fi(2)p Fk(R)3033 632 y Fl(\014)3033 692 y(\014)3033 751 y(\014)3033 811 y(\014)3067 776 y Fq(x)3122 735 y Fn(p)3158 712 y Fe(0)3194 709 y Fq(@)3250 673 y Fn(q)3284 649 y Fe(0)3311 709 y Fq(')p 3194 753 182 4 v 3198 845 a(@)5 b(x)3309 816 y Fn(q)3343 797 y Fe(0)3385 776 y Fs(\()p Fq(x)p Fs(\))3516 632 y Fl(\014)3516 692 y(\014)3516 751 y(\014)3516 811 y(\014)3566 776 y Fq(:)-3711 b Fs(\(4.6\))-118 1099 y(Equations)33 b(\(4.4\))f(and)g(\(4.5\))g(then) i(yield)166 1334 y Fm(j)p Fo(E)12 b Fs(\(\()p Fq(';)6 b(d)p Fm(N)575 1293 y Fn(N)560 1359 y(!)r(;\015)669 1334 y Fs(\)\))22 b Fm(\000)h Fs(\()p Fq(';)17 b(d)p Fm(N)e Fs(\))p Fm(j)26 b(\024)1398 1267 y Fs(1)p 1368 1311 108 4 v 1368 1402 a(4)p Fq(\031)1486 1334 y Fo(E)1563 1194 y Fl(\022)1642 1198 y(Z)1697 1424 y Fk(C)1766 1190 y Fl(\014)1766 1249 y(\014)1766 1309 y(\014)1766 1369 y(\014)1809 1267 y Fq(@)c Fs(~)-66 b Fq(')p 1809 1311 121 4 v 1816 1402 a(@)p 1872 1347 50 4 v 5 w(z)1940 1334 y Fs(\()p Fq(z)t Fs(\))2065 1190 y Fl(\014)2065 1249 y(\014)2065 1309 y(\014)2065 1369 y(\014)2116 1334 y Fm(jh)p Fq(\016)2226 1349 y Fp(0)2265 1334 y Fq(;)2309 1253 y Fl(\000)2354 1334 y Fs(\()p Fq(z)27 b Fm(\000)22 b Fq(H)2652 1293 y Fn(n)2644 1359 y(!)r(;\015)2755 1334 y Fs(\))2793 1293 y Fi(\000)p Fp(1)2909 1334 y Fm(\000)h Fs(\()p Fq(z)j Fm(\000)d Fq(H)3299 1349 y Fn(!)r(;\015)3409 1334 y Fs(\))3447 1293 y Fi(\000)p Fp(1)3541 1253 y Fl(\001)3604 1334 y Fq(\016)3647 1349 y Fp(0)3686 1334 y Fm(ij)p Fq(dxdy)3962 1194 y Fl(\023)454 1670 y Fm(\024)599 1603 y Fs(1)p 569 1647 108 4 v 569 1739 a(4)p Fq(\031)687 1670 y Fo(E)765 1470 y Fl(0)765 1649 y(@)857 1535 y(Z)913 1760 y Fk(C)981 1526 y Fl(\014)981 1586 y(\014)981 1645 y(\014)981 1705 y(\014)1025 1603 y Fq(@)e Fs(~)-65 b Fq(')p 1025 1647 121 4 v 1032 1739 a(@)p 1088 1684 50 4 v 5 w(z)1155 1670 y Fs(\()p Fq(z)t Fs(\))1280 1526 y Fl(\014)1280 1586 y(\014)1280 1645 y(\014)1280 1705 y(\014)1347 1576 y(X)1331 1791 y Fi(j)p Fn(k)r Fi(j\025)p Fn(n)1523 1670 y Fm(j)p Fq(!)1612 1685 y Fn(k)1676 1670 y Fm(\000)23 b Fq(!)1837 1686 y Fp([)p Fn(k)r Fp(])1916 1694 y Fj(n)1961 1670 y Fm(jjh)p Fq(\016)2099 1685 y Fp(0)2137 1670 y Fq(;)17 b Fs(\()p Fq(z)27 b Fm(\000)c Fq(H)2480 1629 y Fn(n)2472 1695 y(!)r(;\015)2582 1670 y Fs(\))2620 1629 y Fi(\000)p Fp(1)2714 1670 y Fq(\016)2757 1685 y Fn(k)2800 1670 y Fm(ijjh)p Fq(\016)2977 1685 y Fn(k)3019 1670 y Fq(;)17 b Fs(\()p Fq(z)26 b Fm(\000)d Fq(H)3353 1685 y Fn(!)r(;\015)3463 1670 y Fs(\))3501 1629 y Fi(\000)p Fp(1)3595 1670 y Fq(\016)3638 1685 y Fp(0)3678 1670 y Fm(ij)p Fq(dxdy)3954 1470 y Fl(1)3954 1649 y(A)-118 1532 y Fs(\(4.7\))-118 1972 y(where)34 b Fq(z)e Fs(=)c Fq(x)22 b Fs(+)g Fq(iy)36 b Fs(and)d([)p Fq(k)s Fs(])935 1987 y Fn(n)1010 1972 y Fm(\021)28 b Fq(k)36 b Fs(mo)s(d)31 b(\(2)p Fq(n)22 b Fs(+)g(1\))32 b(\(the)h(p)s(oin)m(t)f(b)s(eing)g(c)m(hosen)i(in)e([)p Fm(\000)p Fq(n;)17 b(n)p Fs(])3246 1936 y Fn(d)3287 1972 y Fs(\).)-118 2088 y(By)47 b(a)g(Com)m(b)s(es-Thomas)f(argumen)m(t)h (\(see)h(section)f(6.1\),)i(w)m(e)f(kno)m(w)g(that)f(there)g(exists)h Fq(C)59 b(>)51 b Fs(1)c(suc)m(h)h(that,)-118 2205 y(uniformly)30 b(in)i(\()p Fq(!)539 2220 y Fn(n)586 2205 y Fs(\))624 2220 y Fn(n)p Fi(2)p Fk(Z)794 2205 y Fs(and)h Fq(N)38 b Fm(\025)29 b Fs(1,)j(w)m(e)i(ha)m(v)m(e,)g(for)e(Im\()p Fq(z)t Fs(\))27 b Fm(6)p Fs(=)h(0,)621 2430 y Fm(jh)p Fq(\016)731 2445 y Fp(0)770 2430 y Fq(;)17 b Fs(\()p Fq(z)27 b Fm(\000)c Fq(H)1113 2389 y Fn(n)1105 2455 y(!)r(;\015)1215 2430 y Fs(\))1253 2389 y Fi(\000)p Fp(1)1347 2430 y Fq(\016)1390 2445 y Fn(k)1433 2430 y Fm(ij)e Fs(+)h Fm(jh)p Fq(\016)1729 2445 y Fn(k)1772 2430 y Fq(;)17 b Fs(\()p Fq(z)26 b Fm(\000)d Fq(H)2106 2445 y Fn(!)r(;\015)2216 2430 y Fs(\))2254 2389 y Fi(\000)p Fp(1)2348 2430 y Fq(\016)2391 2445 y Fp(0)2431 2430 y Fm(ij)k(\024)2750 2363 y Fq(C)p 2640 2407 298 4 v 2640 2499 a Fm(j)p Fs(Im)o(\()p Fq(z)t Fs(\))p Fm(j)2947 2430 y Fq(e)2992 2389 y Fi(\000j)p Fp(Im)o(\()p Fn(z)s Fp(\))p Fn(k)r Fi(j)p Fn(=C)-118 2664 y Fs(Hence,)34 b(as)f(the)g(random)e(v)-5 b(ariables)32 b(\()p Fq(!)1348 2679 y Fn(n)1394 2664 y Fs(\))1432 2683 y Fn(n)p Fi(2)p Fk(Z)1572 2664 y Fj(d)j Fs(are)e(b)s(ounded)g(b)m(y)h Fq(\025)p Fs(\()p Fq(\015)5 b Fs(\),)32 b(\(4.7\))g(giv)m(es,)h(for)f (some)h Fq(C)h(>)28 b Fs(1,)504 2898 y Fm(j)p Fo(E)12 b Fs(\(\()p Fq(';)17 b(d)p Fm(N)924 2857 y Fn(N)909 2923 y(!)r(;\015)1024 2898 y Fs(\)\))22 b Fm(\000)h Fs(\()p Fq(';)17 b(d)p Fm(N)e Fs(\))p Fm(j)26 b(\024)i Fq(C)7 b(\025)p Fs(\()p Fq(\015)e Fs(\))1996 2763 y Fl(Z)2051 2988 y Fk(C)2119 2754 y Fl(\014)2119 2814 y(\014)2119 2873 y(\014)2119 2933 y(\014)2162 2831 y Fq(@)22 b Fs(~)-66 b Fq(')p 2162 2875 121 4 v 2169 2967 a(@)p 2225 2912 50 4 v 5 w(z)2293 2898 y Fs(\()p Fq(z)t Fs(\))2418 2754 y Fl(\014)2418 2814 y(\014)2418 2873 y(\014)2418 2933 y(\014)2623 2831 y Fs(1)p 2479 2875 337 4 v 2479 2967 a Fm(j)p Fs(Im)o(\()p Fq(z)t Fs(\))p Fm(j)2776 2938 y Fp(3)2826 2898 y Fq(e)2871 2857 y Fi(\000j)p Fp(Im)n(\()p Fn(z)s Fp(\))p Fn(n)p Fi(j)p Fn(=C)3277 2898 y Fq(dxdy)t(:)-118 3133 y Fs(T)-8 b(aking)36 b(in)m(to)g(accoun)m(t)h(the)g(prop)s(erties) f(of)h(almost)d(analytic)i(extensions)h(\(4.6\),)g(for)f(some)h Fq(C)k(>)34 b Fs(1,)k(for)e Fq(K)41 b Fm(\025)35 b Fs(3)-118 3249 y(and)e Fq(N)38 b Fm(\025)28 b Fs(1,)k(w)m(e)i(get)-102 3599 y Fm(j)p Fo(E)13 b Fs(\(\()p Fq(';)k(d)p Fm(N)319 3558 y Fn(N)304 3624 y(!)r(;\015)419 3599 y Fs(\)\))22 b Fm(\000)g Fs(\()p Fq(';)17 b(d)p Fm(N)e Fs(\))p Fm(j)26 b(\024)j Fq(\025)p Fs(\()p Fq(\015)5 b Fs(\))p Fq(C)1374 3558 y Fn(K)g Fp(+1)1548 3463 y Fl(Z)1604 3689 y Fk(R)p Fi(\002)p Fp([)p Fi(\000)p Fp(1)p Fn(;)p Fp(1])2029 3532 y Fs(1)p 1922 3576 264 4 v 1922 3667 a(1)21 b(+)h Fq(x)2145 3639 y Fp(2)2195 3599 y Fm(j)p Fq(y)t Fm(j)2303 3558 y Fn(K)5 b Fi(\000)p Fp(3)2460 3599 y Fq(e)2505 3558 y Fi(\000j)p Fn(y)r(N)i Fi(j)p Fn(=C)2794 3599 y Fq(dxdy)119 b Fs(sup)3124 3681 y Fn(x)p Fi(2)p Fk(R)3019 3739 y Fp(0)p Fi(\024)p Fn(J)6 b Fi(\024)p Fn(K)f Fp(+3)3380 3455 y Fl(\014)3380 3514 y(\014)3380 3574 y(\014)3380 3634 y(\014)3413 3599 y Fs(\(1)22 b(+)g Fq(x)3675 3558 y Fp(2)3715 3599 y Fs(\))3763 3532 y Fq(d)3814 3495 y Fn(J)3862 3532 y Fq(')p 3763 3576 164 4 v 3767 3667 a(d)3818 3639 y Fn(J)3867 3667 y Fq(x)3936 3599 y Fs(\()p Fq(x)p Fs(\))4067 3455 y Fl(\014)4067 3514 y(\014)4067 3574 y(\014)4067 3634 y(\014)1002 3961 y Fm(\024)29 b Fq(\025)p Fs(\()p Fq(\015)5 b Fs(\))1314 3821 y Fl(\022)1396 3894 y Fq(C)i(K)p 1396 3938 168 4 v 1435 4030 a(N)1573 3821 y Fl(\023)1647 3843 y Fn(K)e Fi(\000)p Fp(3)1920 3961 y Fs(sup)1926 4043 y Fn(x)p Fi(2)p Fk(R)1822 4102 y Fp(0)p Fi(\024)p Fn(J)h Fi(\024)p Fn(K)f Fp(+3)2183 3817 y Fl(\014)2183 3877 y(\014)2183 3936 y(\014)2183 3996 y(\014)2216 3961 y Fs(\(1)22 b(+)g Fq(x)2478 3920 y Fp(2)2518 3961 y Fs(\))2566 3894 y Fq(d)2617 3858 y Fn(J)2665 3894 y Fq(')p 2566 3938 164 4 v 2570 4030 a(d)2621 4001 y Fn(J)2669 4030 y Fq(x)2739 3961 y Fs(\()p Fq(x)p Fs(\))2870 3817 y Fl(\014)2870 3877 y(\014)2870 3936 y(\014)2870 3996 y(\014)2920 3961 y Fq(:)-118 4257 y Fs(This)33 b(completes)f(the)h(pro)s(of)f(of)g(the)h (Lemma)e(4.2.)p 4063 4257 4 66 v 4067 4195 59 4 v 4067 4257 V 4125 4257 4 66 v -118 4374 a(Let)36 b(us)h(no)m(w)g(complete)e (the)h(pro)s(of)g(of)f(Lemma)g(4.1.)53 b(First)35 b(note)i(that,)g(as)f Fq(d)p Fm(N)2873 4389 y Fn(!)r(;\015)3019 4374 y Fs(and)g Fq(d)p Fm(N)3345 4389 y Fn(\015)3425 4374 y Fs(are)g(supp)s(orted)h(in) -118 4490 y Fq(I)e Fs(:=)28 b([0)p Fq(;)17 b(C)25 b Fs(+)18 b Fq(\025)p Fs(\()p Fq(\015)5 b Fs(\)])31 b(\(for)f(some)h Fq(C)j(>)28 b Fs(0\),)j(w)m(e)h(only)e(need)i(to)e(pro)m(v)m(e)i(Lemma) d(4.1)i(for)f(energies)h Fq(E)j Fm(2)28 b Fq(I)8 b Fs(.)43 b(Pic)m(k)31 b Fq(')g Fs(b)s(e)-118 4606 y(a)i(Gevrey)h(class)g (function)e(of)h(Gevrey)i(exp)s(onen)m(t)f Fq(\013)c(>)f Fs(1)k(\(see)h([10]\);)f(assume,)i(moreo)m(v)m(er,)f(that)f Fq(')g Fs(has)g(supp)s(ort)-118 4722 y(in)f(\()p Fm(\000)p Fs(1)p Fq(;)17 b Fs(1\),)32 b(that)g(0)c Fm(\024)g Fq(')f Fm(\024)i Fs(1)j(and)h(that)f Fq(')c Fm(\021)g Fs(1)k(on)g(\()p Fm(\000)p Fs(1)p Fq(=)p Fs(2)p Fq(;)17 b Fs(1)p Fq(=)p Fs(2].)43 b(Let)33 b Fq(E)g Fm(2)28 b Fq(I)41 b Fs(and)32 b Fq(\027)j Fm(2)28 b Fs(\(0)p Fq(;)17 b Fs(1\),and)32 b(set)1489 4927 y Fq(')1553 4942 y Fn(E)t(;\027)1671 4927 y Fs(\()p Fm(\001)p Fs(\))27 b(=)h Fy(1)1962 4942 y Fp([0)p Fn(;E)t Fp(])2137 4927 y Fm(\003)22 b Fq(')2290 4816 y Fl(\020)2373 4860 y Fm(\001)p 2359 4904 55 4 v 2359 4995 a Fq(\027)2424 4816 y Fl(\021)2500 4927 y Fq(:)-118 5130 y Fs(Then,)30 b(b)m(y)g(Lemma)d(4.2)h(and)g(the)h(Gevrey)h (estimates)e(on)g(the)h(deriv)-5 b(ativ)m(es)29 b(of)f Fq(')p Fs(,)h(w)m(e)g(get)g(that)f(there)h(exist)g Fq(C)35 b(>)27 b Fs(1)-118 5246 y(suc)m(h)34 b(that,)e(for)g Fq(N)39 b Fm(\025)28 b Fs(1,)k Fq(K)j Fm(\025)28 b Fs(1)k(and)h(0)28 b Fq(<)f(\027)34 b(<)28 b Fs(1,)k(w)m(e)i(ha)m(v)m(e)746 5496 y Fm(j)p Fo(E)12 b Fs(\(\()p Fq(')974 5511 y Fn(E)t(;\027)1098 5496 y Fq(;)17 b(d)p Fm(N)1290 5455 y Fn(N)1275 5521 y(!)r(;\015)1384 5496 y Fs(\)\))22 b Fm(\000)h Fs(\()p Fq(')1684 5511 y Fn(E)t(;\027)1802 5496 y Fq(;)17 b(d)p Fm(N)1979 5511 y Fn(\015)2022 5496 y Fs(\))p Fm(j)28 b(\024)g Fq(\025)p Fs(\()p Fq(\015)5 b Fs(\)\()p Fq(N)10 b(\027)c Fs(\))2628 5455 y Fp(3)2685 5356 y Fl(\022)2768 5429 y Fq(C)h(K)2935 5393 y Fp(1+)p Fn(\013)p 2768 5473 307 4 v 2850 5565 a Fq(N)j(\027)3085 5356 y Fl(\023)3158 5376 y Fn(K)3243 5496 y Fq(:)-3388 b Fs(\(4.8\))1969 5690 y Fg(11)p eop %%Page: 12 12 12 11 bop -118 241 a Fs(W)-8 b(e)34 b(optimize)d(the)j(righ)m(t)e(hand) i(side)f(of)40 b(\(4.8\))33 b(in)f Fq(K)41 b Fs(and)33 b(get)g(that,)h(there)g(exist)g Fq(C)h(>)29 b Fs(1)k(suc)m(h)i(that,)e (for)g Fq(N)40 b Fm(\025)29 b Fs(1)-118 357 y(and)k(0)27 b Fq(<)h(\027)34 b(<)27 b Fs(1,)33 b(w)m(e)g(ha)m(v)m(e)291 539 y Fm(j)p Fo(E)12 b Fs(\(\()p Fq(')519 554 y Fn(E)t(;\027)643 539 y Fq(;)17 b(d)p Fm(N)835 498 y Fn(N)820 564 y(!)r(;\015)930 539 y Fs(\)\))22 b Fm(\000)g Fs(\()p Fq(')1229 554 y Fn(E)t(;\027)1347 539 y Fq(;)17 b(d)p Fm(N)1524 554 y Fn(\015)1568 539 y Fs(\))p Fm(j)27 b(\024)h Fq(C)7 b(\025)p Fs(\()p Fq(\015)e Fs(\)\()p Fq(N)33 b Fs(+)22 b Fq(\027)2333 498 y Fi(\000)p Fp(1)2428 539 y Fs(\))2466 498 y Fp(3)2505 539 y Fq(e)2550 498 y Fi(\000)p Fp(\()p Fn(N)7 b(\027)t(=C)e Fp(\))2851 475 y Fc(1)p Fj(=)p Fc(\(1+)p Fj(\013)p Fc(\))3084 498 y Fp(+)p Fn(C)g Fp(\()p Fn(N)i(\027)t(=C)e Fp(\))3440 475 y Fe(\000)p Fc(1)p Fj(=)p Fc(\(1+)p Fj(\013)p Fc(\))-118 724 y Fs(No)m(w)33 b(if)e(w)m(e)j(c)m(ho)s(ose)g Fq(N)k Fm(\025)28 b Fq(\027)922 683 y Fi(\000)p Fp(1)p Fi(\000)p Fn(\021)1109 724 y Fs(,)33 b(w)m(e)h(get)e(that)h(there)g(exist)g Fq(\027)2215 739 y Fp(0)2282 724 y Fq(>)28 b Fs(0)k(suc)m(h)i(that)e (for)g(0)c Fq(<)f(\027)34 b(<)28 b(\027)3461 739 y Fp(0)3501 724 y Fs(,)k(w)m(e)i(ha)m(v)m(e)973 906 y Fm(j)p Fo(E)12 b Fs(\(\()p Fq(')1201 921 y Fn(E)t(;\027)1325 906 y Fq(;)17 b(d)p Fm(N)1517 865 y Fn(N)1502 931 y(!)r(;\015)1611 906 y Fs(\)\))22 b Fm(\000)h Fs(\()p Fq(')1911 921 y Fn(E)t(;\027)2029 906 y Fq(;)17 b(d)p Fm(N)2206 921 y Fn(\015)2249 906 y Fs(\))p Fm(j)28 b(\024)g Fq(\025)p Fs(\()p Fq(\015)5 b Fs(\))p Fq(e)2682 865 y Fi(\000)p Fn(\027)2776 842 y Fe(\000)p Fj(\021)r(=)p Fc(\(2)p Fj(\013)p Fc(\))3016 906 y Fq(:)-3161 b Fs(\(4.9\))-118 1088 y(By)30 b(de\014nition,)f Fq(')554 1103 y Fn(E)t(;\027)700 1088 y Fm(\021)f Fs(1)h(on)g([0)p Fq(;)17 b(E)6 b Fs(],)30 b(and)f Fq(')1547 1103 y Fn(E)t(;\027)1695 1088 y Fs(has)g(supp)s(ort)h (in)f([)p Fm(\000)p Fq(\027)q(;)17 b(E)22 b Fs(+)16 b Fq(\027)6 b Fs(])29 b(and)g(is)g(b)s(ounded)h(b)m(y)h(1.)42 b(As)30 b Fq(dN)4034 1052 y Fn(N)4024 1113 y(!)r(;\015)-118 1204 y Fs(and)j Fq(d)p Fm(N)205 1219 y Fn(\015)281 1204 y Fs(are)g(p)s(ositiv)m(e)e(measures,)j(w)m(e)g(ha)m(v)m(e)943 1380 y Fo(E)12 b Fs(\()p Fm(N)1138 1339 y Fn(N)1123 1404 y(!)r(;\015)1239 1380 y Fs(\()p Fq(E)6 b Fs(\)\))27 b Fm(\024)i Fo(E)12 b Fs(\(\()p Fq(')1764 1395 y Fn(E)t(;\027)1888 1380 y Fq(;)17 b(d)p Fm(N)2080 1339 y Fn(N)2065 1404 y(!)r(;\015)2174 1380 y Fs(\)\))28 b Fm(\024)g Fo(E)13 b Fs(\()p Fm(N)2579 1339 y Fn(N)2564 1404 y(!)r(;\015)2680 1380 y Fs(\()p Fq(E)28 b Fs(+)22 b Fq(\027)6 b Fs(\)\))p Fq(:)-3191 b Fs(\(4.10\))-118 1555 y(Hence,)34 b(b)m(y)h(\(4.9\))d(and) g(\(4.10\),)g(w)m(e)i(obtain)400 1727 y Fm(N)482 1742 y Fn(\015)526 1727 y Fs(\()p Fq(E)6 b Fs(\))28 b Fm(\024)g Fs(\()p Fq(')915 1742 y Fn(E)t(;\027)1033 1727 y Fq(;)17 b(d)p Fm(N)1210 1742 y Fn(\015)1253 1727 y Fs(\))28 b(=)f Fo(E)13 b Fs(\(\()p Fq(')1623 1742 y Fn(E)t(;\027)1747 1727 y Fq(;)k(d)p Fm(N)1939 1686 y Fn(N)1924 1752 y(!)r(;\015)2033 1727 y Fs(\)\))22 b(+)2229 1647 y Fl(\002)2271 1727 y Fs(\()p Fq(')2373 1742 y Fn(E)t(;\027)2491 1727 y Fq(;)17 b(d)p Fm(N)2668 1742 y Fn(\015)2711 1727 y Fs(\))23 b Fm(\000)f Fo(E)13 b Fs(\(\()p Fq(')3072 1742 y Fn(E)t(;\027)3196 1727 y Fq(;)k(d)p Fm(N)3388 1686 y Fn(N)3373 1752 y(!)r(;\015)3482 1727 y Fs(\)\))3558 1647 y Fl(\003)708 1906 y Fm(\024)28 b Fo(E)12 b Fs(\()p Fm(N)1008 1864 y Fn(N)993 1930 y(!)r(;\015)1109 1906 y Fs(\()p Fq(E)28 b Fs(+)22 b Fq(\027)6 b Fs(\)\))23 b(+)f Fq(\025)p Fs(\()p Fq(\015)5 b Fs(\))p Fq(e)1830 1864 y Fi(\000)p Fn(\027)1924 1841 y Fe(\000)p Fj(\021)r(=)p Fc(\(2)p Fj(\013)p Fc(\))-118 2075 y Fs(and)344 2223 y Fm(N)426 2238 y Fn(\015)471 2223 y Fs(\()p Fq(E)h Fs(\))27 b Fm(\025)h Fs(\()p Fq(')859 2238 y Fn(E)t Fi(\000)p Fn(\027;\027)1067 2223 y Fq(;)17 b(d)p Fm(N)1244 2238 y Fn(\015)1288 2223 y Fs(\))28 b(=)f Fo(E)13 b Fs(\(\()p Fq(')1658 2238 y Fn(E)t Fi(\000)p Fn(\027;\027)1872 2223 y Fq(;)k(d)p Fm(N)2064 2182 y Fn(N)2049 2247 y(!)r(;\015)2158 2223 y Fs(\)\))22 b(+)2354 2142 y Fl(\002)2395 2223 y Fs(\()p Fq(')2497 2238 y Fn(E)t Fi(\000)p Fn(\027;\027)2706 2223 y 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Fp(2)808 2897 y Fs(\([)p Fm(\000)p Fq(\031)t(;)17 b(\031)t Fs(])1139 2861 y Fn(d)1180 2897 y Fs(\))29 b Fm(!)g Fq(`)1417 2861 y Fp(2)1456 2897 y Fs(\()p Fo(Z)1563 2861 y Fn(d)1601 2897 y Fs(\))k(the)h(standard)g(F) -8 b(ourier)32 b(series)i(transform.)45 b(Then,)35 b(w)m(e)f(ha)m(v)m (e,)h(for)-118 3014 y Fq(u)27 b Fm(2)h Fq(L)125 2977 y Fp(2)165 3014 y Fs(\([)p Fm(\000)p Fq(\031)t(;)17 b(\031)t Fs(])496 2977 y Fn(d)537 3014 y Fs(\),)733 3204 y(\()796 3179 y(^)771 3204 y Fq(H)852 3219 y Fn(!)r(;\015)962 3204 y Fq(u)p Fs(\)\()p Fq(\022)s Fs(\))27 b(=)h(\()p Fm(F)10 b Fq(H)1512 3219 y Fn(!)r(;\015)1622 3204 y Fm(F)1704 3163 y Fi(\003)1742 3204 y Fq(u)p Fs(\)\()p Fq(\022)s Fs(\))28 b(=)f Fq(h)p Fs(\()p Fq(\022)s Fs(\))p Fq(u)p Fs(\()p Fq(\022)s Fs(\))22 b(+)g Fq(\015)2652 3110 y Fl(X)2644 3329 y Fn(j)t Fi(2)p Fk(Z)2773 3310 y Fj(d)2820 3204 y Fq(!)2881 3219 y Fn(j)2917 3204 y Fs(\(\005)3028 3219 y Fn(j)3065 3204 y Fq(u)p Fs(\)\()p Fq(\022)s Fs(\))1028 3525 y(where)34 b(\(\005)1421 3540 y 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Fn(N)1428 764 y Fs(\()p Fq(\022)s Fs(\))h(and)h Fq(V)1853 728 y Fn(N)1831 789 y(!)1952 764 y Fs(are)g(non-negativ)m(e)f(matrices.)94 880 y(This)38 b(immediately)c(tells)j(us)h(that)f(the)h(Flo)s(quet)e (eigen)m(v)-5 b(alues)38 b(and)f(eigen)m(v)m(ectors)i(of)e Fq(H)3438 844 y Fn(N)3430 905 y(!)r(;\015)3578 880 y Fs(with)g(Flo)s(quet)-118 1001 y(quasi-momen)m(tum)31 b Fq(\022)k Fs(\(i.e.)43 b(the)33 b(v)m(ectors,)h Fq(u)28 b Fs(=)f(\()p Fq(u)1740 1016 y Fn(j)1776 1001 y Fs(\))1814 1020 y Fn(j)t Fi(2)p Fk(Z)1944 1001 y Fj(d)r Fs(\),)33 b(solution)e(to)h(the)h(problem)966 1109 y Fl(\()1046 1212 y Fq(H)1135 1176 y Fn(N)1127 1237 y(!)r(;\015)1238 1212 y Fq(u)97 b Fs(=)27 b Fq(\025u;)1046 1352 y(u)1102 1367 y Fn(j)t Fp(+)p Fn(k)1391 1352 y Fs(=)g Fq(e)1539 1315 y Fi(\000)p Fn(ik)r(\022)1696 1352 y Fq(u)1752 1367 y Fn(j)1820 1352 y Fs(for)32 b Fq(j)i Fm(2)28 b Fo(Z)2206 1315 y Fn(d)2244 1352 y Fq(;)49 b(k)31 b Fm(2)d Fs(\(2)p Fq(N)k Fs(+)22 b(1\))p Fo(Z)2947 1315 y Fn(d)2985 1352 y Fs(\))-118 1557 y(are)43 b(the)g(eigen)m(v)-5 b(alues)43 b(and)g(eigen)m(v)m(ectors)h(\(con)m(tin)m(ued)g(quasi-p)s(erio)s (dically\))39 b(of)k(the)g(\(2)p Fq(N)c Fs(+)29 b(1\))3521 1521 y Fn(d)3591 1557 y Fm(\002)g Fs(\(2)p Fq(N)40 b Fs(+)29 b(1\))4094 1521 y Fn(d)-118 1674 y Fs(matrix)i Fq(M)303 1638 y Fn(N)293 1698 y(!)r(;\015)404 1674 y Fs(\()p Fq(\022)s Fs(\).)43 b(F)-8 b(or)32 b Fq(E)i Fm(2)28 b Fo(R)5 b Fs(,)38 b(one)33 b(has)686 1919 y Fm(N)783 1877 y Fn(N)768 1943 y(!)r(;\015)878 1919 y Fs(\()p Fq(E)6 b Fs(\))28 b(=)1163 1783 y Fl(Z)1218 2009 y Fp([)p Fi(\000)1364 1981 y Fj(\031)p 1303 1993 162 3 v 1303 2035 a Fc(2)p Fj(N)5 b Fc(+1)1474 2009 y Fn(;)1565 1981 y Fj(\031)p 1503 1993 V 1503 2035 a Fc(2)p Fj(N)g Fc(+1)1674 2009 y Fp(])1694 1990 y Fj(d)1751 1919 y Fq(])p Fm(f)p Fs(eigen)m(v)-5 b(alues)32 b(of)g Fq(M)2558 1877 y Fn(N)2548 1943 y(!)r(;\015)2659 1919 y Fs(\()p Fq(\022)s Fs(\))h(in)e([0)p Fq(;)17 b(E)6 b Fs(])p Fm(g)p Fq(d\022)s(:)-3448 b Fs(\(4.15\))-118 2196 y(Considering)36 b Fq(H)44 b Fs(as)37 b(2)p Fq(N)e Fs(+)25 b(1-p)s(erio)s(dic)35 b(on)h Fo(Z)1603 2159 y Fn(d)1641 2196 y Fs(,)i(w)m(e)g(see)f(that)g(the)g(Flo)s(quet)f(eigen)m (v)-5 b(alues)37 b(of)f Fq(H)44 b Fs(\(for)36 b(the)h(quasi-)-118 2314 y(momen)m(tum)42 b Fq(\022)s Fs(\))i(are)g(\()p Fq(h)p Fs(\()p Fq(\022)33 b Fs(+)1071 2275 y Fp(2)p Fn(\031)r(k)p 1035 2291 189 4 v 1035 2348 a Fp(2)p Fn(N)7 b Fp(+1)1233 2314 y Fs(\)\))1309 2337 y Fn(k)r Fi(2)p Fk(Z)1445 2314 y Fj(d)1445 2360 y Fc(2)p Fj(N)e Fc(+1)1608 2314 y Fs(;)50 b(the)44 b(Flo)s(quet)f(eigen)m(v)-5 b(alue)44 b Fq(h)p Fs(\()p Fq(\022)33 b Fs(+)3038 2275 y Fp(2)p Fn(\031)r(k)p 3002 2291 V 3002 2348 a Fp(2)p Fn(N)7 b Fp(+1)3201 2314 y Fs(\))44 b(is)f(asso)s(ciated)h(to)f(the)-118 2458 y(Flo)s(quet)32 b(eigen)m(v)m(ector)h Fq(u)806 2473 y Fn(k)849 2458 y Fs(\()p Fq(\022)s Fs(\),)f Fq(k)f Fm(2)d Fo(Z)1277 2422 y Fn(d)1277 2483 y Fp(2)p Fn(N)7 b Fp(+1)1500 2458 y Fs(de\014ned)34 b(b)m(y)1134 2693 y Fq(u)1190 2708 y Fn(k)1232 2693 y Fs(\()p 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Fs(\)\))2807 3080 y Fn(l)q Fi(2)p Fk(Z)2926 3057 y Fj(d)2926 3103 y Fc(2)p Fj(N)5 b Fc(+1)3123 3057 y Fs(ha)m(v)m(e)34 b(the)f(follo)m(wing)d(com-)-118 3187 y(p)s(onen)m(ts)k(in)d(this)i (basis)1176 3410 y Fq(v)1223 3425 y Fn(l)1249 3410 y Fs(\()p Fq(\022)s Fs(\))28 b(=)1736 3343 y(1)p 1514 3387 494 4 v 1514 3479 a(\(2)p Fq(N)k Fs(+)22 b(1\))1896 3450 y Fn(d=)p Fp(2)2017 3410 y Fs(\()p Fq(e)2100 3367 y Fn(i)p Fp(\()p Fn(\022)r Fp(+)2281 3339 y Fc(2)p Fj(\031)r(k)p 2251 3351 162 3 v 2251 3393 a Fc(2)p Fj(N)5 b Fc(+1)2422 3367 y Fp(\))p Fn(l)2476 3410 y Fs(\))2514 3433 y Fn(k)r Fi(2)p Fk(Z)2650 3410 y Fj(d)2650 3456 y Fc(2)p Fj(N)g Fc(+1)2813 3410 y Fq(:)-118 3650 y Fs(W)-8 b(e)33 b(de\014ne)h(the)f(v) m(ectors)h(\()p Fq(v)916 3665 y Fn(l)942 3650 y Fs(\))980 3673 y Fn(l)q Fi(2)p Fk(Z)1099 3650 y Fj(d)1099 3696 y Fc(2)p Fj(N)5 b Fc(+1)1295 3650 y Fs(b)m(y)1064 3906 y Fq(v)1111 3921 y Fn(l)1165 3906 y Fs(=)28 b Fq(e)1314 3865 y Fi(\000)p Fn(il)q(\022)1454 3906 y Fq(v)1501 3921 y Fn(l)1527 3906 y Fs(\()p Fq(\022)s Fs(\))f(=)2014 3839 y(1)p 1792 3884 494 4 v 1792 3975 a(\(2)p Fq(N)32 b Fs(+)22 b(1\))2174 3946 y Fn(d=)p Fp(2)2295 3906 y Fs(\()p Fq(e)2378 3863 y Fn(i)2432 3836 y Fc(2)p Fj(\031)r(k)q(l)p 2412 3848 162 3 v 2412 3889 a Fc(2)p Fj(N)5 b Fc(+1)2587 3906 y Fs(\))2625 3930 y Fn(k)r Fi(2)p Fk(Z)2761 3907 y Fj(d)2761 3953 y Fc(2)p Fj(N)g Fc(+1)2924 3906 y Fq(:)-3069 b Fs(\(4.16\))-118 4181 y(4.3.)56 b Fy(Estimating)34 b(the)k(densit)m(y)f(of)g(states)h (of)g(the)f(p)s(erio)s(dic)f(appro)m(ximations.)48 b Fs(De\014ne)33 b(the)g(ev)m(en)m(t)442 4361 y(\012\()p Fq(E)6 b(;)17 b(\015)5 b(;)17 b(N)10 b Fs(\))28 b(=)g Fm(f)p Fq(!)t Fs(;)48 b Fm(9)p Fq(\022)31 b Fm(2)d Fo(R)1511 4319 y Fn(d)1590 4361 y Fs(suc)m(h)34 b(that)f Fq(M)2126 4319 y Fn(N)2116 4385 y(!)r(;\015)2226 4361 y Fs(\()p Fq(\022)s Fs(\))g(has)g(an)f(eigen)m(v)-5 b(alue)32 b(in)g([0)p Fq(;)17 b(E)6 b Fs(])p Fm(g)p Fq(:)-118 4546 y Fs(By)33 b(Lemma)e(4.1)h(and)h(\(4.15\))o(,)g(Theorem)g(3.1)f(is)g(a)g (consequence)k(of)-118 4733 y Fy(Prop)s(osition)f(4.1.)50 b Ff(Fix)39 b Fs(0)e Fq(<)h(\021)j(<)c(\021)1343 4697 y Fi(0)1404 4733 y Fq(<)g(\021)t Fs(\()p Fq(r)s Fs(\))i Ff(and)h Fq(\032)e(>)f(d)p Ff(.)60 b(Then,)41 b(ther)-5 b(e)40 b(exists)g Fq(\015)3130 4748 y Fn(\021)r(;\032)3265 4733 y Fq(>)d Fs(0)j Ff(and)g Fq(")d(>)g Fs(0)j Ff(such)-118 4850 y(that,)35 b(for)g Fq(\015)d Fm(2)c Fs(\(0)p Fq(;)17 b(\015)626 4865 y Fn(\021)r(;\032)723 4850 y Fs(\))p Ff(,)35 b(one)f(has)1360 5046 y Fo(P)p Fs([\012\()p Fq(\015)5 b Fs(\()p 1648 4991 65 4 v Fq(!)28 b Fm(\000)23 b Fq(\015)1893 5005 y Fn(\021)1935 5046 y Fs(\))p Fq(;)17 b(\015)5 b(;)17 b(N)10 b Fs(\)])27 b Fm(\024)h Fq(e)2447 5005 y Fi(\000)p Fn(\015)2542 4981 y Fe(\000)p Fj(")2628 5046 y Fq(:)-2773 b Fs(\(4.17\))-118 5226 y Ff(wher)-5 b(e)837 5405 y Fs(2)p Fq(N)33 b Fs(+)22 b(1)27 b(=)h([)p Fq(\015)1358 5364 y Fi(\000)p Fp(1)p Fn(=)p Fp(2+2)p Fn(\021)1645 5341 y Fe(0)1669 5364 y Fp(+)p Fn(\021)1761 5341 y Fe(0)1784 5364 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\))2002 5405 y Fs(])2029 5420 y Fn(o)2090 5405 y Fm(\001)21 b Fs([)p Fq(\015)2222 5364 y Fi(\000)p Fn(\021)2314 5341 y Fe(0)2338 5364 y Fi(\000)p Fn(\021)2430 5341 y Fe(0)2453 5364 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\))2671 5405 y Fs(])2698 5420 y Fn(o)2758 5405 y Fm(\001)h Fs([)p Fq(\015)2891 5364 y Fi(\000)p Fn(\032)2987 5405 y Fs(])3014 5420 y Fn(o)3152 5405 y Fq(:)-118 5591 y Ff(Her)-5 b(e,)35 b Fs([)p Fm(\001)p Fs(])227 5606 y Fn(o)300 5591 y Ff(denotes)f(the)h (lar)-5 b(gest)35 b(o)-5 b(dd)34 b(inte)-5 b(ger)34 b(smal)5 b(ler)34 b(than)h Fm(\001)p Ff(.)1969 5690 y Fg(13)p eop %%Page: 14 14 14 13 bop -118 241 a Fs(Notice)36 b(that,)i(b)m(y)g(the)f(de\014nition) f(of)g Fq(\021)t Fs(\()p Fq(r)s Fs(\),)h(as)g Fq(\021)h(<)d(\021)1913 205 y Fi(0)1971 241 y Fq(<)g(\021)t Fs(\()p Fq(r)s Fs(\),)i(one)g(has)g Fm(\000)p Fq(\021)2811 205 y Fi(0)2859 241 y Fm(\000)26 b Fq(\021)3014 205 y Fi(0)3037 241 y Fq(=)p Fs(\()p Fq(r)h Fm(\000)f Fs(1\))34 b Fq(<)h Fs(0)h(and)h Fm(\000)p Fs(1)p Fq(=)p Fs(2)25 b(+)-118 357 y(2)p Fq(\021)-17 321 y Fi(0)28 357 y Fs(+)d Fq(\021)178 321 y Fi(0)201 357 y Fq(=)p Fs(\()p Fq(r)i Fm(\000)f Fs(1\))k Fq(<)h Fs(0.)-118 473 y(Let)35 b(us)g(no)m(w)h(deriv)m(e)g(Theorem)f(1.1)f(from)g(Prop)s (osition)f(4.1.)50 b(Pic)m(k)35 b Fq(\021)k Fs(as)c(in)f(Theorem)h(1.1) g(and)g Fq(\021)g(<)c(\021)3770 437 y Fi(0)3825 473 y Fq(<)h(\021)t Fs(\()p Fq(r)s Fs(\).)-118 589 y(Let)37 b Fq(\032)f(>)g(d)h Fs(b)s(e)g(larger)f(than)h(the)h(exp)s(onen)m(t)h Fq(\032)e Fs(obtained)g(in)f(Lemma)g(4.1)h(for)g Fq(\013)f Fs(=)f(1.)58 b(By)39 b(\(4.15\))d(and)h(\(4.17\),)-118 706 y(one)c(has)343 893 y Fo(E)13 b Fs([)p Fm(N)528 852 y Fn(N)513 918 y(!)r(;\015)629 893 y Fs(\()p Fq(\015)5 b Fs(\()p 761 838 65 4 v Fq(!)25 b Fm(\000)e Fq(\015)1003 852 y Fn(\021)1040 829 y Fe(0)1067 893 y Fs(\)\)])28 b(=)1301 757 y Fl(Z)1357 983 y Fp([)p Fi(\000)1503 956 y Fj(\031)p 1441 968 162 3 v 1441 1009 a Fc(2)p Fj(N)5 b Fc(+1)1612 983 y Fn(;)1703 956 y Fj(\031)p 1642 968 V 1642 1009 a Fc(2)p Fj(N)g Fc(+1)1813 983 y Fp(])1833 964 y Fj(d)1889 893 y Fo(E)13 b Fs([)p Fq(])p Fm(f)p Fs(eigen)m(v)-5 b(alues)38 b(of)32 b Fq(M)2790 852 y Fn(N)2780 918 y(!)r(;\015)2891 893 y Fs(\()p Fq(\022)s Fs(\))g(in)g([0)p Fq(;)17 b(\015)5 b Fs(\()p 3375 838 65 4 v Fq(!)25 b Fm(\000)e Fq(\015)3617 852 y Fn(\021)3654 829 y Fe(0)3681 893 y Fs(\)])p Fm(g)p Fs(])p Fq(d\022)1198 1189 y Fm(\024)1303 1053 y Fl(Z)1358 1279 y Fp([)p Fi(\000)1504 1252 y Fj(\031)p 1443 1264 162 3 v 1443 1305 a Fc(2)p Fj(N)5 b Fc(+1)1614 1279 y Fn(;)1705 1252 y Fj(\031)p 1643 1264 V 1643 1305 a Fc(2)p Fj(N)g Fc(+1)1814 1279 y Fp(])1834 1260 y Fj(d)1891 1189 y Fo(P)p Fs([\012\()p Fq(\015)g Fs(\()p 2179 1134 65 4 v Fq(!)28 b Fm(\000)22 b Fq(\015)2423 1148 y Fn(\021)2460 1124 y Fe(0)2487 1189 y Fs(\))p Fq(;)17 b(\015)5 b(;)17 b(N)10 b Fs(\)]\(2)p Fq(N)33 b Fs(+)22 b(1\))3205 1148 y Fn(d)3245 1189 y Fq(d\022)1198 1441 y Fm(\024)28 b Fq(C)7 b Fo(P)p Fs([\012\()p Fq(\015)e Fs(\()p 1668 1387 V Fq(!)28 b Fm(\000)22 b Fq(\015)1912 1400 y Fn(\021)1949 1377 y Fe(0)1977 1441 y Fs(\))p Fq(;)17 b(\015)5 b(;)17 b(N)10 b Fs(\)])p Fq(:)-118 1615 y Fs(Here,)32 b(w)m(e)h(used)f(the)g(fact)f(that)g Fq(M)1174 1578 y Fn(N)1164 1639 y(!)r(;\015)1275 1615 y Fs(\()p Fq(\022)s Fs(\))g(has)h(rank)g(\(2)p Fq(N)e Fs(+)19 b(1\))2204 1578 y Fn(d)2244 1615 y Fs(.)44 b(Applying)30 b(Prop)s(osition)g(4.1,)h(for)g Fq(\015)36 b Fs(su\016cien)m(tly)-118 1731 y(small,)30 b(w)m(e)k(get)1345 1898 y Fo(E)13 b Fs([)p Fm(N)1530 1857 y Fn(N)1515 1923 y(!)r(;\015)1631 1898 y Fs(\()p Fq(\015)5 b Fs(\()p 1763 1844 V Fq(!)25 b Fm(\000)e Fq(\015)2005 1857 y Fn(\021)2042 1834 y Fe(0)2069 1898 y Fs(\)\)])28 b Fm(\024)g Fq(C)7 b(e)2427 1857 y Fi(\000)p Fp(2)p Fn(\015)2557 1834 y Fe(\000)p Fj(")2643 1898 y Fq(:)-118 2076 y Fs(Plugging)34 b(this)i(in)m(to)g(\(4.2\))g (for)g Fq(E)k Fs(=)34 b Fq(\015)5 b Fs(\()p 1395 2022 V Fq(!)28 b Fm(\000)d Fq(\015)1642 2040 y Fn(\021)1684 2076 y Fs(\))36 b(and)h Fq(\027)j Fs(=)34 b Fq(\015)2206 2040 y Fp(1+)p Fn(\021)2333 2017 y Fe(0)2361 2076 y Fs(,)j(as)g Fm(N)2631 2091 y Fn(\015)2711 2076 y Fs(is)f(increasing)g(and)g(as)h Fq(\021)h(<)c(\021)3838 2040 y Fi(0)3861 2076 y Fs(,)j(for)f Fq(\015)-118 2193 y Fs(su\016cien)m(tly)d(small,)e(w)m(e)i(obtain)866 2377 y Fm(N)948 2392 y Fn(\015)993 2377 y Fs([)p Fq(\015)5 b Fs(\()p 1114 2322 V Fq(!)25 b Fm(\000)e Fq(\015)1356 2336 y Fn(\021)1398 2377 y Fs(\)])k Fm(\024)h Fo(E)13 b Fs([)p Fm(N)1780 2336 y Fn(N)1765 2402 y(!)r(;\015)1881 2377 y Fs(\()p Fq(\015)5 b Fs(\()p 2013 2322 V Fq(!)25 b Fm(\000)e Fq(\015)2255 2336 y Fn(\021)2292 2312 y Fe(0)2319 2377 y Fs(\)\)])f(+)g Fq(e)2587 2336 y Fi(\000)p Fn(\015)2682 2312 y Fe(\000)p Fj(")2796 2377 y Fm(\024)28 b Fq(e)2946 2336 y Fi(\000)p Fn(\015)3041 2312 y Fe(\000)p Fj(")3118 2292 y Fe(0)-118 2541 y Fs(with)k Fq(")150 2505 y Fi(0)201 2541 y Fq(<)27 b(")p Fs(.)43 b(This)33 b(completes)f(the)h(pro)s(of)f (of)g(Theorem)h(3.1.)p 4063 2541 4 66 v 4067 2478 59 4 v 4067 2541 V 4125 2541 4 66 v -118 2721 a(4.4.)56 b Fy(The)34 b(pro)s(of)h(of)f(Prop)s(ositions)f(4.1.)49 b Fs(The)31 b(pro)s(of)e(of)g(Prop)s(osition)f(4.1)h(follo)m(ws)g (roughly)g(the)h(lines)f(of)g(the)-118 2837 y(pro)s(of)34 b(of)g(the)h(upp)s(er)h(b)s(ound)f(in)f([11)o(],)i(section)f(2.3.)49 b(The)36 b(main)d(di\013erence)j(comes)f(from)e(the)j(fact)e(that,)h (in)f(the)-118 2953 y(presen)m(t)h(case,)g(the)f(suprem)m(um)f(norm)g (of)g(the)h(random)f(v)-5 b(ariables)32 b(dep)s(ends)j(on)e(the)h (energy)h(in)m(terv)-5 b(al)32 b(on)h(whic)m(h)-118 3070 y(w)m(e)h(w)m(an)m(t)f(to)f(con)m(trol)g(the)h(densit)m(y)h(of)e (states)h(or)f(the)h(probabilit)m(y)e(of)h(the)h(presence)i(of)d(eigen) m(v)-5 b(alues.)-118 3186 y(Pic)m(k)33 b Fq(\021)t Fs(,)f Fq(\032)h Fs(as)g(in)f(Prop)s(osition)f(4.1.)43 b(De\014ne)483 3349 y(2)p Fq(L)598 3308 y Fi(0)643 3349 y Fs(+)22 b(1)28 b(=)f([)p Fq(\015)1004 3308 y Fi(\000)p Fp(1)p Fn(=)p Fp(2+2)p Fn(\021)1291 3285 y Fe(0)1315 3308 y Fp(+)p Fn(\021)1407 3285 y Fe(0)1430 3308 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\))1649 3349 y Fs(])1676 3364 y Fn(o)1844 3349 y Fs(and)130 b(2)p Fq(K)2270 3308 y Fi(0)2316 3349 y Fs(+)22 b(1)27 b(=)h([)p Fq(\015)2677 3308 y Fi(\000)p Fn(\021)2769 3285 y Fe(0)2792 3308 y Fi(\000)p Fn(\021)2884 3285 y Fe(0)2907 3308 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\))3125 3349 y Fs(])3152 3364 y Fn(o)3213 3349 y Fm(\001)21 b Fs([)p Fq(\015)3345 3308 y Fi(\000)p Fn(\032)3441 3349 y Fs(])3468 3364 y Fn(o)3506 3349 y Fq(:)-118 3515 y Fs(Notice)32 b(that)g(2)p Fq(N)h Fs(+)22 b(1)27 b(=)h(\(2)p Fq(K)1014 3479 y Fi(0)1059 3515 y Fs(+)22 b(1\)\(2)p Fq(L)1397 3479 y Fi(0)1442 3515 y Fs(+)g(1\).)-118 3631 y(By)31 b(assumption)f(\(H0\),)h Fq(h)g Fs(is)f(real)g(analytic)g(on)g (the)h(compact)g(torus)g Fo(T)2543 3595 y Fn(d)2587 3631 y Fs(;)g(it)f(admits)f(a)i(compact)f(set)i(of)e(minima.)-118 3747 y(As)41 b(the)h(minima)37 b(are)k(quadratic)f(non-degenerate,)k (they)e(are)f(isolated)e(p)s(oin)m(ts.)68 b(Let)41 b Fq(Z)48 b Fs(b)s(e)41 b(the)g(\014nite)g(set)g(of)-118 3863 y(these)e(minima)c(i.e.)60 b Fq(Z)44 b Fs(=)37 b Fq(h)960 3827 y Fi(\000)p Fp(1)1054 3863 y Fs(\()p Fm(f)p Fs(0)p Fm(g)p Fs(\))g(=)f Fm(f)p Fq(\022)1523 3878 y Fp(1)1563 3863 y Fq(;)17 b(:)g(:)g(:)f(;)h(\022)1827 3878 y Fn(M)1906 3863 y Fm(g)p Fs(.)60 b(By)39 b(assumption)e(\(H1\),)i (w)m(e)h(kno)m(w)f(that,)g(there)g(exists)-118 3980 y Fq(C)c(>)27 b Fs(0)32 b(suc)m(h)i(that,)f(for)f Fq(\022)f Fm(2)d Fo(T)1012 3943 y Fn(d)1056 3980 y Fs(,)1471 4139 y Fq(h)p Fs(\()p Fq(\022)s Fs(\))g Fm(\025)g Fq(C)97 b Fs(inf)1877 4201 y Fp(1)p Fi(\024)p Fn(J)6 b Fi(\024)p Fn(M)2159 4139 y Fm(j)p Fq(\022)25 b Fm(\000)d Fq(\022)2401 4154 y Fn(J)2451 4139 y Fm(j)2479 4098 y Fp(2)2518 4139 y Fq(:)-2663 b Fs(\(4.18\))-118 4343 y(Then,)34 b(w)m(e)f(pro)m(v)m(e) -118 4519 y Fy(Lemma)k(4.3.)49 b Ff(Pick)41 b Fs(0)g Fq(<)f(\021)k(<)c(\021)1195 4483 y Fi(0)1258 4519 y Fq(<)h(\021)t Fs(\()p Fq(r)s Fs(\))g Ff(and)g Fq(N)52 b Ff(as)41 b(in)h(Pr)-5 b(op)g(osition)41 b(4.1.)64 b(L)-5 b(et)43 b Fq(L)3166 4483 y Fi(0)3231 4519 y Ff(and)e Fq(K)3517 4483 y Fi(0)3582 4519 y Ff(b)-5 b(e)42 b(de\014ne)-5 b(d)40 b(as)-118 4636 y(ab)-5 b(ove.)44 b(Ther)-5 b(e)34 b(exists)g Fq(C)h(>)28 b Fs(0)34 b Ff(and)h Fq(\015)1265 4651 y Fp(0)1331 4636 y Fq(>)28 b Fs(0)34 b Ff(such)h(that,)g(for)g Fs(0)27 b Fq(<)h(\015)33 b(<)27 b(\015)2543 4651 y Fp(0)2582 4636 y Ff(,)35 b(we)f(have)987 4881 y Fs(\012\()p Fq(\015)5 b Fs(\()p 1189 4826 65 4 v Fq(!)26 b Fm(\000)c Fq(\015)1431 4840 y Fn(\021)1473 4881 y Fs(\))p Fq(;)17 b(\015)5 b(;)17 b(N)10 b Fs(\))28 b Fm(\032)1980 4787 y Fl([)1914 5002 y Fi(j)p Fn(k)1973 4983 y Fe(0)1995 5002 y Fi(j\024)p Fn(K)2135 4983 y Fe(0)2172 4711 y Fl( )2389 4787 y([)2251 4999 y Fp(1)p Fi(\024)p Fn(J)6 b Fi(\024)p Fn(J)2486 4980 y Fe(0)2508 4999 y Fi(\024)p Fn(M)2654 4881 y Fs(\012)2724 4840 y Fn(J)n(;J)2826 4817 y Fe(0)2849 4840 y Fn(;k)2908 4817 y Fe(0)2933 4711 y Fl(!)-118 4881 y Fs(\(4.19\))-118 5155 y Ff(wher)-5 b(e,)34 b(for)h Fs(1)27 b Fm(\024)h Fq(J)37 b Fm(\024)28 b Fq(J)783 5119 y Fi(0)834 5155 y Fm(\024)h Fq(M)45 b Ff(and)34 b Fm(j)p Fq(k)1350 5119 y Fi(0)1374 5155 y Fm(j)27 b(\024)h Fq(K)1624 5119 y Fi(0)1647 5155 y Ff(,)35 b(we)g(de\014ne)f(the)h(events)489 5436 y Fs(\012)559 5395 y Fn(J)n(;J)661 5372 y Fe(0)683 5395 y Fn(;k)742 5372 y Fe(0)795 5436 y Fs(=)899 5232 y Fl(8)899 5322 y(<)899 5501 y(:)988 5436 y Fq(!)t Fs(;)1328 5369 y(1)p 1140 5413 424 4 v 1140 5505 a(\(2)p Fq(L)1293 5476 y Fi(0)1339 5505 y Fs(+)22 b(1\))1524 5476 y Fn(d)1591 5232 y Fl(\014)1591 5292 y(\014)1591 5352 y(\014)1591 5411 y(\014)1591 5471 y(\014)1591 5531 y(\014)1656 5342 y(X)1624 5558 y Fi(j)p Fn(l)1666 5539 y Fe(0)1688 5558 y Fi(j\024)p Fn(L)1811 5539 y Fe(0)1833 5436 y Fs(\()p Fq(!)1932 5452 y Fn(k)1971 5433 y Fe(0)1992 5452 y Fp(\(2)p Fn(L)2102 5433 y Fe(0)2125 5452 y Fp(+1\)+)p Fn(l)2319 5433 y Fe(0)2368 5436 y Fm(\000)p 2468 5382 65 4 v 23 w Fq(!)s Fs(\))p Fq(e)2615 5395 y Fn(i)p Fp(\()p Fn(\022)2699 5406 y Fj(J)2742 5395 y Fi(\000)p Fn(\022)2830 5414 y Fj(J)2869 5400 y Fe(0)2895 5395 y Fp(\))p Fn(l)2944 5372 y Fe(0)2971 5232 y Fl(\014)2971 5292 y(\014)2971 5352 y(\014)2971 5411 y(\014)2971 5471 y(\014)2971 5531 y(\014)3032 5436 y Fm(\025)3342 5369 y Fq(\015)3398 5333 y Fn(\021)p 3147 5413 487 4 v 3147 5505 a Fs(16\()p Fq(M)33 b Fs(+)22 b(1\))3595 5476 y Fp(2)3644 5232 y Fl(9)3644 5322 y(=)3644 5501 y(;)3749 5436 y Fq(:)1969 5690 y Fg(14)p eop %%Page: 15 15 15 14 bop -118 241 a Fs(Before)29 b(pro)m(ving)f(this)h(lemma,)e(let)h (us)h(use)h(it)e(to)g(complete)g(the)h(pro)s(of)f(of)g(Prop)s(osition)f (4.1.)42 b(Therefore,)31 b(w)m(e)f(need)-118 357 y(to)i(estimate)g(the) h(probabilit)m(y)d(of)j(the)g(ev)m(en)m(ts)h(\012)1703 321 y Fn(J)n(;J)1805 297 y Fe(0)1828 321 y Fn(;k)1887 297 y Fe(0)1912 357 y Fs(.)44 b(This)32 b(is)h(done)g(in)-118 539 y Fy(Prop)s(osition)i(4.2.)50 b Ff(Ther)-5 b(e)35 b(exists)h Fq(")30 b(>)h Fs(0)36 b Ff(such)g(that,)h(for)f Fq(\015)42 b Ff(su\016ciently)36 b(smal)5 b(l,)36 b(for)g Fs(1)30 b Fm(\024)h Fq(J)n(;)17 b(J)3519 503 y Fi(0)3573 539 y Fm(\024)31 b Fq(M)47 b Ff(and)36 b(for)-118 655 y Fm(j)p Fq(k)-36 619 y Fi(0)-13 655 y Fm(j)27 b(\024)h Fq(K)237 619 y Fi(0)261 655 y Ff(,)35 b(we)f(have)1607 839 y Fo(P)p Fs(\(\012)1774 798 y Fn(J)n(;J)1876 774 y Fe(0)1900 798 y Fn(;k)1959 774 y Fe(0)1985 839 y Fs(\))27 b Fm(\024)i Fq(e)2201 798 y Fi(\000)p Fn(\015)2296 774 y Fe(\000)p Fj(")2381 839 y Fq(:)-2526 b Fs(\(4.20\))-118 1031 y(As)35 b Fq(K)118 995 y Fi(0)172 1031 y Fm(\024)30 b Fq(\015)335 995 y Fi(\000)p Fn(\032)p Fi(\000)p Fp(1)555 1031 y Fs(and)k(1)d Fm(\024)f Fq(J)n(;)17 b(J)1092 995 y Fi(0)1146 1031 y Fm(\024)31 b Fq(M)10 b Fs(,)36 b(using)e(Lemma)e (4.3,)j(summing)d(the)j(estimates)f(\(4.20\))f(o)m(v)m(er)i(\()p Fq(J)n(;)17 b(J)3924 995 y Fi(0)3948 1031 y Fq(;)g(k)4046 995 y Fi(0)4069 1031 y Fs(\),)-118 1147 y(w)m(e)34 b(obtain)d(\(4.17\)) o(.)-118 1311 y Fy(Pro)s(of)j(of)g(Prop)s(osition)e(4.2.)43 b Fs(Prop)s(osition)28 b(4.2)h(is)g(a)h(large)e(deviation)h(estimate.) 41 b(The)31 b(only)e(di\016cult)m(y)h(comes)-118 1427 y(from)36 b(the)i(fact)f(that,)i(in)e(the)h(presen)m(t)h(case,)h(the)e (random)e(v)-5 b(ariables)36 b(\()p Fq(!)2651 1442 y Fn(n)2698 1427 y Fs(\))2736 1446 y Fn(n)p Fi(2)p Fk(Z)2876 1427 y Fj(d)k Fs(dep)s(end)f(on)e Fq(\015)5 b Fs(,)39 b(hence,)h(on)d(the)-118 1543 y(size)28 b(of)g(the)g(sample)g(of)f (random)g(v)-5 b(ariables)27 b(w)m(e)i(are)f(considering.)42 b(Nev)m(ertheless,)31 b(the)e(standard)f(ideas)g(to)g(obtain)-118 1659 y(large)j(deviation)h(estimates)g(do)g(apply)h(\(see,)g(e.g.)g ([7]\).)-118 1780 y(Recall)d(that)i Fq(!)447 1795 y Fp(0)514 1780 y Fs(=)27 b Fq(!)678 1795 y Fp(0)717 1780 y Fs(\()p Fq(\025)p Fs(\))32 b(and)g Fq(\025)c Fs(=)f Fq(\015)1315 1744 y Fi(\000)p Fn(\021)1407 1720 y Fe(0)1430 1744 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\))1649 1780 y Fs(.)43 b(Let)32 b(us)g(\014rst)h(estimate)e(the)h(v)-5 b(ariance)31 b(of)h(\()p Fq(!)3366 1795 y Fn(n)3412 1780 y Fs(\))3450 1799 y Fn(n)p Fi(2)p Fk(Z)3591 1780 y Fj(d)r Fs(.)43 b(One)32 b(needs)-118 1896 y(to)g(consider)h(t)m(w)m(o)g(cases.)45 b(If)33 b(the)g(exp)s(onen)m(t)h Fq(r)h Fs(in)d(assumption)g(\(H2\))g (satis\014es)h Fq(r)e Fm(\025)d Fs(2,)k(one)h(has)1368 2078 y(sup)1376 2160 y Fn(\025>)p Fp(1)1532 2078 y Fo(E)12 b Fs(\(\()p Fq(!)1729 2093 y Fp(0)1774 2078 y Fs(\()p Fq(\025)p Fs(\))22 b Fm(\000)p 2029 1991 198 4 v 23 w Fq(!)t Fs(\()p Fq(\025)p Fs(\))o(\))2264 2037 y Fp(2)2303 2078 y Fs(\))28 b Fq(<)f Fs(+)p Fm(1)-2766 b Fs(\(4.21\))-118 2306 y(If)32 b(1)c Fq(<)f(r)k(<)c Fs(2,)33 b(w)m(e)g(compute)348 2476 y Fo(E)13 b Fs(\()p Fq(!)508 2491 y Fp(0)553 2476 y Fs(\()p Fq(\025)p Fs(\))686 2435 y Fp(2)725 2476 y Fs(\))28 b(=)f Fo(E)13 b Fs(\()p Fq(!)1054 2491 y Fp(0)1099 2476 y Fy(1)1155 2491 y Fi(f)p Fn(!)1234 2500 y Fc(0)1269 2491 y Fi(\024)p Fn(\025)p Fi(g)1404 2476 y Fs(\))22 b(+)g Fq(\025)1619 2435 y Fp(2)1658 2476 y Fo(P)p Fs(\()p Fq(!)1816 2491 y Fp(0)1885 2476 y Fq(>)28 b(\025)p Fs(\))f Fm(\024)h Fq(\025)2273 2435 y Fp(2)p Fi(\000)p Fn(r)2401 2476 y Fo(E)13 b Fs(\()p Fq(!)2565 2435 y Fn(r)2561 2501 y Fp(0)2608 2476 y Fs(\))22 b(+)g Fq(\025)2823 2435 y Fp(2)2862 2476 y Fo(E)13 b Fs(\()p Fq(!)3026 2435 y Fn(r)3022 2501 y Fp(0)3069 2476 y Fs(\))p Fq(=\025)3213 2435 y Fi(\000)p Fn(r)3333 2476 y Fm(\024)28 b Fs(2)p Fq(\025)3544 2435 y Fp(2)p Fi(\000)p Fn(r)3672 2476 y Fo(E)13 b Fs(\()p Fq(!)3836 2435 y Fn(r)3832 2501 y Fp(0)3879 2476 y Fs(\))-4035 b(\(4.22\))-118 2665 y(Let)36 b(us)h(no)m(w)g(estimate)f Fo(P)p Fs(\(\012)958 2628 y Fn(J)n(;J)1060 2605 y Fe(0)1084 2628 y Fn(;k)1143 2605 y Fe(0)1168 2665 y Fs(\).)55 b(W)-8 b(e)36 b(only)g(do)g(the)h(explicit)e(computations)g(for)h Fq(J)43 b Fm(6)p Fs(=)34 b Fq(J)3370 2628 y Fi(0)3393 2665 y Fs(;)39 b(the)d(case)h Fq(J)44 b Fs(=)33 b Fq(J)4110 2628 y Fi(0)-118 2781 y Fs(is)f(treated)g(in)f(the)i(same)f(w)m(a)m(y) -8 b(.)44 b(As)33 b(the)g(random)e(v)-5 b(ariables)31 b(\()p Fq(!)2239 2796 y Fn(n)2285 2781 y Fs(\))2323 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v 2886 4439 a Fs(32\()p Fq(M)32 b Fs(+)22 b(1\))3333 4411 y Fp(2)-118 4371 y Fs(\(4.23\))-118 4650 y(and)33 b(the)g(probabilit)m(y)d(that)792 4797 y(1)p 605 4841 424 4 v 605 4933 a(\(2)p Fq(L)758 4904 y Fi(0)803 4933 y Fs(+)22 b(1\))988 4904 y Fn(d)1088 4770 y Fl(X)1055 4985 y Fi(j)p Fn(l)1097 4967 y Fe(0)1119 4985 y Fi(j\024)p Fn(L)1242 4967 y Fe(0)1264 4864 y Fs(\()p Fq(!)1363 4880 y Fn(k)1402 4861 y Fe(0)1423 4880 y Fp(\(2)p Fn(K)1550 4861 y Fe(0)1573 4880 y Fp(+1\)+)p Fn(l)1767 4861 y Fe(0)1816 4864 y Fm(\000)p 1915 4809 65 4 v 22 w Fq(!)t Fs(\))17 b(cos\(\()p Fq(\022)2286 4879 y Fn(J)2357 4864 y Fm(\000)23 b Fq(\022)2502 4879 y Fn(J)2547 4860 y Fe(0)2574 4864 y Fs(\))p Fq(l)2643 4823 y Fi(0)2666 4864 y Fs(\))28 b Fm(\024)g(\000)3119 4797 y Fq(\015)3175 4761 y Fn(\021)p 2924 4841 487 4 v 2924 4933 a Fs(32\()p Fq(M)33 b Fs(+)22 b(1\))3372 4904 y Fp(2)-118 5149 y Fs(Both)32 b(estimates)h(b)s(eing)e(done)i(in)f(the)h(same)g(w)m(a)m(y) -8 b(,)33 b(w)m(e)h(only)e(do)h(the)g(computations)e(for)h(\(4.23\).) -118 5265 y(T)-8 b(o)36 b(simplify)e(the)i(notations,)h(let)e Fq(R)g Fs(=)f(\(2)p Fq(L)1550 5229 y Fi(0)1598 5265 y Fs(+)24 b(1\))1785 5229 y Fn(d)1861 5265 y Fs(and,)38 b(reindex)f(the)f(p)s(oin)m(ts)g Fm(f)p Fq(l)2980 5229 y Fi(0)3003 5265 y Fs(;)53 b Fm(j)p Fq(l)3142 5229 y Fi(0)3165 5265 y Fm(j)34 b(\024)g Fq(L)3404 5229 y Fi(0)3428 5265 y Fm(g)i Fs(as)g Fm(f)p Fq(l)3718 5229 y Fi(0)3741 5265 y Fs(\()p Fq(U)10 b Fs(\);)54 b(1)33 b Fm(\024)-118 5382 y Fq(U)38 b Fm(\024)28 b Fq(R)q Fm(g)p Fs(.)44 b(De\014ne)1025 5552 y(~)-56 b Fq(!)1079 5567 y Fn(U)1165 5552 y Fs(=)28 b(\()p Fq(!)1368 5567 y Fn(k)1407 5548 y Fe(0)1428 5567 y Fp(\(2)p Fn(K)1555 5548 y Fe(0)1577 5567 y Fp(+1\)+)p Fn(l)1771 5548 y Fe(0)1794 5567 y Fp(\()p Fn(U)7 b Fp(\))1930 5552 y Fm(\000)p 2030 5497 65 4 v 23 w Fq(!)s Fs(\))17 b(cos\(\()p Fq(\022)2400 5567 y Fn(J)2472 5552 y Fm(\000)22 b Fq(\022)2616 5567 y Fn(J)2661 5548 y Fe(0)2688 5552 y Fs(\))p Fq(l)2757 5511 y Fi(0)2781 5552 y Fs(\()p Fq(U)10 b Fs(\)\))p Fq(:)1969 5690 y Fg(15)p eop %%Page: 16 16 16 15 bop -118 241 a Fs(The)33 b(random)f(v)-5 b(ariables)31 b(\()8 b(~)-57 b Fq(!)943 256 y Fn(U)1002 241 y Fs(\))1040 256 y Fp(1)p Fi(\024)p Fn(U)7 b Fi(\024)p Fn(R)1330 241 y Fs(are)33 b(indep)s(enden)m(t,)h(cen)m(tered)g(\(i.e.)43 b Fo(E)12 b Fs(\()c(~)-57 b Fq(!)2825 256 y Fn(U)2890 241 y Fs(\))28 b(=)f(0\))32 b(and)h(b)s(ounded)g(b)m(y)h Fq(\025)p Fs(.)-118 357 y(Estimating)c(the)j(probabilit)m(y)e(of)39 b(\(4.23\))32 b(is)g(estimating)e(the)j(probabilit)m(y)e(of)1640 548 y(1)p 1627 592 76 4 v 1627 684 a Fq(R)1774 491 y Fn(R)1729 521 y Fl(X)1729 732 y Fn(U)7 b Fp(=1)1898 615 y Fs(~)-57 b Fq(!)1951 630 y Fn(U)2038 615 y Fm(\025)28 b Fq(\015)2199 574 y Fn(\021)2241 615 y Fq(=C)2360 630 y Fp(0)-118 875 y Fs(where)34 b Fq(C)234 890 y Fp(0)301 875 y Fs(=)27 b(32\()p Fq(M)33 b Fs(+)22 b(1\))852 838 y Fp(2)891 875 y Fs(.)-118 991 y(F)-8 b(or)32 b Fq(t)c(>)f Fs(0,)32 b(as)h(the)g(random)f(v)-5 b(ariables)31 b(\()8 b(~)-57 b Fq(!)1480 1006 y Fn(U)1539 991 y Fs(\))1577 1006 y Fp(1)p Fi(\024)p Fn(U)7 b Fi(\024)p Fn(R)1867 991 y Fs(are)32 b(indep)s(enden)m(t,)i(w)m(e)g(compute)12 1372 y Fo(P)88 1202 y Fl( )191 1305 y Fs(1)p 178 1350 V 178 1441 a Fq(R)326 1248 y Fn(R)280 1278 y Fl(X)280 1489 y Fn(U)7 b Fp(=1)449 1372 y Fs(~)-57 b Fq(!)502 1387 y Fn(U)589 1372 y Fm(\025)28 b Fq(\015)750 1331 y Fn(\021)792 1372 y Fq(=C)911 1387 y Fp(0)950 1202 y Fl(!)1057 1372 y Fm(\024)g Fo(E)1239 1202 y Fl( )1324 1372 y Fs(exp)1489 1202 y Fl(")1547 1372 y Fq(t)1599 1202 y Fl( )1678 1372 y Fm(\000)p Fq(\015)1811 1331 y Fn(\021)1876 1372 y Fs(+)1984 1305 y Fq(C)2054 1320 y Fp(0)p 1984 1350 110 4 v 2001 1441 a Fq(R)2165 1248 y Fn(R)2120 1278 y Fl(X)2119 1489 y Fn(U)7 b Fp(=1)2289 1372 y Fs(~)-57 b Fq(!)2342 1387 y Fn(U)2401 1202 y Fl(!#!)2644 1372 y Fs(=)28 b Fq(e)2793 1331 y Fi(\000)p Fn(t\015)2913 1308 y Fj(\021)3018 1248 y Fn(R)2981 1278 y Fl(Y)2972 1489 y Fn(U)7 b Fp(=1)3134 1372 y Fo(E)3211 1232 y Fl(\022)3290 1372 y Fs(exp)3456 1232 y Fl(\022)3539 1305 y Fq(tC)3644 1320 y Fp(0)p 3539 1350 145 4 v 3574 1441 a Fq(R)3701 1372 y Fs(~)-56 b Fq(!)3755 1387 y Fn(U)3813 1232 y Fl(\023)q(\023)3977 1372 y Fq(:)-118 1145 y Fs(\(4.24\))-118 1622 y(No)m(w,)33 b(pic)m(k)g Fq(t)g Fs(suc)m(h)h(that)e Fq(C)906 1637 y Fp(0)945 1622 y Fq(t\025)c Fm(\024)g Fq(R)q Fs(;)33 b(then,)g(for)f(1)c Fm(\024)g Fq(U)38 b Fm(\024)29 b Fq(R)q Fs(,)j(one)h(has)617 1869 y Fo(E)695 1729 y Fl(\022)774 1869 y Fs(exp)939 1729 y Fl(\022)1022 1802 y Fq(tC)1127 1817 y Fp(0)p 1022 1846 V 1057 1938 a Fq(R)1185 1869 y Fs(~)-57 b Fq(!)1238 1884 y Fn(U)1297 1729 y Fl(\023\023)1471 1869 y Fm(\024)28 b Fo(E)1654 1699 y Fl( )1738 1869 y Fs(1)22 b(+)1917 1802 y Fq(tC)2022 1817 y Fp(0)p 1917 1846 V 1952 1938 a Fq(R)2079 1869 y Fs(~)-57 b Fq(!)2132 1884 y Fn(U)2213 1869 y Fs(+)2311 1729 y Fl(\022)2394 1802 y Fq(tC)2499 1817 y Fp(0)p 2394 1846 V 2429 1938 a Fq(R)2557 1869 y Fs(~)g Fq(!)2610 1884 y Fn(U)2669 1729 y Fl(\023)2742 1751 y Fp(2)2781 1699 y Fl(!)1471 2205 y Fm(\024)28 b Fs(1)22 b(+)1745 2065 y Fl(\022)1828 2138 y Fq(tC)1933 2153 y Fp(0)p 1828 2183 V 1863 2274 a Fq(R)1983 2065 y Fl(\023)2056 2087 y Fp(2)2112 2205 y Fq(C)7 b Fs(\()p Fq(\025)p Fs(\))28 b Fm(\024)g Fs(exp)2620 2035 y Fl( )2699 2065 y(\022)2782 2138 y Fq(tC)2887 2153 y Fp(0)p 2782 2183 V 2817 2274 a Fq(R)2937 2065 y Fl(\023)3010 2087 y Fp(2)3066 2205 y Fq(C)7 b Fs(\()p Fq(\025)p Fs(\))3276 2035 y Fl(!)3371 2205 y Fq(;)-118 2037 y Fs(\(4.25\))-118 2458 y(where)34 b Fq(C)7 b Fs(\()p Fq(\025)p Fs(\))32 b(satis\014es)1499 2621 y Fq(C)7 b Fs(\()p Fq(\025)p Fs(\))28 b Fm(\025)81 b Fs(sup)1842 2702 y Fp(1)p Fi(\024)p Fn(U)7 b Fi(\024)p Fn(R)2112 2621 y Fo(E)12 b Fs(\(\()c(~)-57 b Fq(!)2309 2636 y Fn(U)2374 2621 y Fs(\))2412 2580 y Fp(2)2451 2621 y Fs(\))p Fq(:)-2634 b Fs(\(4.26\))-118 2852 y(Plugging)32 b(\(4.25\))g(in)m(to)g(\(4.24\))o(,)h(w)m(e)g (obtain)820 3110 y Fo(P)896 2939 y Fl( )1000 3042 y Fs(1)p 986 3087 76 4 v 986 3178 a Fq(R)1134 2985 y Fn(R)1089 3015 y Fl(X)1088 3227 y Fn(U)7 b Fp(=1)1257 3110 y Fs(~)-56 b Fq(!)1311 3125 y Fn(U)1397 3110 y Fm(\025)28 b Fq(\015)1558 3069 y Fn(\021)1600 3110 y Fq(=C)1719 3125 y Fp(0)1758 2939 y Fl(!)1865 3110 y Fm(\024)g Fs(exp)2135 2969 y Fl(\022)2209 3110 y Fm(\000)p Fq(t)2338 2969 y Fl(\022)2411 3110 y Fq(\015)2467 3069 y Fn(\021)2532 3110 y Fm(\000)2641 3042 y Fq(tC)2746 3057 y Fp(0)p 2641 3087 145 4 v 2676 3178 a Fq(R)2796 3110 y(C)7 b Fs(\()p Fq(\025)p Fs(\))3006 2969 y Fl(\023\023)3169 3110 y Fq(:)-118 3359 y Fs(Let)33 b(us)g(no)m(w)g(distinguish)e(the)i(t)m(w)m(o)h(cases:)-18 3498 y(1.)41 b(If)32 b(1)c Fq(<)f(r)k(<)c Fs(2,)33 b(one)g(can)f(tak)m (e)1693 3660 y Fq(C)7 b Fs(\()p Fq(\025)p Fs(\))27 b(=)h Fq(C)7 b(\025)2168 3619 y Fp(2)p Fi(\000)p Fn(r)2296 3660 y Fq(:)99 3821 y Fs(Recalling)30 b(that)1160 3987 y Fq(R)18 b Fm(\030)f Fq(\015)1402 3946 y Fi(\000)p Fn(d=)p Fp(2+)p Fn(d\021)1691 3922 y Fe(0)1715 3946 y Fp(\(2+1)p Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\)\))2114 3987 y Fq(;)114 b(\025)17 b Fm(\030)g Fq(\015)2479 3946 y Fi(\000)p Fn(\021)2571 3922 y Fe(0)2594 3946 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\))2812 3987 y Fq(;)99 4167 y Fs(and)33 b(taking)e(2)p Fq(C)708 4182 y Fp(0)747 4167 y Fq(t\025)d Fs(=)g Fq(R)33 b Fs(i.e.)43 b Fq(t)17 b Fm(\030)g Fq(\015)1448 4130 y Fi(\000)p Fn(d=)p Fp(2+2)p Fn(d\021)1772 4107 y Fe(0)1797 4130 y Fp(+\()p Fn(d)p Fp(+1\))p Fn(\021)2069 4107 y Fe(0)2093 4130 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\))2311 4167 y Fs(,)33 b(for)f Fq(\015)38 b Fs(su\016cien)m(tly)33 b(small,)d(w)m(e)k(obtain)571 4384 y Fq(t)623 4243 y Fl(\022)697 4384 y Fq(\015)753 4343 y Fn(\021)817 4384 y Fm(\000)926 4316 y Fq(tC)1031 4331 y Fp(0)p 926 4361 V 961 4452 a Fq(R)1081 4384 y(C)7 b Fs(\()p Fq(\025)p Fs(\))1291 4243 y Fl(\023)1391 4384 y Fm(\025)29 b Fq(t\015)1588 4343 y Fn(\021)1646 4273 y Fl(\020)1706 4384 y Fs(1)22 b Fm(\000)g Fq(\015)1932 4343 y Fi(\000)p Fn(\021)r Fp(+)p Fn(\021)2116 4319 y Fe(0)2144 4273 y Fl(\021)2232 4384 y Fm(\025)2347 4316 y Fs(1)p 2347 4361 49 4 v 2347 4452 a(2)2405 4384 y Fq(\015)2461 4343 y Fi(\000)p Fn(d=)p Fp(2+2)p Fn(d\021)2785 4319 y Fe(0)2810 4343 y Fp(+\()p Fn(d)p Fp(+1\))p Fn(\021)3082 4319 y Fe(0)3106 4343 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\)+)p Fn(\021)3417 4384 y Fq(:)99 4607 y Fs(By)32 b(de\014nition,)e Fm(\000)p Fq(d=)p Fs(2)20 b(+)f(2)p Fq(d\021)t Fs(\()p Fq(r)s Fs(\))g(+)g(\()p Fq(d)g Fs(+)h(1\))p Fq(\021)t Fs(\()p Fq(r)s Fs(\))p Fq(=)p Fs(\()p Fq(r)h Fm(\000)f Fs(1\))f(+)h Fq(\021)t Fs(\()p Fq(r)s Fs(\))27 b(=)g(0;)32 b(hence,)h(as)f(0)27 b Fq(<)h(\021)j(<)c(\021)3600 4571 y Fi(0)3651 4607 y Fq(<)h(\021)t Fs(\()p Fq(r)s Fs(\),)j(one)99 4723 y(has)1142 4885 y Fm(\000)p Fq(d=)p Fs(2)22 b(+)g(2)p Fq(d\021)1640 4843 y Fi(0)1684 4885 y Fs(+)g(\()p Fq(d)g Fs(+)g(1\))p Fq(\021)2130 4843 y Fi(0)2153 4885 y Fq(=)p Fs(\()p Fq(r)i Fm(\000)f Fs(1\))f(+)g Fq(\021)31 b(<)d Fs(0)p Fq(:)99 5046 y Fs(This)33 b(completes)f(the)h(pro)s(of)f(of)g(Prop)s(osition)f (4.2)h(in)g(the)h(case)g(1)28 b Fq(<)f(r)j(<)e Fs(2.)-18 5163 y(2.)41 b(If)34 b Fq(r)f Fm(\025)d Fs(2,)35 b(the)f(v)-5 b(ariance)34 b(of)41 b(~)-57 b Fq(!)1219 5178 y Fn(U)1312 5163 y Fs(is)33 b(b)s(ounded)i(b)m(y)g Fq(C)41 b Fs(uniformly)32 b(in)h(1)d Fm(\024)h Fq(U)40 b Fm(\024)31 b Fq(R)q Fs(.)48 b(T)-8 b(aking)34 b(2)p Fq(C)7 b(C)3694 5178 y Fp(0)3733 5163 y Fq(t\025)30 b Fs(=)g Fq(R)q(\015)4092 5126 y Fn(\021)99 5279 y Fs(i.e.)43 b Fq(t)17 b Fm(\030)g Fq(\015)469 5243 y Fi(\000)p Fn(d=)p Fp(2+2)p Fn(d\021)793 5219 y Fe(0)817 5243 y Fp(+\()p Fn(d)p Fp(+1\))p Fn(\021)1089 5219 y Fe(0)1114 5243 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\)+)p Fn(\021)1425 5279 y Fs(,)32 b(for)g Fq(\015)38 b Fs(su\016cien)m(tly)33 b(small,)d(w)m(e)k(obtain)775 5496 y Fq(t)827 5356 y Fl(\022)900 5496 y Fq(\015)956 5455 y Fn(\021)1020 5496 y Fm(\000)1130 5429 y Fq(tC)1235 5444 y Fp(0)p 1130 5473 145 4 v 1165 5565 a Fq(R)1284 5496 y(C)1361 5356 y Fl(\023)1462 5496 y Fm(\025)28 b Fq(t\015)1658 5455 y Fn(\021)1717 5496 y Fs(\(1)22 b Fm(\000)h Fs(1)p Fq(=)p Fs(2\))k Fm(\025)2253 5429 y Fs(1)p 2253 5473 49 4 v 2253 5565 a(2)2312 5496 y Fq(\015)2368 5455 y Fi(\000)p Fn(d=)p Fp(2+2)p Fn(d\021)2692 5432 y Fe(0)2716 5455 y Fp(+)p Fn(d\021)2844 5432 y Fe(0)2867 5455 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\)+2)p Fn(\021)3214 5496 y Fq(:)1969 5690 y Fg(16)p eop %%Page: 17 17 17 16 bop 99 241 a Fs(By)33 b(de\014nition,)e Fm(\000)p Fq(d=)p Fs(2)21 b(+)g(2)p Fq(d\021)t Fs(\()p Fq(r)s Fs(\))f(+)h Fq(d\021)t Fs(\()p Fq(r)s Fs(\))p Fq(=)p Fs(\()p Fq(r)i Fm(\000)f Fs(1\))f(+)g(2)p Fq(\021)t Fs(\()p Fq(r)s Fs(\))27 b(=)g(0;)32 b(hence,)i(as)e(0)c Fq(<)f(\021)32 b(<)27 b(\021)3426 205 y Fi(0)3477 241 y Fq(<)g(\021)t Fs(\()p Fq(r)s Fs(\),)32 b(one)g(has)1240 416 y Fm(\000)p Fq(d=)p Fs(2)22 b(+)g(2)p Fq(d\021)1738 375 y Fi(0)1782 416 y Fs(+)g Fq(d\021)1983 375 y Fi(0)2006 416 y Fq(=)p Fs(\()p Fq(r)j Fm(\000)d Fs(1\))g(+)g(2)p Fq(\021)31 b(<)d Fs(0)p Fq(:)99 591 y Fs(This)33 b(completes)f(the)h(pro)s(of)f(of)g(Prop)s (osition)f(4.2)h(in)g(the)h(case)g(2)28 b Fm(\024)g Fq(r)s Fs(.)p 4063 737 4 66 v 4067 674 59 4 v 4067 737 V 4125 737 4 66 v -118 947 a(4.5.)56 b Fy(The)35 b(pro)s(of)g(of)g(Lemma)g (4.3.)49 b Fs(T)-8 b(o)30 b(pro)m(v)m(e)i(this)e(Lemma)f(4.3,)h(w)m(e)i (follo)m(w)c(the)j(argumen)m(t)f(sk)m(etc)m(hed)j(at)d(the)-118 1063 y(end)j(of)f(the)h(in)m(tro)s(duction.)-118 1180 y(W)-8 b(e)33 b(de\014ne)519 1355 y(2)p Fq(L)23 b Fs(+)f(1)27 b(=)h([)p Fq(\015)1018 1313 y Fi(\000)p Fp(1)p Fn(=)p Fp(2+2)p Fn(\021)1305 1290 y Fe(0)1329 1313 y Fp(+)p Fn(\021)1421 1290 y Fe(0)1444 1313 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\))1662 1355 y Fs(])1689 1370 y Fn(o)1750 1355 y Fm(\001)22 b Fs([)p Fq(\015)1883 1313 y Fi(\000)p Fn(\021)1975 1290 y Fe(0)1998 1313 y Fi(\000)p Fn(\021)2090 1290 y Fe(0)2113 1313 y Fn(=)p Fp(\()p Fn(r)r Fi(\000)p Fp(1\))2331 1355 y Fs(])2358 1370 y Fn(o)2526 1355 y Fs(and)131 b(2)p Fq(K)29 b Fs(+)22 b(1)27 b(=)h([)p Fq(\015)3336 1313 y Fi(\000)p Fn(\032)3431 1355 y Fs(])3458 1370 y Fn(o)-118 1535 y Fs(Notice)k(that)g(2)p Fq(N)h Fs(+)22 b(1)27 b(=)h(\(2)p Fq(K)h Fs(+)22 b(1\)\(2)p Fq(L)g Fs(+)g(1\).)-118 1673 y(Pic)m(k)33 b Fq(!)e Fm(2)d Fs(\012)356 1688 y Fn(\015)401 1673 y Fs(\()p Fq(\022)s Fs(\))33 b(so)f(there)i(exists)f Fq(a)28 b Fs(=)1446 1578 y Fl(X)1378 1802 y Fn(k)r Fi(2)p Fk(Z)1514 1779 y Fj(d)1514 1823 y Fc(2)p Fj(n)p Fc(+1)1675 1673 y Fq(a)1726 1688 y Fn(k)1769 1673 y Fq(u)1825 1688 y Fn(k)1867 1673 y Fs(\()p Fq(\022)s Fs(\))k(suc)m(h)i(that)8 1984 y Fm(\017)41 b(k)p Fq(a)p Fm(k)250 2007 y Fn(`)279 1988 y Fc(2)313 2007 y Fp(\()p Fk(Z)391 1984 y Fj(d)391 2030 y Fc(2)p Fj(N)5 b Fc(+1)550 2007 y Fp(\))609 1984 y Fs(=)713 1886 y Fl(q)p 812 1886 610 4 v 812 1909 a(P)918 2013 y Fn(k)r Fi(2)p Fk(Z)1053 1990 y Fj(d)1053 2036 y Fc(2)p Fj(N)g Fc(+1)1233 1984 y Fm(j)p Fq(a)1312 1999 y Fn(k)1355 1984 y Fm(j)1383 1955 y Fp(2)1450 1984 y Fs(=)27 b(1,)8 2151 y Fm(\017)41 b(h)p Fq(M)242 2115 y Fn(N)232 2176 y(!)310 2151 y Fs(\()p Fq(\022)s Fs(\))p Fq(a;)17 b(a)p Fm(i)619 2174 y Fn(`)648 2155 y Fc(2)682 2174 y Fp(\()p Fk(Z)760 2151 y Fj(d)760 2197 y Fc(2)p Fj(N)5 b Fc(+1)919 2174 y Fp(\))978 2151 y Fm(\024)28 b Fq(\015)5 b Fs(\()p 1177 2096 65 4 v Fq(!)26 b Fm(\000)c Fq(\015)1419 2115 y Fn(\021)1461 2151 y Fs(\),)8 2287 y Fm(\017)41 b Fq(u)155 2302 y Fn(k)197 2287 y Fs(\()p Fq(\022)s Fs(\))33 b(is)f(de\014ned)i(in)e(section)g(4.2.)-118 2433 y(Using)h(\(4.13\))f(and)h(the)g(p)s(ositivit)m(y)e(of)h Fq(H)1431 2397 y Fn(N)1498 2433 y Fs(\()p Fq(\022)s Fs(\))g(and)h Fq(V)1923 2397 y Fn(N)1901 2457 y(!)1990 2433 y Fs(,)g(w)m(e)g(obtain)f (that)567 2611 y Fm(h)p Fq(H)695 2570 y Fn(N)762 2611 y Fs(\()p Fq(\022)s Fs(\))p Fq(a;)17 b(a)p Fm(i)1071 2634 y Fn(`)1100 2615 y Fc(2)1134 2634 y Fp(\()p Fk(Z)1212 2611 y Fj(d)1212 2657 y Fc(2)p Fj(N)5 b Fc(+1)1371 2634 y Fp(\))1430 2611 y Fm(\024)28 b Fq(\015)5 b Fs(\()p 1629 2556 V Fq(!)26 b Fm(\000)c Fq(\015)1871 2570 y Fn(\021)1913 2611 y Fs(\))98 b(and)f Fm(h)p Fq(V)2421 2570 y Fn(N)2399 2636 y(!)2488 2611 y Fq(a;)17 b(a)p Fm(i)2673 2634 y Fn(`)2702 2615 y Fc(2)2736 2634 y Fp(\()p Fk(Z)2814 2611 y Fj(d)2814 2657 y Fc(2)p Fj(N)5 b Fc(+1)2973 2634 y Fp(\))3032 2611 y Fm(\024)p 3138 2556 V 29 w Fq(!)25 b Fm(\000)e Fq(\015)3380 2570 y Fn(\021)3422 2611 y Fq(:)-3567 b Fs(\(4.27\))-118 2806 y(As)33 b Fq(a)g Fs(is)f(normalized,)f(the)i (second)g(inequalit)m(y)f(in)g(\(4.27\))g(yields)1288 2985 y Fm(h)p Fs(\()p Fq(V)1443 2943 y Fn(N)1422 3009 y(!)1532 2985 y Fm(\000)p 1632 2930 V 23 w Fq(!)s(I)8 b(d)1798 2943 y Fn(N)1865 2985 y Fs(\))p Fq(a;)17 b(a)p Fm(i)2088 3008 y Fn(`)2117 2989 y Fc(2)2151 3008 y Fp(\()p Fk(Z)2229 2985 y Fj(d)2229 3031 y Fc(2)p Fj(N)5 b Fc(+1)2388 3008 y Fp(\))2447 2985 y Fm(\024)29 b(\000)p Fq(\015)2686 2943 y Fn(\021)-118 2985 y Fs(\(4.28\))-118 3186 y(where)34 b Fq(I)8 b(d)266 3150 y Fn(N)365 3186 y Fs(is)32 b(the)h(iden)m(tit)m (y)f(matrix.)-118 3303 y(W)-8 b(e)46 b(no)m(w)h(use)g(the)g(\014rst)g 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b(\()p Fq(\033)t Fs(\()p Fq(H)1332 4178 y Fn(!)r(;\015)t Fi(j)p Fp(\003)1511 4162 y Fs(\)\).)61 b(Pic)m(k)39 b Fq(a)f Fm(2)g Fo(R)2158 4126 y Fn(d)2205 4162 y Fs(,)i Fm(j)p Fq(a)p Fm(j)d Fq(<)h(c)g Fs(where)i Fq(c)e Fs(is)g(de\014ned)i (in)e(\(H0\).)61 b(On)38 b(the)-118 4285 y(domain)c Fq(`)273 4300 y Fn(c)308 4285 y Fs(\()p Fo(Z)415 4249 y Fn(d)452 4285 y Fs(\))g(=)f Fm(f)p Fq(u)f Fs(=)h(\()p Fq(u)974 4300 y Fn(n)1021 4285 y Fs(\))1059 4304 y Fn(n)p Fi(2)p Fk(Z)1199 4285 y Fj(d)r Fs(;)52 b Fm(9)p Fq(N)45 b(>)33 b Fs(0)p Fq(;)52 b(u)1783 4300 y Fn(n)1862 4285 y Fs(=)34 b(0)h(if)g Fm(j)p Fq(n)p Fm(j)d Fq(>)i(N)10 b Fm(g)p Fs(,)37 b(consider)f(the)g(op)s(erator)f Fq(T)3614 4300 y Fn(a)3692 4285 y Fs(de\014ned)i(b)m(y)-118 4401 y(\()p Fq(T)-23 4416 y Fn(a)19 4401 y Fq(u)p Fs(\))113 4416 y Fn(n)187 4401 y Fs(=)27 b Fq(e)335 4365 y Fn(a)p Fi(\001)p Fn(n)440 4401 y Fq(u)496 4416 y Fn(n)573 4401 y Fs(\()p Fq(u)g Fs(=)h(\()p Fq(u)892 4416 y Fn(n)938 4401 y Fs(\))976 4420 y Fn(n)p Fi(2)p 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Fs(,)j(a)d(trigonometric)f(p)s(olynomial,)g(one)i(has) -118 4955 y(\()p Fq(T)-23 4970 y Fn(a)19 4955 y Fq(u)p Fs(\)\()p Fq(\022)s Fs(\))32 b(=)g Fq(u)p Fs(\()p Fq(\022)27 b Fm(\000)e Fq(ia)p Fs(\).)53 b Fq(T)904 4970 y Fn(a)981 4955 y Fs(can)36 b(b)s(e)f(extended)j(to)d(the)h(space)h(of)e (functions)g(analytic)g(in)f(the)i(strip)f Fm(j)p Fs(Im)p Fq(\022)s Fm(j)d Fq(<)h(c)p Fs(.)-118 5071 y(By)h(\(6.2\),)e(one)h(has) 1271 5231 y Fq(T)1328 5246 y Fi(\000)p Fn(a)1425 5231 y Fs(\()p Fq(h)22 b Fs(+)g Fq(\015)5 b(V)1752 5246 y Fn(!)1802 5231 y Fs(\))p Fq(T)1897 5246 y Fn(a)1966 5231 y Fs(=)28 b Fq(h)p Fs(\()p Fm(\001)22 b Fs(+)g Fq(ia)p Fs(\))h(+)f Fq(\015)5 b(V)2668 5246 y Fn(!)2718 5231 y Fq(:)-118 5392 y Fs(As)33 b Fq(h)g Fs(is)f(real)f(analytic,)h(one)h (has)842 5552 y Fq(h)p Fs(\()p Fq(\022)26 b Fs(+)c Fq(ia)p Fs(\))28 b(=)f Fq(h)p Fs(\()p Fq(\022)s Fs(\))c(+)f Fq(a)1710 5511 y Fp(2)1749 5552 y Fq(g)1796 5567 y Fn(r)1834 5552 y Fs(\()p Fq(\022)s(;)17 b(a)p Fs(\))22 b(+)g Fq(i)p Fs(\()p Fq(a)h Fm(\001)f(r)p Fq(h)p Fs(\()p Fq(\022)s Fs(\))g(+)g Fq(a)2802 5511 y Fp(2)2842 5552 y Fq(g)2889 5567 y Fn(i)2916 5552 y Fs(\()p Fq(\022)s(;)17 b(a)p Fs(\)\))-3291 b(\(6.3\))1969 5690 y Fg(20)p eop %%Page: 21 21 21 20 bop -118 241 a Fs(where)34 b Fq(g)211 256 y Fn(i)271 241 y Fs(and)f Fq(g)508 256 y Fn(r)578 241 y Fs(are)g(b)s(ounded)g (uniformly)d(for)i Fm(j)p Fq(a)p Fm(j)c Fq(<)f(c=)p Fs(2.)-118 357 y(Equations)33 b(\(6.2\))f(and)g(\(6.3\))g(implies)f(that,)h(for)g Fq(u)27 b Fs(=)h(\005)1976 372 y Fp(\003)2029 357 y Fq(u)f Fm(2)h Fq(`)2247 321 y Fp(2)2287 357 y Fs(\()p Fo(Z)2394 321 y Fn(d)2432 357 y Fs(\),)339 537 y Fm(jh)p Fq(T)463 552 y Fi(\000)p Fn(a)560 537 y Fs(\005)633 552 y Fp(\003)686 537 y Fs(\()p Fq(h)22 b Fs(+)g Fq(V)957 552 y Fn(!)1029 537 y Fm(\000)h Fq(E)6 b Fs(\)\005)1318 552 y Fp(\003)1371 537 y Fq(T)1428 552 y Fn(a)1470 537 y Fq(u;)17 b(u)p Fm(ij)1693 496 y Fp(2)1758 537 y Fs(=)28 b Fm(jh)p Fq(T)1986 552 y Fi(\000)p Fn(a)2082 537 y Fs(\()p Fq(h)22 b Fs(+)g Fq(V)2353 552 y Fn(!)2426 537 y Fm(\000)g Fq(E)6 b Fs(\))p Fq(T)2698 552 y Fn(a)2740 537 y Fq(u;)17 b(u)p Fm(ij)2963 496 y Fp(2)1758 697 y Fs(=)28 b Fm(jh)p Fq(T)1986 712 y Fi(\000)p Fn(a)2082 697 y Fs(\()p Fq(h)22 b Fs(+)g Fq(V)2353 712 y Fn(!)2426 697 y Fm(\000)g Fq(E)28 b Fs(+)22 b Fq(a)2774 656 y Fp(2)2814 697 y Fq(g)2861 712 y Fn(r)2899 697 y Fs(\()p Fm(\001)p Fq(;)17 b(a)p Fs(\)\))p Fq(T)3193 712 y Fn(a)3234 697 y Fq(u;)g(u)p Fm(ij)3457 656 y Fp(2)2225 857 y Fs(+)22 b Fm(jh)p Fq(T)2447 872 y Fi(\000)p Fn(a)2543 857 y Fs(\()p Fq(a)h Fm(\001)e(r)p Fq(h)p Fs(\()p Fq(\022)s Fs(\))i(+)f Fq(a)3139 816 y Fp(2)3178 857 y Fq(g)3225 872 y Fn(i)3253 857 y Fs(\()p Fm(\001)p Fq(;)17 b(a)p Fs(\)\))p Fq(T)3547 872 y Fn(a)3589 857 y Fm(j)p Fq(u;)g(u)p Fm(ij)3840 816 y Fp(2)1758 1018 y Fm(\025)28 b(jh)p Fq(T)1987 1033 y Fi(\000)p Fn(a)2084 1018 y Fs(\005)2157 1033 y Fp(\003)2210 1018 y Fs(\()p Fq(h)22 b Fs(+)g Fq(V)2481 1033 y Fn(!)2553 1018 y Fm(\000)h Fq(E)28 b Fs(+)22 b Fq(a)2902 977 y Fp(2)2942 1018 y Fq(g)2989 1033 y Fn(r)3027 1018 y Fs(\()p Fm(\001)p Fq(;)17 b(a)p Fs(\)\)\005)3337 1033 y Fp(\003)3389 1018 y Fq(T)3446 1033 y Fn(a)3488 1018 y Fq(u;)g(u)p Fm(ij)3711 977 y Fp(2)3749 1018 y Fq(:)-118 780 y Fs(\(6.4\))-118 1203 y(Pic)m(k)33 b Fq(a)g Fs(suc)m(h)h(that)e Fm(j)p Fq(a)p Fm(j)27 b Fs(=)h Fq(\032)p Fs(\()p Fq(E)6 b Fs(\))p Fq(=C)39 b Fs(where)34 b Fq(\032)p Fs(\()p Fq(E)6 b Fs(\))33 b(is)f(de\014ned)i(in)e(Lemma)f(6.1)h(and)h Fq(C)h(>)28 b Fs(0)k(is)g(suc)m(h)i(that)1398 1382 y Fq(\032)p Fs(\()p Fq(E)6 b Fs(\))60 b(sup)1619 1468 y Fi(j)p Fn(a)p Fi(j\024)p Fn(c=)p Fp(2)1652 1552 y Fn(\022)r Fi(2)p Fk(R)1782 1528 y Fj(d)1868 1382 y Fm(j)p Fq(g)1943 1397 y Fn(r)1981 1382 y Fs(\()p Fq(\022)s(;)17 b(a)p Fs(\))p Fm(j)27 b(\024)h Fq(C)2437 1341 y Fp(2)2476 1382 y Fq(=)p Fs(2)p Fq(;)-118 1706 y Fs(then)993 1909 y(\005)1066 1924 y Fp(\003)1119 1909 y Fs(\()p Fq(h)22 b Fs(+)g Fq(V)1390 1924 y Fn(!)1463 1909 y Fm(\000)g Fq(E)28 b Fs(+)22 b Fq(a)1811 1868 y Fp(2)1851 1909 y Fq(g)1898 1924 y Fn(r)1936 1909 y Fs(\()p Fm(\001)p Fq(;)17 b(a)p Fs(\)\)\005)2246 1924 y Fp(\003)2326 1909 y Fm(\025)2441 1841 y Fs(1)p 2441 1886 49 4 v 2441 1977 a(2)2500 1909 y(\005)2573 1924 y Fp(\003)2626 1909 y Fq(\032)p Fs(\()p Fq(E)6 b Fs(\))2830 1868 y Fp(2)2870 1909 y Fs(\005)2943 1924 y Fp(\003)2996 1909 y Fq(:)-118 2116 y Fs(Hence,)922 2367 y Fm(jh)p Fq(T)1046 2382 y Fi(\000)p Fn(a)1142 2367 y Fs(\005)1215 2382 y Fp(\003)1268 2367 y Fs(\()p Fq(h)22 b Fs(+)g Fq(V)1539 2382 y Fn(!)1612 2367 y Fm(\000)g Fq(E)6 b Fs(\)\005)1900 2382 y Fp(\003)1953 2367 y Fq(T)2010 2382 y Fn(a)2052 2367 y Fq(u;)17 b(u)p Fm(ij)2275 2326 y Fp(2)2340 2367 y Fm(\025)2446 2227 y Fl(\022)2529 2300 y Fq(\032)p Fs(\()p Fq(E)6 b Fs(\))p 2529 2345 205 4 v 2607 2436 a(2)2743 2227 y Fl(\023)2816 2249 y Fp(4)2873 2367 y Fm(k)p Fq(u)p Fm(k)3029 2326 y Fp(2)3067 2367 y Fq(:)-118 2612 y Fs(This)34 b(implies)e(that)i Fq(T)709 2627 y Fi(\000)p Fn(a)805 2612 y Fs(\005)878 2627 y Fp(\003)932 2612 y Fs(\()p Fq(h)23 b Fs(+)g Fq(V)1205 2627 y Fn(!)1278 2612 y Fm(\000)h Fq(E)6 b Fs(\)\005)1568 2627 y Fp(\003)1621 2612 y Fq(T)1678 2627 y Fn(a)1754 2612 y Fs(is)34 b(in)m(v)m(ertible)f (and)h(its)g(in)m(v)m(erse)h(is)f(b)s(ounded)g(b)m(y)h(4)p Fq(=\032)3723 2576 y Fp(2)3763 2612 y Fs(\()p Fq(E)6 b Fs(\).)48 b(Let)-118 2728 y(us)33 b(no)m(w)g(compute)352 2903 y Fm(jh)p Fq(\016)462 2918 y Fn(n)509 2903 y Fq(;)17 b Fs(\(\005)664 2918 y Fp(\003)717 2903 y Fs(\()p Fq(h)22 b Fs(+)g Fq(V)988 2918 y Fn(!)1060 2903 y Fm(\000)h Fq(E)6 b Fs(\)\005)1349 2918 y Fp(\003)1402 2903 y Fs(\))1440 2861 y Fi(\000)p Fp(1)1534 2903 y Fq(\016)1577 2918 y Fn(m)1644 2903 y Fm(ij)27 b Fs(=)h Fm(jh)p Fq(T)1966 2918 y Fn(a)2007 2903 y Fq(\016)2050 2918 y Fn(n)2097 2903 y Fq(;)17 b Fs(\()p Fq(T)2236 2918 y Fi(\000)p Fn(a)2333 2903 y Fs(\(\005)2444 2918 y Fp(\003)2497 2903 y Fs(\()p Fq(h)22 b Fs(+)g Fq(V)2768 2918 y Fn(!)2840 2903 y Fm(\000)h Fq(E)6 b Fs(\)\005)3129 2918 y Fp(\003)3182 2903 y Fs(\))p Fq(T)3277 2918 y Fn(a)3319 2903 y Fs(\))3357 2861 y Fi(\000)p Fp(1)3451 2903 y Fq(T)3508 2918 y Fi(\000)p Fn(a)3605 2903 y Fq(\016)3648 2918 y Fn(m)3715 2903 y Fm(ij)1738 3140 y(\024)1843 2999 y Fl(\022)2005 3073 y Fs(2)p 1927 3117 V 1927 3208 a Fq(\032)p Fs(\()p Fq(E)g Fs(\))2141 2999 y Fl(\023)2214 3022 y Fp(2)2270 3140 y Fm(k)p Fq(T)2377 3155 y Fi(\000)p Fn(a)2474 3140 y Fq(\016)2517 3155 y Fn(m)2584 3140 y Fm(kk)p Fq(T)2741 3155 y Fn(a)2782 3140 y Fq(\016)2825 3155 y Fn(n)2872 3140 y Fm(k)28 b Fs(=)3053 2999 y Fl(\022)3214 3073 y Fs(2)p 3137 3117 V 3137 3208 a Fq(\032)p Fs(\()p Fq(E)6 b Fs(\))3351 2999 y Fl(\023)3424 3022 y Fp(2)3480 3140 y Fq(e)3525 3099 y Fi(\000)p Fn(a)p Fp(\()p Fn(m)p Fi(\000)p Fn(n)p Fp(\))3837 3140 y Fq(:)-118 3448 y Fs(T)-8 b(aking)32 b Fq(a)c Fs(=)496 3380 y Fq(\032)p Fs(\()p Fq(E)6 b Fs(\))p 399 3425 398 4 v 399 3516 a Fq(C)h Fm(j)p Fq(m)23 b Fm(\000)f Fq(n)p Fm(j)807 3448 y Fs(\()p Fq(m)g Fm(\000)h Fq(n)p Fs(\),)33 b(w)m(e)g(obtain)f(\(6.1\)) o(.)44 b(This)33 b(completes)f(the)h(pro)s(of)f(of)g(Lemma)f(6.1.)p 4063 3448 4 66 v 4067 3385 59 4 v 4067 3448 V 4125 3448 4 66 v -118 3721 a(6.2.)56 b Fy(The)c(k)m(ey)h(lemma.)48 b Fs(The)e(follo)m(wing)d(lemma)h(is)h(a)g(v)m(ery)j(simple)c(but)i (quite)g(con)m(v)m(enien)m(t)h(quan)m(titativ)m(e)-118 3837 y(v)m(ersion)32 b(of)f(the)i(uncertain)m(t)m(y)f(principle)e(for)i (the)g(F)-8 b(ourier)30 b(transformation)g(on)h(\014nite)h(Ab)s(elian)e (discrete)i(groups.)-118 4024 y Fy(Lemma)37 b(6.2)h Fs(\([11)o(]\))p Fy(.)49 b Ff(Assume)35 b Fq(N)10 b Ff(,)35 b Fq(L)p Ff(,)g Fq(K)7 b Ff(,)35 b Fq(K)1660 3988 y Fi(0)1719 4024 y Fq(L)1785 3988 y Fi(0)1843 4024 y Ff(ar)-5 b(e)35 b(p)-5 b(ositive)34 b(inte)-5 b(gers)34 b(such)h(that)8 4171 y Fm(\017)41 b Fs(2)p Fq(N)32 b Fs(+)22 b(1)28 b(=)f(\(2)p Fq(K)i Fs(+)22 b(1\)\(2)p Fq(L)g Fs(+)g(1\))28 b(=)f(\(2)p Fq(K)1588 4135 y Fi(0)1634 4171 y Fs(+)22 b(1\)\(2)p Fq(L)1972 4135 y Fi(0)2017 4171 y Fs(+)g(1\))34 b Ff(such)8 4288 y Fm(\017)41 b Fq(K)35 b(<)27 b(K)410 4251 y Fi(0)469 4288 y Ff(and)34 b Fq(L)724 4251 y Fi(0)775 4288 y Fq(<)28 b(L)p Ff(.)-118 4435 y(Pick)34 b Fq(a)28 b Fs(=)g(\()p Fq(a)374 4450 y Fn(n)421 4435 y Fs(\))459 4458 y Fn(n)p Fi(2)p Fk(Z)599 4435 y Fj(d)599 4481 y Fc(2)p Fj(N)5 b Fc(+1)790 4435 y Fm(2)28 b Fq(`)925 4399 y Fp(2)964 4435 y Fs(\()p Fo(Z)1071 4399 y Fn(d)1071 4460 y Fp(2)p Fn(N)7 b Fp(+1)1262 4435 y Fs(\))34 b Ff(such)h(that,)1573 4634 y(for)g Fm(j)p Fq(n)p Fm(j)27 b Fq(>)h(K)r(;)51 b(a)2188 4649 y Fn(n)2263 4634 y Fs(=)28 b(0)p Fq(:)-118 4826 y Ff(Then,)34 b(ther)-5 b(e)35 b(exists)g Fs(~)-50 b Fq(a)28 b Fm(2)g Fq(`)889 4790 y Fp(2)929 4826 y Fs(\()p Fo(Z)1036 4790 y Fn(d)1036 4852 y Fp(2)p Fn(N)7 b Fp(+1)1226 4826 y Fs(\))34 b Ff(such)h(that)-18 4976 y Fs(1.)41 b Fm(k)p Fq(a)22 b Fm(\000)i Fs(~)-50 b Fq(a)p Fm(k)423 4999 y Fn(`)452 4980 y Fc(2)486 4999 y Fp(\()p Fk(Z)564 4976 y Fj(d)564 5022 y Fc(2)p Fj(N)5 b Fc(+1)723 4999 y Fp(\))782 4976 y Fm(\024)28 b Fq(C)957 4991 y Fn(K)q(;K)1103 4972 y Fe(0)1128 4976 y Fm(k)p Fq(a)p Fm(k)1279 4999 y Fn(`)1308 4980 y Fc(2)1342 4999 y Fp(\()p Fk(Z)1420 4976 y Fj(d)1420 5022 y Fc(2)p Fj(N)5 b Fc(+1)1579 4999 y Fp(\))1645 4976 y Ff(wher)-5 b(e)35 b Fq(C)1991 4991 y Fn(K)q(;K)2137 4972 y Fe(0)2283 4976 y Fm(\020)2178 5043 y Fn(K)q(=K)2339 5024 y Fe(0)2360 5043 y Fi(!)p Fp(0)2482 4976 y Fq(K)r(=K)2706 4935 y Fi(0)2729 4976 y Ff(.)-18 5160 y Fs(2.)41 b Ff(for)35 b Fq(l)286 5124 y Fi(0)337 5160 y Fm(2)28 b Fo(Z)500 5124 y Fn(d)500 5186 y Fp(2)p Fn(L)583 5167 y Fe(0)603 5186 y Fp(+1)732 5160 y Ff(and)34 b Fq(k)975 5124 y Fi(0)1026 5160 y Fm(2)29 b Fo(Z)1190 5124 y Fn(d)1190 5186 y Fp(2)p Fn(K)1290 5167 y Fe(0)1309 5186 y Fp(+1)1403 5160 y Ff(,)35 b(we)f(have)1012 5339 y Fm(h)q Fs(~)-50 b Fq(a;)17 b(v)1193 5354 y Fn(l)1215 5335 y Fe(0)1237 5354 y Fp(+)p Fn(k)1331 5335 y Fe(0)1353 5354 y Fp(\(2)p Fn(L)1463 5335 y Fe(0)1486 5354 y Fp(+1\))1608 5339 y Fm(i)1647 5362 y Fn(`)1676 5343 y Fc(2)1710 5362 y Fp(\()p Fk(Z)1788 5339 y Fj(d)1788 5385 y Fc(2)p Fj(N)5 b Fc(+1)1947 5362 y Fp(\))2006 5339 y Fs(=)27 b Fm(h)q Fs(~)-50 b Fq(a;)17 b(v)2290 5354 y Fn(k)2329 5335 y Fe(0)2351 5354 y Fp(\(2)p Fn(L)2461 5335 y Fe(0)2484 5354 y Fp(+1\))2606 5339 y Fm(i)2645 5362 y Fn(`)2674 5343 y Fc(2)2708 5362 y Fp(\()p Fk(Z)2786 5339 y Fj(d)2786 5385 y Fc(2)p Fj(N)5 b Fc(+1)2945 5362 y Fp(\))2977 5339 y Fq(:)-18 5538 y Fs(3.)41 b Fm(k)p Fq(a)p Fm(k)250 5561 y Fn(`)279 5542 y Fc(2)313 5561 y Fp(\()p Fk(Z)391 5538 y Fj(d)391 5584 y Fc(2)p Fj(N)5 b Fc(+1)550 5561 y Fp(\))609 5538 y Fs(=)28 b Fm(k)q Fs(~)-50 b Fq(a)p Fm(k)864 5561 y Fn(`)893 5542 y Fc(2)927 5561 y Fp(\()p Fk(Z)1005 5538 y Fj(d)1005 5584 y Fc(2)p Fj(N)5 b Fc(+1)1164 5561 y Fp(\))1196 5538 y Ff(.)1969 5690 y Fg(21)p eop %%Page: 22 22 22 21 bop -118 241 a Fs(A)30 b(one-dimensional)d(v)m(ersion)k(of)f (this)f(result)h(w)m(as)h(pro)m(v)m(ed)h(in)d([11].)42 b(The)31 b(pro)s(of)f(in)f(arbitrary)g(dimension)g(is)g(v)m(ery)-118 357 y(similar)f(to)j(that)g(in)g(dimension)f(1;)h(for)g(the)h(reader's) g(con)m(v)m(enience,)i(as)e(its)f(pro)s(of)f(is)h(quite)g(short,)h(w)m (e)h(repro)s(duce)-118 473 y(it)e(here.)-118 589 y Fy(Pro)s(of.)43 b Fs(By)33 b(de\014nition)f(of)g(\()p Fq(v)1011 604 y Fn(m)1078 589 y Fs(\))g(\(see)i(\(4.16\))o(\),)f(for)f Fq(m)c Fm(2)g Fo(Z)2116 553 y Fn(d)2116 615 y Fp(2)p Fn(N)7 b Fp(+1)2306 589 y Fs(,)32 b(one)h(has)994 816 y Fm(h)p Fq(a;)17 b(v)1175 831 y Fn(m)1241 816 y Fm(i)1280 839 y Fn(`)1309 820 y Fc(2)1343 839 y Fp(\()p Fk(Z)1421 816 y Fj(d)1421 862 y Fc(2)p Fj(N)5 b Fc(+1)1580 839 y Fp(\))1639 816 y Fs(=)1975 748 y(1)p 1753 793 494 4 v 1753 884 a(\(2)p Fq(N)32 b Fs(+)22 b(1\))2135 855 y Fn(d=)p Fp(2)2351 721 y Fl(X)2272 945 y Fn(n)p Fi(2)p Fk(Z)2413 922 y Fj(d)2413 968 y Fc(2)p Fj(K)s Fc(+1)2589 816 y Fq(a)2640 831 y Fn(n)2688 816 y Fq(e)2733 772 y Fi(\000)2798 745 y Fc(2)p Fj(i\031)r(nm)p 2798 757 184 3 v 2809 798 a Fc(2)p Fj(N)5 b Fc(+1)2995 816 y Fq(:)-118 1130 y Fs(W)-8 b(e)33 b(decomp)s(ose)g Fq(m)28 b Fs(=)f Fq(l)788 1093 y Fi(0)834 1130 y Fs(+)22 b Fq(k)986 1093 y Fi(0)1009 1130 y Fs(\(2)p Fq(L)1162 1093 y Fi(0)1208 1130 y Fs(+)g(1\))32 b(where)i Fq(k)1761 1093 y Fi(0)1812 1130 y Fm(2)28 b Fo(Z)1975 1093 y Fn(d)1975 1156 y Fp(2)p Fn(K)2075 1137 y Fe(0)2094 1156 y Fp(+1)2221 1130 y Fs(and)33 b Fq(l)2442 1093 y Fi(0)2493 1130 y Fm(2)28 b Fo(Z)2656 1093 y Fn(d)2656 1156 y Fp(2)p Fn(L)2739 1137 y Fe(0)2759 1156 y Fp(+1)2886 1130 y Fs(so)33 b(that)310 1357 y Fm(h)p Fq(a;)17 b(v)491 1372 y Fn(m)558 1357 y Fm(i)597 1380 y Fn(`)626 1361 y Fc(2)660 1380 y Fp(\()p Fk(Z)738 1357 y Fj(d)738 1403 y Fc(2)p Fj(N)5 b Fc(+1)897 1380 y Fp(\))956 1357 y Fs(=)1551 1290 y(1)p 1069 1334 1013 4 v 1069 1426 a(\(2)p Fq(K)1246 1397 y Fi(0)1292 1426 y Fs(+)22 b(1\))1477 1397 y Fn(d=)p Fp(2)1588 1426 y Fs(\(2)p Fq(L)1741 1397 y Fi(0)1786 1426 y Fs(+)g(1\))1971 1397 y Fn(d=)p Fp(2)2187 1263 y Fl(X)2108 1486 y Fn(n)p Fi(2)p Fk(Z)2249 1463 y Fj(d)2249 1509 y Fc(2)p Fj(K)s Fc(+1)2425 1357 y Fq(a)2476 1372 y Fn(n)2523 1357 y Fq(e)2568 1310 y Fi(\000)2633 1283 y Fc(2)p Fj(i\031)r(nk)2795 1262 y Fe(0)p 2634 1295 186 3 v 2634 1340 a Fc(2)p Fj(K)2718 1326 y Fe(0)2741 1340 y Fc(+1)2833 1357 y Fq(e)2878 1314 y Fi(\000)2943 1286 y Fc(2)p Fj(i\031)r(nl)3091 1265 y Fe(0)p 2943 1298 172 3 v 2948 1340 a Fc(2)p Fj(N)5 b Fc(+1)956 1798 y Fs(=)1292 1730 y(1)p 1069 1775 495 4 v 1069 1866 a(\(2)p Fq(L)1222 1837 y Fi(0)1268 1866 y Fs(+)22 b(1\))1453 1837 y Fn(d=)p Fp(2)1590 1567 y Fl(0)1590 1743 y(B)1590 1807 y(@)1922 1730 y Fs(1)p 1687 1775 519 4 v 1687 1866 a(\(2)p Fq(K)1864 1837 y Fi(0)1910 1866 y Fs(+)g(1\))2095 1837 y Fn(d=)p Fp(2)2321 1703 y Fl(X)2232 1926 y Fn(n)p Fi(2)p Fk(Z)2372 1904 y Fj(d)2372 1957 y Fc(2)p Fj(K)2456 1943 y Fe(0)2474 1957 y Fc(+1)2571 1798 y Fq(a)2622 1813 y Fn(n)2670 1798 y Fq(e)2715 1750 y Fi(\000)2780 1723 y Fc(2)p Fj(i\031)r(nk)2942 1702 y Fe(0)p 2780 1735 186 3 v 2780 1780 a Fc(2)p Fj(K)2864 1766 y Fe(0)2887 1780 y Fc(+1)1664 2253 y Fs(+)1984 2186 y(1)p 1750 2230 519 4 v 1750 2322 a(\(2)p Fq(K)1927 2293 y Fi(0)1972 2322 y Fs(+)g(1\))2157 2293 y Fn(d=)p Fp(2)2383 2159 y Fl(X)2294 2382 y Fn(n)p Fi(2)p Fk(Z)2434 2359 y Fj(d)2434 2413 y Fc(2)p Fj(K)2518 2399 y Fe(0)2535 2413 y Fc(+1)2617 2253 y Fs(\()p Fq(e)2700 2210 y Fi(\000)2765 2183 y Fc(2)p Fj(i\031)r(nl)2913 2162 y Fe(0)p 2765 2195 172 3 v 2770 2236 a Fc(2)p Fj(N)5 b Fc(+1)2972 2253 y Fm(\000)23 b Fs(1\))p Fq(a)3210 2268 y Fn(n)3257 2253 y Fq(e)3302 2206 y Fi(\000)3367 2179 y Fc(2)p Fj(i\031)r(nk)3529 2158 y Fe(0)p 3367 2191 186 3 v 3367 2236 a Fc(2)p Fj(K)3451 2222 y Fe(0)3475 2236 y Fc(+1)3566 2023 y Fl(1)3566 2199 y(C)3566 2262 y(A)956 2607 y Fs(=)1292 2539 y(1)p 1069 2584 495 4 v 1069 2675 a(\(2)p Fq(L)1222 2646 y Fi(0)1268 2675 y Fs(+)f(1\))1453 2646 y Fn(d=)p Fp(2)1574 2607 y Fm(h)p Fq(a;)17 b(v)1759 2565 y Fn(K)1824 2542 y Fe(0)1755 2631 y Fn(k)1794 2612 y Fe(0)1849 2607 y Fm(i)1888 2630 y Fn(`)1917 2611 y Fc(2)1951 2630 y Fp(\()p Fk(Z)2029 2607 y Fj(d)2029 2661 y Fc(2)p Fj(K)2113 2647 y Fe(0)2130 2661 y Fc(+1)2211 2630 y Fp(\))2265 2607 y Fs(+)2596 2539 y(1)p 2373 2584 V 2373 2675 a(\(2)p Fq(L)2526 2646 y Fi(0)2571 2675 y Fs(+)22 b(1\))2756 2646 y Fn(d=)p Fp(2)2877 2607 y Fm(h)p Fq(D)3000 2565 y Fn(l)3022 2542 y Fe(0)3048 2607 y Fq(a;)17 b(v)3194 2565 y Fn(K)3259 2542 y Fe(0)3190 2631 y Fn(k)3229 2612 y Fe(0)3284 2607 y Fm(i)3323 2630 y Fn(`)3352 2611 y Fc(2)3387 2630 y Fp(\()p Fk(Z)3464 2607 y Fj(d)3464 2661 y Fc(2)p Fj(K)3548 2647 y Fe(0)3566 2661 y Fc(+1)3647 2630 y Fp(\))3678 2607 y Fq(:)-118 2832 y Fs(where)8 2975 y Fm(\017)41 b Fq(a)33 b Fs(is)f(seen)i(as)e(an)h(elemen)m(t)f(of)g Fq(`)1260 2939 y Fp(2)1300 2975 y Fs(\()p Fo(Z)1407 2939 y Fn(d)1407 3002 y Fp(2)p Fn(K)1507 2983 y Fe(0)1526 3002 y Fp(+1)1620 2975 y Fs(\))8 3104 y Fm(\017)41 b Fs(\()p Fq(v)188 3068 y Fn(K)253 3044 y Fe(0)184 3130 y Fn(k)223 3111 y Fe(0)278 3104 y Fs(\))316 3127 y Fn(k)355 3108 y Fe(0)377 3127 y Fi(2)p Fk(Z)474 3104 y Fj(d)474 3158 y Fc(2)p Fj(K)558 3144 y Fe(0)576 3158 y Fc(+1)694 3104 y Fs(is)32 b(the)h(orthonormal)d(basis)i(of)h Fq(`)1910 3068 y Fp(2)1949 3104 y Fs(\()p Fo(Z)2056 3068 y Fn(d)2056 3130 y Fp(2)p Fn(K)2156 3111 y Fe(0)2175 3130 y Fp(+1)2269 3104 y Fs(\))g(de\014ned)h(b)m(y)1277 3359 y Fq(v)1328 3318 y Fn(K)1393 3295 y Fe(0)1324 3384 y Fn(k)1363 3365 y Fe(0)1446 3359 y Fs(=)1794 3292 y(1)p 1559 3337 519 4 v 1559 3428 a(\(2)p Fq(K)1736 3399 y Fi(0)1781 3428 y Fs(+)22 b(1\))1966 3399 y Fn(d=)p Fp(2)2087 3359 y Fs(\()p Fq(e)2182 3285 y Fc(2)p Fj(i\031)r(k)q(k)2340 3264 y Fe(0)p 2180 3297 185 3 v 2180 3342 a Fc(2)p Fj(K)2264 3328 y Fe(0)2288 3342 y Fc(+1)2379 3359 y Fs(\))2417 3383 y Fn(k)r Fi(2)p Fk(Z)2553 3360 y Fj(d)2553 3414 y Fc(2)p Fj(K)2637 3400 y Fe(0)2654 3414 y Fc(+1)8 3603 y Fm(\017)41 b Fs(for)34 b Fq(l)281 3566 y Fi(0)335 3603 y Fm(2)d Fo(Z)501 3566 y Fn(d)501 3629 y Fp(2)p Fn(L)584 3610 y Fe(0)604 3629 y Fp(+1)698 3603 y Fs(,)k Fq(D)844 3566 y Fn(l)866 3543 y Fe(0)926 3603 y Fs(is)f(the)g(\(2)p Fq(K)1372 3566 y Fi(0)1419 3603 y Fs(+)23 b(1\))1605 3566 y Fn(d)1668 3603 y Fm(\002)h Fs(\(2)p Fq(K)1946 3566 y Fi(0)1993 3603 y Fs(+)f(1\))2179 3566 y Fn(d)2253 3603 y Fs(diagonal)32 b(matrix)h(\(acting)g(on)h Fq(`)3474 3566 y Fp(2)3514 3603 y Fs(\()p Fo(Z)3621 3566 y Fn(d)3621 3629 y Fp(2)p Fn(K)3721 3610 y Fe(0)3740 3629 y Fp(+1)3834 3603 y Fs(\)\))g(with)99 3719 y(the)f(diagonal)d(en)m(tries)1308 3976 y Fq(d)1359 3935 y Fn(l)1381 3912 y Fe(0)1359 4001 y Fn(nn)1476 3976 y Fs(=)1579 3806 y Fl(\()1757 3920 y Fq(e)1802 3877 y Fi(\000)1867 3849 y Fc(2)p Fj(i\031)r(nl)2015 3829 y Fe(0)p 1867 3861 172 3 v 1872 3903 a Fc(2)p Fj(N)5 b Fc(+1)2075 3920 y Fm(\000)22 b Fs(1)33 b(if)e Fm(j)p Fq(n)p Fm(j)c(\024)h Fq(K)1757 4060 y Fs(0)k(if)g(not.)-118 4233 y(Ob)m(viously)-8 b(,)901 4435 y(sup)818 4528 y Fn(l)840 4509 y Fe(0)862 4528 y Fi(2)p Fk(Z)960 4506 y Fj(d)960 4559 y Fc(2)p Fj(L)1032 4545 y Fe(0)1049 4559 y Fc(+1)1146 4435 y Fm(k)p Fq(D)1280 4394 y Fn(l)1302 4370 y Fe(0)1328 4435 y Fm(k)1378 4458 y Fi(L)p Fp(\()p Fn(`)1483 4439 y Fc(2)1518 4458 y Fp(\()p Fk(Z)1595 4435 y Fj(d)1595 4489 y Fc(2)p Fj(K)1679 4475 y Fe(0)1697 4489 y Fc(+1)1778 4458 y Fp(\)\))1864 4435 y Fm(\024)111 b Fs(sup)1977 4529 y Fn(k)r Fi(2)p Fk(Z)2113 4506 y Fj(d)2113 4552 y Fc(2)p Fj(K)s Fc(+1)1970 4625 y Fn(l)1992 4602 y Fe(0)2014 4625 y Fi(2)p Fk(Z)2111 4602 y Fj(d)2111 4656 y Fc(2)p Fj(L)2183 4642 y Fe(0)2200 4656 y Fc(+1)2298 4320 y Fl(\014)2298 4380 y(\014)2298 4440 y(\014)2331 4435 y Fq(e)2376 4391 y Fi(\000)2441 4364 y Fc(2)p Fj(i\031)r(k)q(l) 2585 4343 y Fe(0)p 2441 4376 167 3 v 2444 4417 a Fc(2)p Fj(N)5 b Fc(+1)2644 4435 y Fm(\000)23 b Fs(1)2793 4320 y Fl(\014)2793 4380 y(\014)2793 4440 y(\014)2854 4435 y Fs(=)k Fq(C)3027 4450 y Fn(K)q(;K)3173 4431 y Fe(0)-118 4805 y Fs(and)1647 5003 y Fq(C)1717 5018 y Fn(K)q(;K)1863 4999 y Fe(0)2009 5003 y Fm(\020)1904 5069 y Fn(K)q(=K)2065 5050 y Fe(0)2086 5069 y Fi(!)p Fp(0)2230 4935 y Fq(K)p 2218 4980 114 4 v 2218 5071 a(K)2308 5042 y Fi(0)2342 5003 y Fq(:)-118 5228 y Fs(W)-8 b(e)33 b(de\014ne)1091 5422 y(~)-50 b Fq(a)28 b Fs(=)1350 5327 y Fl(X)1272 5551 y Fn(n)p Fi(2)p Fk(Z)1412 5528 y Fj(d)1412 5574 y Fc(2)p Fj(N)5 b Fc(+1)1821 5355 y Fs(1)p 1598 5399 495 4 v 1598 5490 a(\(2)p Fq(L)1751 5462 y Fi(0)1797 5490 y Fs(+)22 b(1\))1982 5462 y Fn(d=)p Fp(2)2102 5422 y Fm(h)p Fq(a;)17 b(v)2287 5381 y Fn(K)2352 5357 y Fe(0)2283 5447 y Fp([)p Fn(n)p Fp(])2366 5466 y Fj(L)2408 5452 y Fe(0)2438 5422 y Fm(i)2477 5445 y Fn(`)2506 5426 y Fc(2)2540 5445 y Fp(\()p Fk(Z)2618 5422 y Fj(d)2618 5476 y Fc(2)p Fj(K)2702 5462 y Fe(0)2719 5476 y Fc(+1)2800 5445 y Fp(\))2832 5422 y Fq(v)2879 5437 y Fn(n)1969 5690 y Fg(22)p eop %%Page: 23 23 23 22 bop -118 241 a Fs(where)43 b([)p Fq(n)p Fs(])285 256 y Fn(L)333 237 y Fe(0)401 241 y Fs(is)e(the)h(unique)g(p)s(oin)m(t) e Fq(k)1328 205 y Fi(0)1395 241 y Fm(2)j Fo(Z)1573 205 y Fn(d)1652 241 y Fs(suc)m(h)g(that)e Fq(n)29 b Fm(\000)g Fq(k)2348 205 y Fi(0)2371 241 y Fs(\(2)p Fq(L)2524 205 y Fi(0)2575 241 y Fs(+)g(1\))42 b Fm(2)h Fs(\()p Fm(\000)p Fq(L)3099 205 y Fi(0)3123 241 y Fq(;)17 b(L)3233 205 y Fi(0)3257 241 y Fs(])3284 205 y Fn(d)3324 241 y Fs(.)70 b(Then,)47 b(~)-51 b Fq(a)42 b Fs(satis\014es)-118 357 y(Lemma)36 b(6.2.)58 b(Indeed,)41 b(as)d(the)g(v)m(ectors)h(\()p Fq(v)1515 372 y Fn(n)1562 357 y Fs(\))1600 380 y Fn(n)p Fi(2)p Fk(Z)1740 357 y Fj(d)1740 403 y Fc(2)p Fj(N)5 b Fc(+1)1941 357 y Fs(form)36 b(an)i(orthonormal)d(basis)j(of)f Fq(`)3282 321 y Fp(2)3321 357 y Fs(\()p Fo(Z)3428 321 y Fn(d)3428 382 y Fp(2)p Fn(N)7 b Fp(+1)3618 357 y Fs(\),)39 b(p)s(oin)m(t)e(2)g(is)-118 488 y(ob)m(vious)c(b)m(y)g(the)g (de\014nition)f(of)h(~)-50 b Fq(a)p Fs(.)43 b(P)m(oin)m(t)33 b(3)f(follo)m(ws)g(from)351 697 y Fm(k)q Fs(~)-50 b Fq(a)p Fm(k)502 656 y Fp(2)502 729 y Fn(`)531 710 y Fc(2)565 729 y Fp(\()p Fk(Z)643 706 y Fj(d)643 752 y Fc(2)p Fj(N)5 b Fc(+1)802 729 y Fp(\))861 697 y Fs(=)1043 602 y Fl(X)965 826 y Fn(n)p Fi(2)p Fk(Z)1105 803 y Fj(d)1105 849 y Fc(2)p Fj(N)g Fc(+1)1478 629 y Fs(1)p 1291 674 424 4 v 1291 765 a(\(2)p Fq(L)1444 736 y Fi(0)1489 765 y Fs(+)22 b(1\))1674 736 y Fn(d)1724 697 y Fm(jh)p Fq(a;)17 b(v)1937 656 y Fn(K)2002 632 y Fe(0)1933 721 y Fp([)p Fn(n)p Fp(])2016 741 y Fj(L)2058 727 y Fe(0)2088 697 y Fm(i)2127 720 y Fn(`)2156 701 y Fc(2)2190 720 y Fp(\()p Fk(Z)2268 697 y Fj(d)2268 751 y Fc(2)p Fj(K)2352 737 y Fe(0)2369 751 y Fc(+1)2450 720 y Fp(\))2482 697 y Fm(j)2510 656 y Fp(2)861 1035 y Fs(=)1162 967 y(1)p 975 1012 V 975 1103 a(\(2)p Fq(L)1128 1074 y Fi(0)1173 1103 y Fs(+)22 b(1\))1358 1074 y Fn(d)1431 1035 y Fm(\001)f Fs(\(2)p Fq(L)1633 994 y Fi(0)1679 1035 y Fs(+)h(1\))1864 994 y Fn(d)2010 940 y Fl(X)1921 1164 y Fn(n)p Fi(2)p Fk(Z)2061 1141 y Fj(d)2061 1195 y Fc(2)p Fj(K)2145 1181 y Fe(0)2162 1195 y Fc(+1)2260 1035 y Fm(jh)p Fq(a;)17 b(v)2473 994 y Fn(K)2538 970 y Fe(0)2469 1059 y Fn(k)2508 1040 y Fe(0)2563 1035 y Fm(i)2602 1058 y Fn(`)2631 1039 y Fc(2)2665 1058 y Fp(\()p Fk(Z)2743 1035 y Fj(d)2743 1089 y Fc(2)p Fj(K)2827 1075 y Fe(0)2844 1089 y Fc(+1)2925 1058 y Fp(\))2957 1035 y Fm(j)2985 994 y Fp(2)3052 1035 y Fs(=)27 b Fm(k)p Fq(a)p Fm(k)3306 994 y Fp(2)3306 1067 y Fn(`)3335 1048 y Fc(2)3369 1067 y Fp(\()p Fk(Z)3447 1044 y Fj(d)3447 1090 y Fc(2)p Fj(N)5 b Fc(+1)3606 1067 y Fp(\))3638 1035 y Fq(:)-118 1339 y Fs(Let)33 b(us)g(c)m(hec)m(k)i(p)s(oin)m(t)c(1;)i(w) m(e)g(compute)403 1577 y Fm(k)p Fq(a)22 b Fm(\000)i Fs(~)-50 b Fq(a)p Fm(k)727 1536 y Fp(2)727 1610 y Fn(`)756 1591 y Fc(2)790 1610 y Fp(\()p Fk(Z)868 1587 y Fj(d)868 1633 y Fc(2)p Fj(N)5 b Fc(+1)1027 1610 y Fp(\))1086 1577 y Fs(=)1278 1483 y Fl(X)1190 1706 y Fn(N)i Fi(2)p Fk(Z)1350 1683 y Fj(d)1350 1729 y Fc(2)p Fj(N)e Fc(+1)1526 1433 y Fl(\014)1526 1493 y(\014)1526 1552 y(\014)1526 1612 y(\014)1792 1510 y Fs(1)p 1569 1554 495 4 v 1569 1646 a(\(2)p Fq(L)1722 1617 y Fi(0)1768 1646 y Fs(+)22 b(1\))1953 1617 y Fn(d=)p Fp(2)2074 1577 y Fm(h)p Fq(a;)17 b(v)2259 1536 y Fn(K)2324 1513 y Fe(0)2255 1602 y Fp([)p Fn(n)p Fp(])2338 1621 y Fj(L)2380 1607 y Fe(0)2409 1577 y Fm(i)2448 1600 y Fn(`)2477 1582 y Fc(2)2511 1600 y Fp(\()p Fk(Z)2589 1578 y Fj(d)2589 1632 y Fc(2)p Fj(K)2673 1618 y Fe(0)2690 1632 y Fc(+1)2772 1600 y Fp(\))2825 1577 y Fm(\000)23 b(h)p Fq(a;)17 b(v)3106 1592 y Fn(n)3153 1577 y Fm(i)3192 1600 y Fn(`)3221 1582 y Fc(2)3255 1600 y Fp(\()p Fk(Z)3333 1578 y Fj(d)3333 1624 y Fc(2)p Fj(N)5 b Fc(+1)3492 1600 y Fp(\))3523 1433 y Fl(\014)3523 1493 y(\014)3523 1552 y(\014)3523 1612 y(\014)3557 1459 y Fp(2)1086 1915 y Fs(=)1274 1821 y Fl(X)1190 2044 y Fn(l)1212 2025 y Fe(0)1234 2044 y Fi(2)p Fk(Z)1332 2021 y Fj(d)1332 2075 y Fc(2)p Fj(L)1404 2061 y Fe(0)1421 2075 y Fc(+1)1716 1848 y Fs(1)p 1528 1892 424 4 v 1528 1983 a(\(2)p Fq(L)1681 1955 y Fi(0)1727 1983 y Fs(+)22 b(1\))1912 1955 y Fn(d)2077 1821 y Fl(X)1979 2044 y Fn(k)2018 2025 y Fe(0)2039 2044 y Fi(2)p Fk(Z)2137 2021 y Fj(d)2137 2075 y Fc(2)p Fj(K)2221 2061 y Fe(0)2238 2075 y Fc(+1)2336 1915 y Fm(jh)p Fq(D)2487 1874 y Fn(l)2509 1851 y Fe(0)2534 1915 y Fq(a;)17 b(v)2680 1874 y Fn(K)2745 1851 y Fe(0)2676 1940 y Fn(k)2715 1921 y Fe(0)2771 1915 y Fm(i)2810 1938 y Fn(`)2839 1919 y Fc(2)2873 1938 y Fp(\()p Fk(Z)2951 1915 y Fj(d)2951 1969 y Fc(2)p Fj(K)3035 1955 y Fe(0)3052 1969 y Fc(+1)3133 1938 y Fp(\))3165 1915 y Fm(j)3193 1874 y Fp(2)1086 2261 y Fs(=)1274 2166 y Fl(X)1190 2390 y Fn(l)1212 2371 y Fe(0)1234 2390 y Fi(2)p Fk(Z)1332 2367 y Fj(d)1332 2421 y Fc(2)p Fj(L)1404 2407 y Fe(0)1421 2421 y Fc(+1)1716 2194 y Fs(1)p 1528 2238 V 1528 2329 a(\(2)p Fq(L)1681 2300 y Fi(0)1727 2329 y Fs(+)22 b(1\))1912 2300 y Fn(d)1962 2261 y Fm(k)p Fq(D)2096 2220 y Fn(l)2118 2196 y Fe(0)2144 2261 y Fq(a)p Fm(k)2245 2220 y Fp(2)2245 2293 y Fn(`)2274 2274 y Fc(2)2308 2293 y Fp(\()p Fk(Z)2386 2270 y Fj(d)2386 2324 y Fc(2)p Fj(K)2470 2310 y Fe(0)2487 2324 y Fc(+1)2568 2293 y Fp(\))1086 2559 y Fm(\024)29 b Fq(C)1269 2518 y Fp(2)1262 2584 y Fn(K)q(;K)1408 2565 y Fe(0)1432 2559 y Fm(k)p Fq(a)p Fm(k)1583 2518 y Fp(2)1583 2591 y Fn(`)1612 2572 y Fc(2)1646 2591 y Fp(\()p Fk(Z)1724 2568 y Fj(d)1724 2614 y Fc(2)p Fj(N)5 b Fc(+1)1883 2591 y Fp(\))1915 2559 y Fq(:)-118 2749 y Fs(This)33 b(completes)f(the)h(pro)s(of)f(of)g (Lemma)f(6.2.)p 4063 2749 4 66 v 4067 2686 59 4 v 4067 2749 V 4125 2749 4 66 v -118 2941 a(6.3.)56 b Fy(The)26 b Fq(s)p Fy(-decoupling)f(prop)s(ert)m(y.)49 b Fs(Let)22 b(us)h(recall)e(the)h Fq(s)p Fs(-decoupling)f(prop)s(ert)m(y)i(\(1.7\)) f(in)f([4].)40 b(A)23 b(probabilit)m(y)-118 3058 y(distribution)33 b Fq(dP)48 b Fs(satis\014es)35 b(the)g Fq(s)p Fs(-decoupling)f(prop)s (ert)m(y)h(with)g(some)f(0)d Fq(<)h(s)f(<)g Fs(1,)36 b(if)d(there)j(exists)f Fq(C)k(>)31 b Fs(0)j(suc)m(h)-118 3174 y(that)318 3379 y Fm(8)p Fq(a;)17 b(b;)g(c)28 b Fm(2)g Fo(C)20 b Fq(;)865 3243 y Fl(Z)920 3469 y Fk(R)1135 3311 y Fs(1)p 999 3356 321 4 v 999 3447 a Fm(j)p Fq(x)i Fm(\000)h Fq(a)p Fm(j)1283 3418 y Fn(s)1346 3234 y Fl(\014)1346 3294 y(\014)1346 3354 y(\014)1346 3414 y(\014)1390 3311 y Fq(x)f Fm(\000)h Fq(b)p 1389 3356 220 4 v 1389 3447 a(x)g Fm(\000)g Fq(c)1619 3234 y Fl(\014)1619 3294 y(\014)1619 3354 y(\014)1619 3414 y(\014)1652 3261 y Fn(s)1705 3379 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))27 b Fm(\024)h Fq(C)2190 3243 y Fl(Z)2245 3469 y Fk(R)2460 3311 y Fs(1)p 2324 3356 321 4 v 2324 3447 a Fm(j)p Fq(x)22 b Fm(\000)h Fq(a)p Fm(j)2608 3418 y Fn(s)2654 3379 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))2930 3243 y Fl(Z)2985 3469 y Fk(R)3054 3234 y Fl(\014)3054 3294 y(\014)3054 3354 y(\014)3054 3414 y(\014)3097 3311 y Fq(x)23 b Fm(\000)f Fq(b)p 3097 3356 220 4 v 3097 3447 a(x)g Fm(\000)h Fq(c)3326 3234 y Fl(\014)3326 3294 y(\014)3326 3354 y(\014)3326 3414 y(\014)3359 3261 y Fn(s)3412 3379 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))p Fq(:)-3816 b Fs(\(6.5\))-118 3600 y(W)-8 b(e)33 b(pro)m(v)m(e)-118 3779 y Fy(Lemma)k(6.3.)49 b Ff(If)38 b Fq(dP)14 b Ff(,)37 b(the)h(c)-5 b(ommon)37 b(pr)-5 b(ob)g(ability)38 b(distribution)g(of)g (the)g(r)-5 b(andom)37 b(variables)g Fs(\()p Fq(!)3506 3794 y Fn(n)3552 3779 y Fs(\))3590 3798 y Fn(n)p Fi(2)p Fk(Z)3730 3780 y Fj(d)3803 3779 y Ff(satis\014es)-118 3896 y(assumption)j(\(H2'\))h(and)g(\(H3\))g(and)f(the)h(essential)g (in\014mum)f(of)h Fq(dP)54 b Ff(is)41 b Fs(0)p Ff(,)h(it)g(is)f Fq(s)p Ff(-r)-5 b(e)g(gular)40 b(for)h Fs(0)e Fq(<)g(s)h(<)f(\034)11 b(=)p Fs(4)-118 4012 y Ff(\(wher)-5 b(e)34 b Fq(\034)46 b Ff(is)35 b(the)g(exp)-5 b(onent)34 b(de\014ne)-5 b(d)34 b(in)g(assumption)g(\(H3\)\).)-118 4191 y Fs(The)e(purp)s(ose)h(of)e (this)g(result)g(is)h(only)f(to)g(b)s(e)g(used)i(in)e(section)h(2;)f(w) m(e)i(did)e(not)g(try)h(to)f(extend)i(it)e(to)g(its)g(maximal)-118 4308 y(generalit)m(y)-8 b(.)-118 4471 y Fy(Pro)s(of.)43 b Fs(De\014ne)692 4676 y Fq(\036)750 4691 y Fn(s)787 4676 y Fs(\()p Fq(a)p Fs(\))28 b(=)1045 4540 y Fl(Z)1101 4766 y Fk(R)1315 4609 y Fs(1)p 1179 4653 321 4 v 1179 4744 a Fm(j)p Fq(x)22 b Fm(\000)h Fq(a)p Fm(j)1463 4716 y Fn(s)1510 4676 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))97 b(and)g Fq( )2183 4691 y Fn(s)2221 4676 y Fs(\()p Fq(b;)17 b(c)p Fs(\))27 b(=)2555 4540 y Fl(Z)2610 4766 y Fk(R)2679 4532 y Fl(\014)2679 4591 y(\014)2679 4651 y(\014)2679 4711 y(\014)2722 4609 y Fq(x)c Fm(\000)g Fq(b)p 2722 4653 220 4 v 2722 4744 a(x)g Fm(\000)f Fq(c)2951 4532 y Fl(\014)2951 4591 y(\014)2951 4651 y(\014)2951 4711 y(\014)2984 4558 y Fn(s)3038 4676 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))p Fq(:)-118 4904 y Fs(By)35 b(the)g(Cauc)m(h)m(y-Sc)m(h)m (w)m(artz)k(inequalit)m(y)-8 b(,)34 b(as)h(in)f(Lemma)f(C.3)i(of)f ([4],)i(it)d(su\016ces)k(to)e(sho)m(w)g(that)g(the)g(con)m(tin)m(uous) -118 5030 y(functions)302 4945 y Fl(p)p 402 4945 257 4 v 85 x Fq(\036)460 5045 y Fp(2)p Fn(s)532 5030 y Fs(\()p Fq(a)p Fs(\))p Fq(=\036)766 5045 y Fn(s)802 5030 y Fs(\()p Fq(a)p Fs(\))d(and)1151 4945 y Fl(p)p 1250 4945 339 4 v 1250 5030 a Fq( )1313 5045 y Fp(2)p Fn(s)1386 5030 y Fs(\()p Fq(b;)17 b(c)p Fs(\))p Fq(= )1701 5045 y Fn(s)1738 5030 y Fs(\()p Fq(b;)g(c)p Fs(\))32 b(are)g(b)s(ounded.)44 b(T)-8 b(o)33 b(pro)m(v)m(e)g(this,)f(w)m(e)i(only)d(need)j(to)e(study) -118 5146 y(the)h(b)s(eha)m(vior)f(of)g(the)h Fq(\036)785 5161 y Fn(s)854 5146 y Fs(and)g Fq( )1107 5161 y Fn(s)1177 5146 y Fs(at)f(in\014nit)m(y)-8 b(.)-118 5262 y(Let)33 b(us)g(start)f(with)h Fq(\036)696 5277 y Fn(s)732 5262 y Fs(.)44 b(W)-8 b(e)33 b(ma)m(y)f(assume)h Fm(j)p Fq(a)p Fm(j)f Fs(large.)42 b(One)33 b(has)928 5362 y Fl(Z)1027 5388 y Fp(+)p Fi(1)983 5588 y Fp(0)1320 5430 y Fs(1)p 1184 5475 321 4 v 1184 5566 a Fm(j)p Fq(x)22 b Fm(\000)h Fq(a)p Fm(j)1468 5537 y Fn(s)1514 5498 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))27 b Fm(\025)1905 5362 y Fl(Z)2005 5388 y Fp(1)1961 5588 y(0)2207 5430 y Fs(1)p 2071 5475 V 2071 5566 a Fm(j)p Fq(x)22 b Fm(\000)h Fq(a)p Fm(j)2355 5537 y Fn(s)2402 5498 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))27 b Fm(\025)2817 5430 y Fs(1)p 2803 5475 77 4 v 2803 5566 a Fq(C)2890 5498 y Fm(j)p Fq(a)p Fm(j)2997 5456 y Fi(\000)p Fn(s)-118 5498 y Fs(\(6.6\))1969 5690 y Fg(23)p eop %%Page: 24 24 24 23 bop -118 241 a Fs(where)34 b Fq(C)g(>)28 b Fs(0)k(is)g(indep)s (enden)m(t)i(of)e Fq(a)p Fs(,)h Fm(j)p Fq(a)p Fm(j)27 b Fq(>)h Fs(1.)43 b(On)33 b(the)g(other)f(hand,)1276 360 y Fl(Z)1376 386 y Fp(+)p Fi(1)1331 585 y Fp(0)1668 428 y Fs(1)p 1532 472 321 4 v 1532 563 a Fm(j)p Fq(x)22 b Fm(\000)h Fq(a)p Fm(j)1816 535 y Fn(s)1862 495 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))27 b(=)h Fq(I)2295 510 y Fp(1)2357 495 y Fs(+)22 b Fq(I)2498 510 y Fp(2)2559 495 y Fs(+)g Fq(I)2700 510 y Fp(3)-118 743 y Fs(where,)34 b(b)m(y)f(assumption)f (\(H2'\),)393 1007 y Fq(I)436 1022 y Fp(1)503 1007 y Fs(=)607 872 y Fl(Z)706 898 y Fi(j)p Fn(a)p Fi(j)p Fn(=)p Fp(2)662 1097 y(0)1020 940 y Fs(1)p 884 984 V 884 1076 a Fm(j)p Fq(x)23 b Fm(\000)f Fq(a)p Fm(j)1168 1047 y Fn(s)1215 1007 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))27 b Fm(\024)h Fq(C)7 b Fm(j)p Fq(a)p Fm(j)1790 966 y Fi(\000)p Fn(s)1881 1007 y Fq(;)114 b(I)2065 1022 y Fp(3)2133 1007 y Fs(=)2236 872 y Fl(Z)2336 898 y Fp(+)p Fi(1)2291 1097 y Fp(2)p Fi(j)p Fn(a)p Fi(j)2628 940 y Fs(1)p 2492 984 V 2492 1076 a Fm(j)p Fq(x)22 b Fm(\000)h Fq(a)p Fm(j)2776 1047 y Fn(s)2822 1007 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))28 b Fm(\024)g Fq(C)7 b Fm(j)p Fq(a)p Fm(j)3398 966 y Fi(\000)p Fn(s)p Fi(\000)p Fp(1)3579 1007 y Fq(;)1472 1310 y(I)1515 1325 y Fp(2)1582 1310 y Fs(=)1686 1175 y Fl(Z)1785 1201 y Fp(2)p Fi(j)p Fn(a)p Fi(j)1741 1400 y(j)p Fn(a)p Fi(j)p Fn(=)p Fp(2)2064 1243 y Fs(1)p 1928 1287 V 1928 1379 a Fm(j)p Fq(x)22 b Fm(\000)h Fq(a)p Fm(j)2212 1350 y Fn(s)2258 1310 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))p Fq(:)-118 1572 y Fs(Let)33 b(us)g(estimate)f Fq(I)616 1587 y Fp(2)655 1572 y Fs(.)43 b(By)34 b(\(H2'\))e(and)h(\(H3\),)f(w)m(e)i(kno)m(w)f (that,)g(for)f(0)27 b Fm(\024)h Fq(u)g Fm(\024)g Fq(v)t Fs(,)886 1760 y Fq(P)14 b Fs(\([)p Fq(u;)j Fs(+)p Fm(1)p Fs(\)\))26 b Fm(\024)i Fq(C)7 b Fm(j)p Fq(u)p Fm(j)1700 1719 y Fi(\000)p Fp(1)1923 1760 y Fs(and)130 b Fq(P)14 b Fs(\([)p Fq(u;)j(v)t Fs(]\))26 b Fm(\024)i Fq(C)7 b Fm(j)p Fq(u)22 b Fm(\000)g Fq(v)t Fm(j)3060 1719 y Fn(\034)3103 1760 y Fq(:)-118 1954 y Fs(In)m(terp)s(olating)31 b(b)s(et)m(w)m(een)k (these)e(t)m(w)m(o)h(estimates,)e(for)g Fq(\026)27 b Fm(2)h Fs([0)p Fq(;)17 b Fs(1],)32 b(w)m(e)i(get)1348 2150 y Fq(P)14 b Fs(\([)p Fq(u;)j(v)t Fs(]\))26 b Fm(\024)i Fq(C)7 b Fm(j)p Fq(u)p Fm(j)2026 2109 y Fi(\000)p Fn(\026)2127 2150 y Fm(j)p Fq(u)21 b Fm(\000)i Fq(v)t Fm(j)2411 2109 y Fn(\034)8 b Fp(\(1)p Fi(\000)p Fn(\026)p Fp(\))2641 2150 y Fq(:)-2786 b Fs(\(6.7\))-118 2338 y(W)-8 b(e)33 b(rewrite)484 2577 y Fq(I)527 2592 y Fp(2)594 2577 y Fs(=)698 2442 y Fl(Z)798 2468 y Fi(j)p Fn(a)p Fi(j)753 2667 y(j)p Fn(a)p Fi(j)p Fn(=)p Fp(2)1067 2510 y Fs(1)p 931 2554 V 931 2645 a Fm(j)p Fq(x)22 b Fm(\000)h Fq(a)p Fm(j)1215 2617 y Fn(s)1262 2577 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))21 b(+)1640 2442 y Fl(Z)1740 2468 y Fp(2)p Fi(j)p Fn(a)p Fi(j)1696 2667 y(j)p Fn(a)p Fi(j)2018 2510 y Fs(1)p 1883 2554 V 1883 2645 a Fm(j)p Fq(x)h Fm(\000)g Fq(a)p Fm(j)2166 2617 y Fn(s)2213 2577 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))594 2896 y(=)707 2771 y Fp(+)p Fi(1)698 2801 y Fl(X)703 3010 y Fn(n)p Fp(=1)858 2760 y Fl(Z)958 2786 y Fp(\(1)p Fi(\000)p Fp(2)1110 2763 y Fe(\000)p Fj(k)q Fe(\000)p Fc(1)1275 2786 y Fp(\))p Fi(j)p Fn(a)p Fi(j)914 2986 y Fp(\(1)p Fi(\000)p Fp(2)1066 2967 y Fe(\000)p Fj(k)1153 2986 y Fp(\))p Fi(j)p Fn(a)p Fi(j)1546 2828 y Fs(1)p 1410 2873 V 1410 2964 a Fm(j)p Fq(x)22 b Fm(\000)h Fq(a)p Fm(j)1694 2935 y Fn(s)1741 2896 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))21 b(+)2129 2771 y Fp(+)p Fi(1)2119 2801 y Fl(X)2125 3010 y Fn(n)p Fp(=1)2280 2760 y Fl(Z)2379 2786 y Fp(\(1+2)2531 2763 y Fe(\000)p Fj(k)q Fc(+1)2695 2786 y Fp(\))p Fi(j)p Fn(a)p Fi(j)2335 2986 y Fp(\(1+2)2487 2967 y Fe(\000)p Fj(k)2574 2986 y Fp(\))p Fi(j)p Fn(a)p Fi(j)2966 2828 y Fs(1)p 2830 2873 V 2830 2964 a Fm(j)p Fq(x)i Fm(\000)f Fq(a)p Fm(j)3114 2935 y Fn(s)3161 2896 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))594 3232 y Fm(\024)709 3108 y Fp(+)p Fi(1)699 3138 y Fl(X)705 3347 y Fn(n)p Fp(=1)860 3097 y Fl(Z)960 3123 y Fp(\(1)p Fi(\000)p Fp(2)1112 3099 y Fe(\000)p Fj(k)q Fe(\000)p Fc(1)1277 3123 y Fp(\))p Fi(j)p Fn(a)p Fi(j)915 3322 y Fp(\(1)p Fi(\000)p Fp(2)1067 3303 y Fe(\000)p Fj(k)1154 3322 y Fp(\))p Fi(j)p Fn(a)p Fi(j)1575 3165 y Fs(1)p 1412 3209 376 4 v 1412 3301 a Fm(j)p Fq(x)22 b Fm(\000)h(j)p Fq(a)p Fm(jj)1752 3272 y Fn(s)1798 3232 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))21 b(+)2186 3108 y Fp(+)p Fi(1)2176 3138 y Fl(X)2182 3347 y Fn(n)p Fp(=1)2337 3097 y Fl(Z)2436 3123 y Fp(\(1+2)2588 3099 y Fe(\000)p Fj(k)q Fc(+1)2752 3123 y Fp(\))p Fi(j)p Fn(a)p Fi(j)2392 3322 y Fp(\(1+2)2544 3303 y Fe(\000)p Fj(k)2631 3322 y Fp(\))p Fi(j)p Fn(a)p Fi(j)3051 3165 y Fs(1)p 2887 3209 V 2887 3301 a Fm(j)p Fq(x)i Fm(\000)f(j)p Fq(a)p Fm(jj)3227 3272 y Fn(s)3273 3232 y Fq(dP)14 b Fs(\()p Fq(x)p Fs(\))-118 3511 y(Hence,)34 b(using)g(\(6.7\))o(,)f(if)e Fm(\000)p Fq(s)23 b Fs(+)f Fq(\034)11 b Fs(\(1)22 b Fm(\000)h Fq(\026)p Fs(\))k Fq(<)g Fs(0,)33 b(w)m(e)g(get)724 3791 y Fq(I)767 3806 y Fp(2)834 3791 y Fm(\024)28 b Fq(C)1042 3666 y Fp(+)p Fi(1)1033 3696 y Fl(X)1038 3906 y Fn(n)p Fp(=1)1177 3791 y Fs(\(2)1264 3750 y Fi(\000)p Fn(k)r Fi(\000)p Fp(1)1451 3791 y Fm(j)p Fq(a)p Fm(j)p Fs(\))1596 3750 y Fi(\000)p Fn(s)1688 3791 y Fs(\(\(1)21 b Fm(\000)i Fs(2)1983 3750 y Fi(\000)p Fn(k)2080 3791 y Fs(\))p Fm(j)p Fq(a)p Fm(j)p Fs(\))2263 3750 y Fi(\000)p Fn(\026)2364 3791 y Fs(\(2)2451 3750 y Fi(\000)p Fn(k)r Fi(\000)p Fp(1)2639 3791 y Fm(j)p Fq(a)p Fm(j)p Fs(\))2784 3750 y Fn(\034)8 b Fp(\(1)p Fi(\000)p Fn(\026)p Fp(\))1301 4128 y Fs(+)22 b Fq(C)1502 4003 y Fp(+)p Fi(1)1493 4033 y Fl(X)1498 4242 y Fn(n)p Fp(=1)1637 4128 y Fs(\(2)1724 4086 y Fi(\000)p Fn(k)1821 4128 y Fm(j)p Fq(a)p Fm(j)p Fs(\))1966 4086 y Fi(\000)p Fn(s)2057 4128 y Fs(\(\(1)g(+)g(2)2351 4086 y Fi(\000)p Fn(k)2448 4128 y Fs(\))p Fm(j)p Fq(a)p Fm(j)p Fs(\))2631 4086 y Fi(\000)p Fn(\026)2732 4128 y Fs(\(2)2819 4086 y Fi(\000)p Fn(k)2916 4128 y Fm(j)p Fq(a)p Fm(j)p Fs(\))3061 4086 y Fn(\034)8 b Fp(\(1)p Fi(\000)p Fn(\026)p Fp(\))834 4421 y Fm(\024)28 b Fq(C)7 b Fm(j)p Fq(a)p Fm(j)1123 4380 y Fi(\000)p Fn(s)p Fi(\000)p Fn(\026)p Fp(+)p Fn(\034)h Fp(\(1)p Fi(\000)p Fn(\026)p Fp(\))1875 4354 y Fs(1)p 1603 4398 593 4 v 1603 4489 a(1)22 b Fm(\000)h Fs(2)1823 4461 y 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Fs(\(6.8\))-118 5271 y(This)33 b(and)f(\(6.6\))g (pro)m(v)m(es)j(that,)d(if)g Fq(s)27 b(<)h(\034)11 b(=)p Fs(2,)32 b(for)g(some)h Fq(C)2008 5286 y Fn(s)2072 5271 y Fq(>)28 b Fs(1)k(indep)s(enden)m(t)i(of)e Fq(a)p Fs(,)h Fm(j)p Fq(a)p Fm(j)27 b(\025)h Fs(1,)k(one)h(has)1475 5440 y(1)p 1446 5485 107 4 v 1446 5576 a Fq(C)1516 5591 y Fn(s)1562 5508 y Fm(j)p Fq(a)p Fm(j)1669 5466 y Fi(\000)p Fn(s)1788 5508 y Fm(\024)28 b Fq(\036)1951 5523 y Fn(s)1988 5508 y Fs(\()p Fq(a)p Fs(\))g Fm(\024)g Fq(C)2318 5523 y Fn(s)2354 5508 y Fm(j)p Fq(a)p Fm(j)2461 5466 y Fi(\000)p Fn(s)2553 5508 y Fq(:)1969 5690 y Fg(24)p eop %%Page: 25 25 25 24 bop -118 251 a Fs(So,)32 b(for)h Fq(s)27 b(<)h(\034)11 b(=)p Fs(4,)32 b(the)h(con)m(tin)m(uous)h(function)1620 166 y Fl(p)p 1719 166 257 4 v 1719 251 a Fq(\036)1777 266 y Fp(2)p Fn(s)1849 251 y Fs(\()p Fq(a)p Fs(\))p Fq(=\036)2083 266 y Fn(s)2119 251 y Fs(\()p Fq(a)p Fs(\))f(is)f(b)s(ounded.)-118 367 y(Let)e(us)h(no)m(w)g(study)h Fq( )708 382 y Fn(s)745 367 y Fs(.)43 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(calization)h(at)h(large)f(disorder)f(and)i(at)g(extreme)g(energies:)33 b(an)21 b(elemen)n(tary)g(deriv)-5 b(ation.)53 3897 y Fh(Communic)l(ations)30 b(in)g(Mathematic)l(al)i(Physics)p Fx(,)d(157:245{278,)23 b(1993.)-77 3997 y([4])42 b(M.)24 b(Aizenman,)i(F.)f(Sc)n(henk)n(er,)f(R.)g(F)-7 b(riedric)n(h,)25 b(and)f(D.)h(Hundertmark.)g(Finite-v)n(olume)f(Criteria)f(for)h (Anderson)g(Lo)r(calization.)53 4096 y(1999.)h(T)-7 b(o)28 b(app)r(ear)e(in)i(Comm.)g(Math.)f(Ph)n(ys.)-77 4196 y([5])42 b(J.)31 b(M.)h(Com)n(b)r(es)f(and)h(L.)f(Thomas.)g(Asymptotic) h(b)r(eha)n(vior)f(of)g(eigenfunctions)h(for)f(m)n(ulti-particle)g(Sc)n (hr\177)-42 b(odinger)30 b(op)r(erators.)53 4296 y Fh(Communic)l (ations)g(in)g(Mathematic)l(al)i(Physics)p Fx(,)d(34:251{270,)23 b(1973.)-77 4395 y([6])42 b(H.L.)27 b(Cycon,)h(R.G.)g(F)-7 b(ro)r(ese,)26 b(W.)j(Kirsc)n(h,)d(and)h(B.)h(Simon.)g Fh(Schr\177)-42 b(odinger)31 b(Op)l(er)l(ators)p Fx(.)d(Springer)f(V)-7 b(erlag,)26 b(Berlin,)h(1987.)-77 4495 y([7])42 b(A.)27 b(Dem)n(b)r(o)h(and)f(O.)g(Zeitouni.)g Fh(L)l(ar)l(ge)j(deviation)i(te) l(chniques)d(and)h(applic)l(ations)p Fx(.)g(Jones)c(and)i(Bartlett)f (Publi-shers,)f(Boston,)53 4595 y(1992.)-77 4694 y([8])42 b(B.)31 b(Hel\013er)h(and)g(J.)g(Sj\177)-42 b(ostrand.)31 b(On)h(diamagnetism)f(and)h(the)g(De)h(Haas-V)-7 b(an)30 b(Alphen)j(e\013ect.)f Fh(A)n(nnales)i(de)g(l'Institut)f(Henri)53 4794 y(Poinc)l(ar)n(\023)-40 b(e,)31 b(s)n(\023)-40 b(erie)31 b(Physique)g(Th)n(\023)-40 b(eorique)p Fx(,)29 b(52:303{375,)24 b(1990.)-77 4893 y([9])42 b(P)-7 b(.)27 b(Hislop.)g(Exp)r(onen)n(tial)g (deca)n(y)g(of)g(t)n(w)n(o-b)r(o)r(dy)g(eigenfunctions:)37 b(A)28 b(review.)e(Av)-5 b(ailable)28 b(on)f(mp-arc,)g(2001.)-118 4993 y([10])41 b(L.)27 b(H\177)-42 b(ormander.)27 b Fh(The)j(Analysis)h (of)f(Line)l(ar)h(Partial)g(Di\013er)l(ential)f(Op)l(er)l(ators)p Fx(.)e(Springer)f(V)-7 b(erlag,)26 b(Heidelb)r(erg,)h(1983.)-118 5093 y([11])41 b(F.)32 b(Klopp.)g(Band)g(edge)g(b)r(eha)n(viour)g(for)f 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1137 y([19])41 b(J.)21 b(N.)i(Mather.)f(On)g(Niren)n(b)r(erg's)f(pro)r(of)g(of)h (Malgrange's)e(preparation)h(theorem.)g(In)i Fh(Pr)l(o)l(c)l(e)l(e)l (dings)i(of)h(Liverp)l(o)l(ol)g(Singularities-)53 1237 y(Symp)l(osium)k(I)p Fx(,)d(n)n(um)n(b)r(er)h(192)e(in)i(Lecture)f (Notes)g(in)h(Mathematics,)g(Berlin,)f(1971.)f(Springer)g(V)-7 b(erlag.)-118 1337 y([20])41 b(N.)23 b(Minami.)g(Lo)r(cal)g (\015uctuation)g(of)g(the)h(sp)r(ectrum)f(of)g(a)g(m)n(ultidimensional) g(Anderson)g(tigh)n(t)g(binding)g(mo)r(del.)h Fh(Comm.)j(Math.)53 1436 y(Phys.)p Fx(,)i(177\(3\):709{725,)23 b(1996.)-118 1536 y([21])41 b(L.)27 b(P)n(astur)f(and)i(A.)g(Figotin.)f Fh(Sp)l(e)l(ctr)l(a)j(of)g(R)l(andom)g(and)g(A)n(lmost-Perio)l(dic)i (Op)l(er)l(ators)p Fx(.)c(Springer)e(V)-7 b(erlag,)27 b(Berlin,)g(1992.)-118 1636 y([22])41 b(M.)c(Reed)h(and)f(B.)g(Simon.)g Fh(Metho)l(ds)j(of)g(Mo)l(dern)f(Mathematic)l(al)i(Physics,)i(V)-6 b(ol)39 b(IV:)f(Analysis)i(of)f(Op)l(er)l(ators)p Fx(.)f(Academic)53 1735 y(Press,)26 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