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b(analysis)g([12])-201 7850 y(of)57 b(the)f(form)g([10)o(])h(applies.)79 b(Thereb)-5 b(y)-15 b(,)57 b(Dynamical)f(Lo)5 b(calization)58 b(w)-5 b(as)57 b(pro)-5 b(v)g(ed)56 b(for)h(those)f(mo)5 b(dels.)78 b(V)-15 b(ery)-201 8076 y(recen)-5 b(tly)-15 b(,)75 b(in)e([5],)h(this) f(result)g(has)f(b)5 b(een)72 b(strenghened)g(to)g Ff(Str)-9 b(ong)73 b Fo(Dynamical)e(Lo)5 b(calization,)76 b(namely)-201 8302 y(it)f(is)g(sho)-5 b(wn)74 b(that)g(for)g(random)f(Hamiltonians)h (the)g(prop)5 b(ert)-5 b(y)74 b(SULE)g(in)h(the)f(form)f(obtained)h(in) h([14])-201 8528 y(actually)c(implies)g(\(4\).)110 b(In)71 b([15)o(],)i(an)e(ev)-5 b(en)70 b(more)f(straigh)-5 b(tforw)g(ard)71 b(argumen)-5 b(t)69 b(is)i(giv)-5 b(en.)111 b(W)-15 b(e)71 b(men)-5 b(tion)-201 8754 y(that)69 b(these)g(results)g(also)h(apply)g (to)f(the)g(random)f(dimer)h(op)5 b(erator)69 b([7)o(],)i(a)f(discrete) f(1-D)g(mo)5 b(del)68 b(where)-201 8979 y(the)60 b(Aizenman-Molc)-5 b(hano)g(v)59 b(approac)-5 b(h)60 b(fails.)81 9205 y(Besides)81 b(random,)i(another)e(imp)5 b(ortan)-5 b(t)79 b(class)h(of)g(ergo)5 b(dic)81 b(op)5 b(erators)80 b(for)h(whic)-5 b(h)81 b(results)f(on)h (the)-201 9431 y(pure)j(p)5 b(oin)-5 b(t)83 b(sp)5 b(ectrum)81 b(ha)-5 b(v)g(e)83 b(b)5 b(een)83 b(obtained)g(is)h(the)e(class)h(of)g (quasi-p)5 b(erio)g(dic)84 b(op)5 b(erators.)148 b(As)83 b(w)-5 b(ell)-201 9657 y(as)73 b(with)h(the)e(p)5 b(oin)-5 b(t)74 b(sp)5 b(ectrum)71 b(itself,)k(establishing)f(Dynamical)e(Lo)5 b(calization)73 b(in)h(this)f(case)f(requires)-201 9883 y(approac)-5 b(hes)71 b(that)h(are)g(quite)g(di\013eren)-5 b(t)72 b(from)e(that)h(of)g(the)h(random)e(case.)114 b(There)72 b(ha)-5 b(v)g(e)72 b(b)5 b(een)71 b(sev)-5 b(eral)-201 10108 y(new)44 b(imp)5 b(ortan)-5 b(t)43 b(lo)5 b(calization)45 b(results)e(for)h(quasip)5 b(erio)g(dic)45 b(mo)5 b(dels)42 b(recen)-5 b(tly:)72 b(Anderson)44 b(lo)5 b(calization)45 b(for)-201 10334 y(the)53 b(almost)f(Mathieu)h(op)5 b(erator)53 b(with)i(Diophan)-5 b(tine)53 b(frequencies)g([21)o(],)h (whic)-5 b(h)54 b(is)g(the)f(op)5 b(erator)54 b(studied)-201 10560 y(in)72 b(this)g(pap)5 b(er,)74 b(and)e(ev)-5 b(en)71 b(more)f(recen)-5 b(tly)-15 b(,)74 b(Bourgain-Goldstein)d([3])g(pro)-5 b(v)g(ed)71 b(Anderson)h(lo)5 b(calization)-201 10786 y(for)81 b(a)g(large)h(class)f(of)g(one)g(and)g(t)-5 b(w)g(o-frequencies)80 b(p)5 b(oten)-5 b(tials)81 b(in)h(the)f(regime)f (of)h(p)5 b(ositiv)-5 b(e)81 b(Ly)-5 b(apuno)g(v)-201 11012 y(exp)5 b(onen)-5 b(ts.)83 b(Also,)62 b(pure)f(p)5 b(oin)-5 b(t)62 b(sp)5 b(ectrum)59 b(w)-5 b(as)62 b(pro)-5 b(v)g(ed)62 b(b)-5 b(y)61 b(Eliasson)h([11)o(])f(for)h(an)f(ev)-5 b(en)61 b(larger)i(class)e(of)3439 11631 y(2)p eop %%Page: 3 3 3 2 bop -201 1158 a Fo(one-frequency)50 b(p)5 b(oten)-5 b(tials)52 b(at)f(high)h(disorder.)78 b(These)51 b(mo)5 b(dels)50 b(are)i(natural)g(candidates)f(for)g(dynamical)-201 1384 y(strengthening.)81 1610 y(F)-15 b(rom)64 b(the)h(dynamical)g(p)5 b(oin)-5 b(t)65 b(of)g(view,)i(for)e(quasi-p)5 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Fn(H)1073 11153 y Fl([)p Fm(x;y)5 b Fl(])p Fm(;\022)1502 11116 y Fj(\000)40 b Fn(E)10 b Fo(.)81 b(W)-15 b(e)60 b(need)h(the)f(follo)-5 b(wing)61 b(de\014nition.)3439 11631 y(3)p eop %%Page: 4 4 4 3 bop -201 1158 a Fg(De\014nition)71 b(2.1.)90 b Ff(L)-9 b(et)70 b Fn(\015)h(>)60 b Fo(0)p Ff(,)72 b Fn(k)66 b(>)61 b Fo(0)p Ff(,)71 b(and)g Fn(E)h Fj(62)61 b Fn(\033)7 b Fo(\()p Fn(H)3759 1195 y Fl([)p Fm(x;y)e Fl(])p Fm(;\022)4147 1158 y Fo(\))p Ff(.)100 b(A)71 b(p)-9 b(oint)70 b Fn(z)f Fj(2)60 b Fk(Z)88 b Ff(is)71 b(c)-9 b(al)9 b(le)-9 b(d)69 b Fo(\()p Fn(\015)10 b(;)30 b(E)10 b(;)30 b(\022)5 b(;)30 b(k)6 b Fo(\))p Ff(-)-201 1384 y(r)-9 b(e)g(gular)66 b(at)j(ener)-9 b(gy)68 b Fn(E)80 b Ff(if)68 b(ther)-9 b(e)68 b(exists)i(an)f(interval)e Fo([)p Fn(x;)30 b(y)7 b Fo(])p Ff(,)68 b(with)h Fj(j)p Fn(x)43 b Fj(\000)f Fn(y)7 b Fj(j)57 b(\024)g Fn(k)6 b Ff(,)69 b(c)-9 b(ontaining)68 b Fn(z)77 b Ff(and)69 b(such)-201 1610 y(that)2184 1627 y Fe(\014)2184 1736 y(\014)2184 1845 y(\014)2245 1836 y 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Fo(\()p Fn(x)p Fo(\))p Fn(')6580 10812 y Fm(\022)6580 10932 y(E)6698 10878 y Fo(\()p Fn(y)7 b Fo(\))p Fj(j)60 b Fo(is)-201 11104 y(to)50 b(replace)g Fn(')736 11038 y Fm(\022)736 11158 y(E)905 11104 y Fo(b)-5 b(y)50 b(the)f(generalized) i(eigenfunctions)73 b Fe(e)-125 b Fn(')3675 11038 y Fm(\022)3675 11158 y(E)3844 11104 y Fo(that)49 b(supplies)i(the)e(eigenfunctions)g (expansion)3439 11631 y(4)p eop %%Page: 5 5 5 4 bop -201 1158 a Ff(\022)-93 b(a)67 b(la)62 b Fo(Berezanskii)g([2].) 85 b(This)63 b(has)f(t)-5 b(w)g(o)63 b(crucial)g(consequences.)84 b(The)63 b(\014rst)f(one)g(is)h(that)f(the)g(generalized)-201 1384 y(eigenfunctions)90 b Fe(e)-125 b Fn(')1112 1318 y Fm(\022)1112 1438 y(E)1297 1384 y Fo(are)66 b(p)5 b(olynomially)66 b(b)5 b(ounded,)67 b(and)f(this,)h(uniformly)e(in)i(energy)e(\()p Fj(j)24 b Fe(e)-125 b Fn(')6147 1318 y Fm(\022)6147 1438 y(E)6266 1384 y Fo(\()p Fn(x)p Fo(\))p Fj(j)58 b(\024)h Fo(\(1)44 b(+)-201 1610 y Fj(j)p Fn(x)p Fj(j)p Fo(\))76 1544 y Fm(\016)151 1610 y Fo(,)67 b(see)f(\(8\))g(b)5 b(elo)-5 b(w\).)99 b(The)67 b(exp)5 b(onen)-5 b(tial)66 b(deca)-5 b(y)66 b(of)g(pro)5 b(ducts)67 b(of)f(the)g(form)f Fj(j)24 b Fe(e)-125 b Fn(')5526 1544 y Fm(\022)5526 1664 y(E)5644 1610 y Fo(\()p Fn(x)p Fo(\))24 b Fe(e)-125 b Fn(')6009 1544 y Fm(\022)6009 1664 y(E)6128 1610 y Fo(\()p Fn(y)7 b Fo(\))p Fj(j)65 b Fo(pro)-5 b(vided)-201 1836 y(b)g(y)54 b(Lemma)c(2.1)j(will)j(not)d(then)g(b)5 b(e)54 b(a\013ected)f(b)-5 b(y)53 b(that)g(c)-5 b(hange.)78 b(Second,)54 b(and)f(crucial,)j(the)d(sum)f(o)-5 b(v)g(er)54 b Fn(E)7091 1770 y Fm(\022)-201 2062 y Fo(no)-5 b(w)59 b(con)-5 b(v)g(erges.)78 b(In)59 b(other)f(w)-5 b(ords,)58 b(the)g(loss)h(of)e(con)-5 b(trol)59 b(on)f(the)g(eigenfunctions)g(\()p Fj(j)24 b Fe(e)-125 b Fn(')5786 1996 y Fm(\022)5786 2116 y(E)5904 2062 y Fo(\()p Fn(x)p Fo(\))p Fj(j)50 b(\024)g Fo(\(1)35 b(+)h Fj(j)p Fn(x)p Fj(j)p Fo(\))7095 1996 y Fm(\016)-201 2288 y Fo(instead)57 b(of)f Fj(j)p Fn(')796 2222 y Fm(\022)796 2341 y(E)914 2288 y Fo(\()p 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y(w)-5 b(e)67 b(kno)-5 b(w)68 b(that)e(under)h(the)g(h)-5 b(yp)5 b(otheses)65 b(of)i(the)f(theorem)f(the)i(sp)5 b(ectrum)65 b(of)i Fn(H)5462 4880 y Fm(!)5 b(;\025;\022)5860 4850 y Fo(is)67 b(pure)g(p)5 b(oin)-5 b(t)68 b(for)-201 5099 y(all)d Fn(\022)k Fo(in)64 b(a)f(set)885 5053 y Fe(e)865 5099 y Fo(\002)g(of)g(full)h(Leb)5 b(esgue)62 b(measure)g([21)o(].)89 b(In)63 b(addition)h(the)f(sp)5 b(ectrum)62 b(is)i(simple)e(b)-5 b(y)64 b(a)f(usual)-201 5325 y(1-D)55 b(W)-15 b(ronskian)55 b(argumen)-5 b(t.)76 b(Let)54 b(us)h(denote)g(b)-5 b(y)55 b Fn(')3422 5259 y Fm(\022)3422 5379 y(E)3596 5325 y Fo(the)f(orthonormalized)g(eigenfunctions)h(of)f Fn(H)6787 5355 y Fm(!)5 b(;\025;\022)7119 5325 y Fo(,)-201 5575 y(with)61 b(the)f(energy)h Fn(E)1252 5509 y Fm(\022)1331 5575 y Fo(,)f Fn(\022)c Fj(2)1774 5529 y Fe(e)1754 5575 y Fo(\002.)80 b(F)-15 b(urthermore,)58 b(as)i(in)h([13],)e(de\014ne) 1431 5920 y Fe(e)-126 b Fn(')1525 5845 y Fm(\022)1525 5965 y(E)1695 5920 y Fo(:=)49 b Fn(')2056 5845 y Fm(\022)2056 5965 y(E)2176 5920 y Fn(=)p Fj(k)p Fn(B)9 b(')2624 5845 y Fm(\022)2624 5965 y(E)2743 5920 y Fj(k)2834 5959 y Fm(`)2892 5921 y Fd(2)2969 5920 y Fn(;)272 b Fo(so)60 b(that)242 b Fn(')4210 5845 y Fm(\022)4210 5965 y(E)4379 5920 y Fo(=)50 b Fj(k)p Fn(B)9 b(')4928 5845 y Fm(\022)4928 5965 y(E)5047 5920 y Fj(k)5138 5959 y Fm(`)5196 5921 y Fd(2)5298 5920 y Fe(e)-126 b Fn(')5392 5845 y Fm(\022)5392 5965 y(E)5512 5920 y Fn(;)1374 b Fo(\(7\))-201 6265 y(where)64 b Fn(B)73 b Fo(is)64 b(the)f(m)-5 b(ultiplication)64 b(op)5 b(erator)63 b(b)-5 b(y)64 b Fn(b)p Fo(\()p Fn(x)p Fo(\))55 b(:=)g(\(1)42 b(+)g Fj(j)p Fn(x)p Fj(j)p Fo(\))4495 6199 y Fi(\000)p Fm(\016)4679 6265 y Fo(,)64 b Fn(\016)e(>)55 b Fo(1)p Fn(=)p Fo(2.)90 b(Note)63 b(that)88 b Fe(e)-125 b Fn(')6507 6199 y Fm(\022)6507 6319 y(E)6689 6265 y Fo(is)64 b(still)-201 6491 y(in)f Fn(`)89 6425 y Fl(2)168 6491 y Fo(\()p Fk(Z)8 b Fo(\).)94 b(As)63 b(brie\015y)g(men)-5 b(tioned)61 b(ab)5 b(o)-5 b(v)g(e,)62 b(what)g(mak)-5 b(es)60 b(those)i(eigenfunctions)g(of)g(particular)h(in)-5 b(terest)-201 6717 y(is)71 b(the)g(follo)-5 b(wing:)102 b(\014rst,)72 b Fj(k)p Fn(B)33 b Fe(e)-125 b Fn(')1970 6651 y Fm(\022)1970 6771 y(E)2089 6717 y Fj(k)2180 6756 y Fm(`)2238 6718 y Fd(2)2383 6717 y Fo(=)67 b(1)k(so)f(that)h (uniformly)f(in)h Fn(\022)76 b Fo(and)71 b(in)h(energy)e Fn(E)5936 6651 y Fm(\022)6015 6717 y Fo(,)j(the)d(follo)-5 b(wing)-201 6943 y(holds)61 b(:)2318 7168 y Fj(8)30 b Fn(x)51 b Fj(2)f Fk(Z)9 b Fn(;)99 b Fj(j)24 b Fe(e)-125 b Fn(')3208 7093 y Fm(\022)3208 7213 y(E)3327 7168 y Fo(\()p Fn(x)p Fo(\))p Fj(j)49 b(\024)h Fo(\(1)40 b(+)g Fj(j)p Fn(x)p Fj(j)p Fo(\))4526 7093 y Fm(\016)4600 7168 y Fn(:)2286 b Fo(\(8\))-201 7470 y(Second,)59 b(while)j(considering)e (a)g(sum)e(of)h(the)h(form)3373 7334 y Fe(P)3565 7524 y Fm(n)p Fi(\025)p Fl(0)3870 7470 y Fj(j)p Fn(')4040 7405 y Fm(\022)4040 7524 y(E)4158 7470 y Fo(\()p Fn(x)p Fo(\))p Fn(')4523 7405 y Fm(\022)4523 7524 y(E)4642 7470 y Fo(\()p Fn(y)7 b Fo(\))p Fj(j)58 b 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b(exclusion)g(of)g(singular)-201 9760 y(b)5 b(o)-5 b(xes)58 b(together)f(with)i(\(7\)-\(9\))e(supplies)h(the) g(k)-5 b(ey)58 b(dynamical)g(step)f(that)h(implies)g(\(Strong\))f (Dynamical)-201 9986 y(Lo)5 b(calization.)-201 10308 y Fg(Lemma)68 b(2.2.)90 b Ff(L)-9 b(et)56 b(us)h(pick)e Fn(L)50 b(>)g(L)2353 10242 y Fi(\003)2432 10308 y Fo(\()p Fn(\013)q(;)30 b(s)p Fo(\))56 b Ff(and)g Fn(y)h Fj(2)51 b Fk(Z)8 b Ff(.)90 b(Then)56 b(ther)-9 b(e)56 b(exists)h(a)g(c)-9 b(onstant)57 b Fn(C)6228 10335 y Fl(1)6307 10308 y Fo(\()p Fn(\016)m(;)30 b(\013)q(;)g(\015)6829 10335 y Fl(0)6907 10308 y Fo(\))50 b Fn(<)-201 10534 y Fj(1)p Ff(,)68 b(such)f(that)i (for)e(any)h Fn(\022)61 b Fj(62)55 b Fo(\002)2004 10562 y Fm(L)2108 10534 y Fo(\()p Fn(y)7 b Fo(\))67 b Ff(\(given)g(by)g(L)-9 b(emma)67 b(2.1\),)h(any)g Fn(E)e Fo(=)55 b Fn(E)5182 10468 y Fm(\022)5260 10534 y Ff(,)68 b(and)g(for)f(al)9 b(l)67 b Fn(x)55 b Fj(2)h Fk(Z)85 b Ff(such)-201 10760 y(that)191 10688 y Fm(L)p Fl(+1)p 191 10718 277 8 v 294 10823 a(2)538 10760 y Fj(\024)50 b(j)p Fn(x)39 b Fj(\000)h Fn(y)7 b Fj(j)50 b Fn(<)g(L)1619 10694 y Fm(\013)1717 10760 y Ff(,)65 b(the)g(fol)9 b(lowing)64 b(holds:)1328 11133 y Fj(j)p Fn(')1498 11058 y Fm(\022)1498 11178 y(E)1616 11133 y Fo(\()p Fn(x)p Fo(\))p Fn(')1981 11058 y Fm(\022)1981 11178 y(E)2100 11133 y Fo(\()p Fn(y)7 b Fo(\))p Fj(j)49 b(\024)h Fn(C)2760 11160 y Fl(1)2838 11133 y Fo(\()p Fn(\016)m(;)30 b(\013)q(;)g(\015)3360 11160 y Fl(0)3438 11133 y Fo(\))p Fj(k)p Fn(B)9 b(')3866 11058 y Fm(\022)3866 11178 y(E)3985 11133 y Fj(k)4076 11058 y Fl(2)4076 11189 y Fm(`)4134 11151 y Fd(2)4211 11133 y Fo(\(1)40 b(+)g Fj(j)p Fn(y)7 b Fj(j)p Fo(\))4864 11058 y Fl(2)p Fm(\016)5009 11133 y Fn(e)5094 11058 y Fi(\000)p Fm(\015)5277 11076 y Fd(0)5345 11058 y Fm(L=)p Fl(4)5590 11133 y Fn(:)1205 b Fo(\(10\))3439 11631 y(5)p eop %%Page: 6 6 6 5 bop -201 1158 a Ff(As)66 b(a)f(c)-9 b(onse)g(quenc)g(e,)62 b(for)i(some)h(c)-9 b(onstant)66 b Fn(C)2906 1185 y Fl(2)2984 1158 y Fo(\()p Fn(\016)m(;)30 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y(generalized)c(eigenfunctions)84 b Fe(e)-125 b Fn(')2046 2457 y Fm(\022)2046 2577 y(E)2165 2523 y Fo(:)1902 2930 y Fj(j)p Fn(')2072 2855 y Fm(\022)2072 2975 y(E)2190 2930 y Fo(\()p Fn(x)p Fo(\))p Fn(')2555 2855 y Fm(\022)2555 2975 y(E)2674 2930 y Fo(\()p Fn(y)7 b Fo(\))p Fj(j)49 b Fo(=)h Fj(k)p Fn(B)9 b(')3561 2855 y Fm(\022)3561 2975 y(E)3680 2930 y Fj(k)3771 2855 y Fl(2)3771 2986 y Fm(`)3829 2948 y Fd(2)3957 2930 y Fj(j)24 b Fe(e)-125 b Fn(')4127 2855 y Fm(\022)4127 2975 y(E)4245 2930 y Fo(\()p Fn(x)p Fo(\))24 b Fe(e)-125 b Fn(')4610 2855 y Fm(\022)4610 2975 y(E)4729 2930 y Fo(\()p Fn(y)7 b Fo(\))p Fj(j)p Fn(:)1777 b Fo(\(12\))-201 3337 y(No)-5 b(w,)64 b(from)e(Lemma)d(2.1,)k(if)g Fn(\022)e Fj(62)55 b Fo(\002)2331 3365 y Fm(L)2434 3337 y Fo(\()p Fn(y)7 b Fo(\),)63 b(then)g(at)f(least)h(one)g(of)g Fn(x)g Fo(and)g Fn(y)7 b Fo(,)63 b(with)5571 3265 y Fm(L)p Fl(+1)p 5571 3295 277 8 v 5674 3399 a(2)5922 3337 y Fj(\024)55 b(j)p Fn(x)41 b Fj(\000)h Fn(y)7 b Fj(j)54 b Fn(<)g(L)7020 3271 y Fm(\013)7119 3337 y Fo(,)-201 3563 y(is)68 b(\()p Fn(\015)154 3590 y Fl(0)233 3563 y Fn(=)p Fo(2)p Fn(;)30 b(\022)5 b(;)30 b(L)p Fo(\)-regular.)101 b(Let)67 b(us)g(denote)g(b)-5 b(y)68 b Fn(u)62 b Fo(=)g Fn(x)67 b Fo(or)h Fn(y)7 b(;)67 b Fo(the)g(regular)h(p)5 b(oin)-5 b(t,)70 b(and)d Fn(v)75 b Fo(the)67 b(other)g(one.)-201 3788 y(Then,)75 b(applying)e(the)e (iden)-5 b(tit)g(y)72 b(\(5\))g(to)96 b Fe(e)-126 b Fn(')2757 3722 y Fm(\022)2757 3842 y(E)2949 3788 y Fo(at)71 b(the)h(p)5 b(oin)-5 b(t)72 b Fn(u)p Fo(,)j(and)d(taking)g(in)-5 b(to)72 b(accoun)-5 b(t)71 b(the)h(uniform)-201 4014 y(p)5 b(olynomial)61 b(b)5 b(ound)60 b(\(8\),)g(one)g(has)990 4421 y Fj(j)24 b Fe(e)-125 b Fn(')1160 4346 y Fm(\022)1160 4466 y(E)1279 4421 y Fo(\()p Fn(u)p Fo(\))p Fj(j)49 b(\024)h Fo(2\(2)40 b(+)g Fj(j)p Fn(u)p Fj(j)g Fo(+)f Fn(L)p Fo(\))2914 4346 y Fm(\016)2990 4421 y Fn(e)3075 4346 y Fi(\000)p Fm(\015)3258 4364 y Fd(0)3326 4346 y Fm(L=)p Fl(2)3622 4421 y Fj(\024)50 b Fn(C)13 b Fo(\(1)40 b(+)f Fj(j)p 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b(in)h(an)-5 b(y)61 b(case,)e(whether)i Fn(x)f Fo(or)h Fn(y)67 b Fo(is)61 b(\()p Fn(\015)2682 5688 y Fl(0)2761 5661 y Fn(=)p Fo(2)p Fn(;)30 b(\022)5 b(;)30 b(L)p Fo(\)-regular,)59 b(one)i(gets)e(that)h (for)h(all)g Fn(\022)56 b Fj(62)51 b Fo(\002)6205 5689 y Fm(L)6308 5661 y Fo(\()p Fn(y)7 b Fo(\))279 6068 y Fj(j)24 b Fe(e)-125 b Fn(')449 5993 y Fm(\022)449 6113 y(E)568 6068 y Fo(\()p Fn(x)p Fo(\))24 b Fe(e)-125 b Fn(')933 5993 y Fm(\022)933 6113 y(E)1051 6068 y Fo(\()p Fn(y)7 b Fo(\))p Fj(j)49 b(\024)h Fn(C)13 b Fo(\(1)40 b(+)g Fj(j)p Fn(y)7 b Fj(j)p Fo(\))2377 5993 y Fl(2)p Fm(\016)2521 6068 y Fo(\(1)40 b(+)g Fn(L)p Fo(\))3100 5993 y Fl(\()p Fm(\013)p Fl(+1\))p Fm(\016)3556 6068 y Fn(e)3641 5993 y Fi(\000)p Fm(\015)3824 6011 y Fd(0)3893 5993 y Fm(L=)p Fl(2)4188 6068 y Fj(\024)50 b Fn(C)4510 6095 y Fl(1)4589 6068 y Fo(\()p Fn(\016)m(;)30 b(\013)q(;)g(\015)5111 6095 y Fl(0)5189 6068 y Fo(\)\(1)40 b(+)g Fj(j)p Fn(y)7 b Fj(j)p Fo(\))5913 5993 y Fl(2)p Fm(\016)6057 6068 y Fn(e)6142 5993 y Fi(\000)p Fm(\015)6325 6011 y Fd(0)6394 5993 y Fm(L=)p Fl(4)6639 6068 y Fn(:)-201 6475 y Fo(T)-15 b(ogether)60 b(with)h(\(12\),)f(this)g(pro)-5 b(v)g(es)60 b(\(10\).)80 b(Com)-5 b(bining)60 b(\(10\))g(with)h(\(9\),)f(w)-5 b(e)60 b(obtain)236 6909 y Fj(jh)p Fn(e)443 6834 y Fi(\000)p Fm(iH)717 6857 y Fh(\022)786 6834 y Fm(t)845 6909 y Fn(\016)926 6936 y Fm(x)1014 6909 y Fn(;)30 b(\016)1176 6936 y Fm(y)1258 6909 y Fj(ij)166 b(\024)1853 6736 y Fe(X)1894 7146 y Fm(E)2004 7108 y Fh(\022)2146 6909 y Fj(j)24 b Fe(e)-125 b Fn(')2316 6834 y Fm(\022)2316 6954 y(E)2435 6909 y Fo(\()p Fn(x)p Fo(\))24 b Fe(e)-125 b Fn(')2800 6834 y Fm(\022)2800 6954 y(E)2918 6909 y Fo(\()p Fn(y)7 b Fo(\))p Fj(j)49 b(\024)h Fn(C)3578 6936 y Fl(1)3657 6909 y Fo(\()p Fn(\016)m(;)30 b(\013)q(;)g(\015)4179 6936 y Fl(0)4257 6909 y Fo(\)\(1)40 b(+)f Fj(j)p Fn(y)7 b Fj(j)p Fo(\))4980 6834 y Fl(2)p Fm(\016)5125 6909 y Fn(e)5210 6834 y Fi(\000)p Fm(\015)5393 6852 y Fd(0)5461 6834 y Fm(L=)p Fl(4)5737 6736 y Fe(X)5778 7146 y Fm(E)5888 7108 y Fh(\022)6030 6909 y Fj(k)p Fn(B)i(')6387 6834 y Fm(\022)6387 6954 y(E)6506 6909 y Fj(k)6597 6834 y Fl(2)6597 6965 y Fm(`)6655 6927 y Fd(2)1546 7406 y Fj(\024)165 b Fn(C)1983 7433 y Fl(2)2062 7406 y Fo(\()p Fn(\016)m(;)30 b(\013)q(;)g(\015)2584 7433 y Fl(0)2662 7406 y Fo(\)\(1)40 b(+)f Fj(j)p Fn(y)7 b Fj(j)p Fo(\))3385 7331 y Fl(2)p Fm(\016)3530 7406 y Fn(e)3615 7331 y Fi(\000)p Fm(\015)3798 7349 y Fd(0)3866 7331 y Fm(L=)p Fl(4)4112 7406 y Fn(:)7028 7813 y Fb(\003)-201 8137 y Fg(Pro)f(of)69 b(of)g(Theorem)f(1.1.)81 8589 y Fo(Pic)-5 b(k)54 b(some)e Fn(\013)g Fj(2)e Fo(\(1)p Fn(;)30 b Fo(2\),)54 b(and)g(for)f(an)-5 b(y)54 b(giv)-5 b(en)54 b Fn(q)j(>)50 b Fo(0,)55 b(tak)-5 b(e)53 b Fn(s)e(>)f(\013)q Fo(\()p Fn(q)33 b Fo(+)27 b(2\).)77 b(Lemma)50 b(2.1)k(then)f(supplies) -201 8815 y(a)h(sequence)e(of)h(scales,)i Fn(L)1567 8842 y Fl(1)1696 8815 y Fo(=)50 b Fn(L)2012 8749 y Fi(\003)2090 8815 y Fo(\()p Fn(\013)q(;)30 b(s)p Fo(\),)54 b Fn(L)2744 8845 y Fm(k)t Fl(+1)3060 8815 y Fo(=)c(\(2)p Fn(L)3538 8845 y Fm(k)3649 8815 y Fj(\000)27 b Fo(1\))3980 8749 y Fm(\013)4131 8815 y Fo(\(so)53 b(that)g(if)h Fn(L)5089 8845 y Fm(k)5225 8815 y Fo(=)5436 8743 y Fm(L)p Fl(+1)p 5436 8773 V 5539 8878 a(2)5786 8815 y Fo(then)g Fn(L)6318 8845 y Fm(k)t Fl(+1)6633 8815 y Fo(=)c Fn(L)6949 8749 y Fm(\013)7048 8815 y Fo(\),)-201 9041 y(and)k(for)f(an)-5 b(y)53 b Fn(y)k Fj(2)51 b Fk(Z)70 b Fo(some)52 b(sets)g(\002)2184 9069 y Fm(L)2280 9092 y Fh(k)2364 9041 y Fo(\()p Fn(y)7 b Fo(\))52 b(suc)-5 b(h)53 b(that)g(\(6\))g(and)g(consequen)-5 b(tly)53 b(\(10\))f(hold)i(b)5 b(et)-5 b(w)g(een)53 b Fn(L)6738 9071 y Fm(k)6876 9041 y Fo(and)-201 9267 y Fn(L)-77 9297 y Fm(k)t Fl(+1)189 9267 y Fo(,)60 b Fn(k)c Fj(\025)50 b Fo(1.)80 b(Hence)60 b(one)g(has)h(for)f(all)i Fn(t)50 b Fj(2)g Fk(R)27 b Fo(,)69 b Fn(k)56 b Fj(\025)50 b Fo(1)60 b(and)h Fn(x;)30 b(y)67 b Fo(so)60 b(that)g Fn(L)5054 9297 y Fm(k)5189 9267 y Fj(\024)50 b(j)p Fn(x)40 b Fj(\000)g Fn(y)7 b Fj(j)50 b Fn(<)g(L)6271 9297 y Fm(k)t Fl(+1)6536 9267 y Fo(,)880 9510 y Fe(Z)981 9922 y Fc(T)1108 9757 y Fo(d)p Fn(\022)1330 9610 y Fe(\000)1413 9757 y Fj(jh)p Fn(e)1620 9682 y Fi(\000)p Fm(iH)1894 9705 y Fh(\022)1963 9682 y Fm(t)2022 9757 y Fn(\016)2103 9784 y Fm(x)2191 9757 y Fn(;)30 b(\016)2353 9784 y Fm(y)2435 9757 y Fj(ij)2557 9610 y Fe(\001)2691 9757 y Fj(\024)49 b Fn(C)3012 9784 y Fl(2)3091 9757 y Fo(\()p Fn(\016)m(;)30 b(\013)q(;)g(\015)3613 9784 y Fl(0)3691 9757 y Fo(\)\(1)40 b(+)g Fj(j)p Fn(y)7 b Fj(j)p Fo(\))4415 9682 y Fl(2)p Fm(\016)4559 9757 y Fn(e)4644 9682 y Fi(\000)p Fm(\015)4827 9700 y Fd(0)4896 9682 y Fm(L)4992 9705 y Fh(k)5068 9682 y Fm(=)p Fl(4)5258 9757 y Fo(+)40 b(9)p Fn(L)5655 9681 y Fi(\000)p Fm(s)p Fl(+)p Fm(\013)5655 9816 y(k)6038 9757 y Fn(:)757 b Fo(\(15\))-201 10256 y(Indeed,)54 b(splitting)f(the)g (in)-5 b(tegral)53 b(in)h(t)-5 b(w)g(o)52 b(pieces)h(:)77 b(the)52 b(\014rst)h(one)f(o)-5 b(v)g(er)53 b(\002)4814 10190 y Fm(c)4814 10310 y(L)4910 10333 y Fh(k)4994 10256 y Fo(\()p Fn(y)7 b Fo(\))52 b(and)h(the)g(second)f(one)h(o)-5 b(v)g(er)-201 10482 y(\002)-59 10510 y Fm(L)37 10533 y Fh(k)121 10482 y Fo(\()p Fn(y)7 b Fo(\),)59 b(one)i(has,)e(from)g (Lemma)e(2.1)j(that,)g(pro)-5 b(vided)61 b Fn(L)3866 10512 y Fm(k)4001 10482 y Fj(\024)50 b(j)p Fn(x)40 b Fj(\000)g Fn(y)7 b Fj(j)49 b Fn(<)h(L)5082 10512 y Fm(k)t Fl(+1)5348 10482 y Fo(,)60 b Fn(k)c Fj(\025)50 b Fo(1,)1076 10731 y Fe(Z)1177 11143 y Fc(T)1304 10979 y Fo(d)p Fn(\022)1526 10831 y Fe(\000)1610 10979 y Fj(jh)p Fn(e)1817 10904 y Fi(\000)p Fm(iH)2091 10927 y Fh(\022)2159 10904 y Fm(t)2219 10979 y Fn(\016)2300 11006 y Fm(x)2387 10979 y Fn(;)30 b(\016)2549 11006 y Fm(y)2632 10979 y Fj(ij)2754 10831 y Fe(\001)2887 10979 y Fj(\024)3079 10731 y Fe(Z)3180 11143 y Fl(\002)3290 11098 y Fh(c)3290 11190 y(L)3374 11225 y(k)3458 11143 y Fl(\()p Fm(x)p Fl(\))3686 10979 y Fj(jh)p Fn(e)3893 10904 y Fi(\000)p Fm(iH)4167 10927 y Fh(\022)4236 10904 y Fm(t)4295 10979 y Fn(\016)4376 11006 y Fm(x)4464 10979 y Fn(;)g(\016)4626 11006 y Fm(y)4708 10979 y Fj(ij)p Fo(d)p Fn(\022)45 b Fo(+)40 b(9)p Fn(L)5458 10902 y Fi(\000)p Fm(s)p Fl(+)p Fm(\013)5458 11037 y(k)5842 10979 y Fn(:)3439 11631 y Fo(6)p eop %%Page: 7 7 7 6 bop -201 1158 a Fo(W)-15 b(e)85 b(used)g Fj(jh)p Fn(e)784 1092 y Fi(\000)p Fm(iH)1058 1115 y Fh(\022)1127 1092 y Fm(t)1187 1158 y Fn(x;)30 b(\016)1453 1185 y Fm(y)1535 1158 y Fj(ij)91 b(\024)h Fo(1)85 b(for)g(the)g(second)g(term.)154 b(Lemma)81 b(2.2)k(then)g(yields)h(\(15\).)155 b(One)85 b(can)-201 1384 y(therefore)60 b(end)g(the)g(pro)5 b(of)61 b(of)f(the)g(theorem)e(as)j(follo)-5 b(ws:)185 1554 y Fe(Z)286 1966 y Fc(T)413 1801 y Fo(d)p Fn(\022)635 1592 y Fe(\015)635 1701 y(\015)635 1810 y(\015)766 1801 y Fj(j)p Fn(X)14 b Fj(j)1033 1726 y Fm(q)5 b(=)p Fl(2)1249 1801 y Fn(e)1334 1726 y Fi(\000)p Fm(iH)1608 1749 y Fh(\022)1678 1726 y Fm(t)1737 1801 y Fn( )1863 1592 y Fe(\015)1863 1701 y(\015)1863 1810 y(\015)1963 1645 y Fl(2)533 2224 y Fj(\024)840 2051 y Fe(X)843 2445 y Fm(y)g Fi(2)p Fc(Z)1133 2224 y Fj(j)p Fn( )i Fo(\()p Fn(y)g Fo(\))p Fj(j)1629 2076 y Fe(\002)1702 2224 y Fo(2\()p Fj(j)p Fn(y)g Fj(j)39 b Fo(+)h Fn(L)2407 2148 y Fi(\003)2485 2224 y Fo(\()p Fn(\013)q(;)30 b(s)p Fo(\)\))2981 2148 y Fm(q)3097 2224 y Fo(+)40 b Fj(jh)p Fn(e)3486 2148 y Fi(\000)p Fm(iH)3760 2171 y Fh(\022)3829 2148 y Fm(t)3888 2224 y Fn( )7 b(;)79 b Fj(j)p Fn(X)14 b Fj(j)4411 2148 y Fm(q)4486 2224 y Fn(e)4571 2148 y Fi(\000)p Fm(iH)4845 2171 y Fh(\022)4915 2148 y Fm(t)4974 2224 y Fn(\016)5055 2251 y Fm(y)5138 2224 y Fj(ij)5260 2076 y Fe(\003)533 2929 y Fj(\024)840 2756 y Fe(X)843 3150 y Fm(y)5 b Fi(2)p Fc(Z)1133 2929 y Fj(j)p Fn( )i Fo(\()p Fn(y)g Fo(\))p Fj(j)1629 2563 y Fe(2)1629 2890 y(4)1747 2929 y Fo(2\()p Fj(j)p Fn(y)g Fj(j)39 b Fo(+)h Fn(L)2452 2853 y Fi(\003)2531 2929 y Fo(\()p Fn(\013)q(;)30 b(s)p Fo(\)\))3027 2853 y Fm(q)3142 2929 y Fo(+)3324 2756 y Fe(X)3327 3151 y Fm(k)t Fi(\025)p Fl(1)4016 2756 y Fe(X)3617 3158 y Fm(L)3713 3181 y Fh(k)3789 3158 y Fi(\024j)p Fm(x)p Fi(\000)p Fm(y)5 b Fi(j)p Fm()f(\013)61 b Fo(and)g(let)g(us)f(de\014ne,)g(for)g Fn(L)50 b Fj(\025)g Fo(1)1248 9740 y Fe(e)1227 9786 y Fo(\002)1369 9691 y Fl(\(1\))1369 9844 y Fm(L)1608 9786 y Fo(:=)1850 9530 y Fe(\032)1987 9786 y Fn(\022)56 b Fj(2)50 b Fk(T)74 b Fj(j)50 b(9j)p Fn(j)10 b Fj(j)50 b(\024)g Fo(2)p Fn(L)3317 9711 y Fm(\013)3415 9786 y Fn(;)3607 9523 y Fe(\014)3607 9632 y(\014)3607 9741 y(\014)3607 9850 y(\014)3668 9786 y Fo(sin)31 b(2)p Fn(\031)4153 9530 y Fe(\022)4287 9786 y Fn(\022)46 b Fo(+)4622 9663 y Fn(j)p 4619 9745 91 8 v 4619 9911 a Fo(2)4730 9786 y Fn(!)4850 9530 y Fe(\023)4984 9523 y(\014)4984 9632 y(\014)4984 9741 y(\014)4984 9850 y(\014)5095 9786 y Fj(\024)5360 9663 y Fo(1)p 5307 9745 198 8 v 5307 9911 a Fn(L)5431 9859 y Fm(s)5524 9530 y Fe(\033)5691 9786 y Fn(:)1104 b Fo(\(17\))-201 10303 y(Since)66 b(for)f(an)-5 b(y)66 b(\014xed)f Fn(j)k Fj(2)59 b Fk(Z)9 b Fo(,)1918 10093 y Fe(\014)1918 10203 y(\014)1918 10312 y(\014)1979 10303 y Fj(f)p Fn(\022)56 b Fj(2)50 b Fk(T)74 b Fj(j)2635 10093 y Fe(\014)2635 10203 y(\014)2635 10312 y(\014)2695 10303 y Fo(sin)31 b(2)p Fn(\031)3180 10101 y Fe(\020)3289 10303 y Fn(\022)46 b Fo(+)3624 10220 y Fm(j)p 3621 10261 71 8 v 3621 10365 a Fl(2)3712 10303 y Fn(!)3832 10101 y Fe(\021)3940 10093 y(\014)3940 10203 y(\014)3940 10312 y(\014)4051 10303 y Fj(\024)k Fn(L)4367 10237 y Fi(\000)p Fm(s)4550 10303 y Fj(g)4641 10093 y Fe(\014)4641 10203 y(\014)4641 10312 y(\014)4761 10303 y Fj(\024)58 b Fo(2)p Fn(L)5176 10237 y Fi(\000)p Fm(s)5359 10303 y Fo(,)66 b(one)g(immediately)d(gets)-201 10646 y(that)198 10437 y Fe(\014)198 10546 y(\014)198 10655 y(\014)278 10600 y(e)258 10646 y Fo(\002)400 10551 y Fl(\(1\))400 10704 y Fm(L)588 10437 y Fe(\014)588 10546 y(\014)588 10655 y(\014)707 10646 y Fj(\024)57 b Fo(8)p Fn(L)1121 10580 y Fi(\000)p Fm(s)p Fl(+)p Fm(\013)1505 10646 y Fo(.)94 b(Then)65 b(de\014ne)g(the)f(set)h(of)f(\\bad")h (angles)g Fn(\022)71 b Fo(asso)5 b(ciated)64 b(to)h Fn(L)58 b Fj(\025)f Fo(1)65 b(and)g Fn(y)g Fj(2)58 b Fk(Z)-201 10907 y Fo(as)i(follo)-5 b(ws:)2670 11133 y(\002)2812 11037 y Fl(\(1\))2812 11190 y Fm(L)3000 11133 y Fo(\()p Fn(y)7 b Fo(\))50 b(=)3500 11087 y Fe(e)3480 11133 y Fo(\002)3622 11037 y Fl(\(1\))3622 11190 y Fm(L)3850 11133 y Fj(\000)40 b Fn(y)7 b(!)g(:)2547 b Fo(\(18\))3439 11631 y(7)p eop %%Page: 8 8 8 7 bop -201 1158 a Fo(Ob)-5 b(viously)-15 b(,)60 b(one)g(still)i(has) 2775 1241 y Fe(\014)2775 1351 y(\014)2775 1460 y(\014)2835 1451 y Fo(\002)2977 1355 y Fl(\(1\))2977 1508 y Fm(L)3165 1451 y Fo(\()p Fn(y)7 b Fo(\))3403 1241 y Fe(\014)3403 1351 y(\014)3403 1460 y(\014)3514 1451 y Fj(\024)3879 1328 y Fo(8)p 3726 1409 398 8 v 3726 1575 a Fn(L)3850 1523 y Fm(s)p Fi(\000)p Fm(\013)4143 1451 y Fn(:)2652 b Fo(\(19\))81 1818 y(Lemma)57 b(2.1)j(will)j(b)5 b(e)60 b(a)h(consequence)e(of)h(the)g(follo)-5 b(wing)62 b(t)-5 b(w)g(o)60 b(lemmas.)-201 2181 y Fg(Lemma)68 b(3.1.)90 b([21)o(])60 b Ff(L)-9 b(et)61 b Fn(")50 b Fj(2)h Fo(\(0)p Fn(;)30 b(\015)2296 2208 y Fl(0)2374 2181 y Fn(=)p Fo(2\))p Ff(,)61 b Fn(\013)51 b Fj(2)g Fo(\(1)p Fn(;)30 b Fo(2\))p Ff(,)60 b Fn(s)51 b(>)f(\013)61 b Ff(and)g Fn(!)67 b Ff(b)-9 b(e)59 b(a)i(Diophantine)f(irr)-9 b(ational.)81 b(L)-9 b(et)-201 2439 y Fn(L)76 b Ff(b)-9 b(e)75 b(a)h(p)-9 b(ositive)75 b(inte)-9 b(ger)75 b(and)h(assume)g(that)h(ther)-9 b(e)75 b(exists)i(a)f(set)g Fo(\002)4654 2343 y Fl(\(2\))4654 2496 y Fm(L)4918 2439 y Ff(such)g(that)h(for)e(al)9 b(l)76 b Fn(k)f Fj(\025)70 b Fn(L)76 b Ff(and)-201 2708 y Fn(\022)56 b Fj(62)51 b Fo(\002)254 2612 y Fl(\(2\))254 2765 y Fm(L)442 2708 y Ff(,)65 b Fj(k)p Fn(M)831 2738 y Fm(k)915 2708 y Fo(\()p Fn(\022)5 b(;)30 b(E)10 b Fo(\))p Fj(k)51 b(\024)f Fn(e)1791 2642 y Fm(k)t Fl(\()p Fm(\015)8 b Fl(\()p Fm(E)f Fl(\)+)p Fm(")p Fl(\))2462 2708 y Ff(.)-201 2934 y(Then)87 b(ther)-9 b(e)85 b(exists)j Fn(L)1413 2961 y Fl(1)1491 2934 y Fo(\()p Fn(";)30 b(\013)q(;)g(!)7 b Fo(\))85 b Ff(such)i(that)g(for)f(any)h Fn(L)i Fj(\025)g Fn(L)4281 2961 y Fl(1)4360 2934 y Fo(\()p Fn(";)30 b(\013)q(;)g(!)7 b Fo(\))85 b Ff(and)i(any)g(given)e Fn(y)93 b Ff(and)87 b Fn(\022)95 b Fj(62)-201 3003 y Fe(\020)-92 3205 y Fo(\002)50 3109 y Fl(\(1\))50 3262 y Fm(L)238 3205 y Fo(\()p Fn(y)7 b Fo(\))39 b Fj([)i Fo(\002)819 3109 y Fl(\(2\))819 3262 y Fm(L)1007 3003 y Fe(\021)1116 3205 y Ff(,)64 b(wher)-9 b(e)64 b Fo(\002)1892 3109 y Fl(\(1\))1892 3262 y Fm(L)2080 3205 y Fo(\()p Fn(y)7 b Fo(\))64 b Ff(is)h(given)f(by)h(\(18\),)g(the)g (fol)9 b(lowing)64 b(holds)g(for)h(al)9 b(l)64 b Fn(E)10 b Ff(:)8 3514 y Fe(\022)193 3771 y Fn(x)50 b Fo(and)h Fn(y)57 b Fo(are)50 b(\()p Fn(\015)10 b Fo(\()p Fn(E)g Fo(\))41 b Fj(\000)e Fn(";)30 b(E)10 b(;)30 b(\022)5 b(;)30 b(L)p Fo(\))41 b Fj(\000)e Fo(singular)66 b(and)51 b Fj(j)p Fn(x)40 b Fj(\000)f Fn(y)7 b Fj(j)50 b Fn(>)4614 3648 y(L)40 b Fo(+)g(1)p 4614 3729 437 8 v 4787 3896 a(2)5071 3514 y Fe(\023)5255 3771 y Fo(=)-30 b Fj(\))49 b(j)p Fn(x)40 b Fj(\000)g Fn(y)7 b Fj(j)50 b Fn(>)g(L)6488 3696 y Fm(\013)6586 3771 y Fn(:)209 b Fo(\(20\))-201 4264 y Fg(Lemma)68 b(3.2.)90 b([3)o(,)72 b(16)o(])66 b Ff(Assume)g(that)h Fn(!)73 b Ff(is)67 b(Diophantine.)87 b(Then)66 b(for)g Fn(\033)59 b(<)53 b Fo(1)p Fn(=)p Fo(2)p Ff(,)66 b(ther)-9 b(e)65 b(exists)i Fn(L)p Fo(\()p Fn(\033)l(;)30 b(!)7 b Fo(\))-201 4490 y Ff(such)65 b(that,)g(for)g(al)9 b(l)64 b Fn(E)10 b Ff(,)65 b Fn(k)56 b Fj(\025)50 b Fn(L)p Fo(\()p Fn(\033)l(;)30 b(!)7 b Fo(\))64 b Ff(implies)1344 4719 y Fe(\014)1344 4828 y(\014)1344 4938 y(\014)1344 5047 y(\014)1405 4727 y(\032)1541 4983 y Fn(\022)56 b Fj(2)51 b Fo(\002)2090 4719 y Fe(\014)2090 4828 y(\014)2090 4938 y(\014)2090 5047 y(\014)2176 4860 y Fo(1)p 2171 4941 101 8 v 2171 5108 a Fn(k)2322 4983 y Fo(ln)30 b Fj(k)p Fn(M)2771 5013 y Fm(k)2856 4983 y Fo(\()p Fn(\022)5 b(;)30 b(E)10 b Fo(\))p Fj(k)41 b(\000)e Fn(\015)10 b Fo(\()p Fn(E)g Fo(\))4016 4719 y Fe(\014)4016 4828 y(\014)4016 4938 y(\014)4016 5047 y(\014)4128 4983 y Fj(\025)4391 4860 y Fo(1)p 4340 4941 195 8 v 4340 5108 a Fn(k)4441 5055 y Fm(\033)4554 4727 y Fe(\033)4690 4719 y(\014)4690 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b(precise)i(what)g(these)g (required)g(\\sligh)-5 b(t)78 b(mo)5 b(di\014cations")-201 6605 y(are.)80 b(W)-15 b(e)61 b(\014rst)f(sho)-5 b(w)60 b(ho)-5 b(w)61 b(to)f(deriv)-5 b(e)61 b(Lemma)c(2.1)j(and)h(then)f (turn)g(to)h(those)e(commen)-5 b(ts.)-201 6926 y Fg(Pro)6 b(of)69 b(of)g(Lemma)e(2.1)p Fo(.)-201 7151 y(Let)76 b Fn(\013)q(;)30 b(!)7 b(;)30 b(s)77 b Fo(and)g Fn(")g Fo(b)5 b(e)77 b(as)f(in)i(Lemma)73 b(3.1.)129 b(Lemma)73 b(3.2)k(will)i(pro)-5 b(vide)77 b(the)f(con)-5 b(trol)77 b(on)g Fj(k)p Fn(M)6536 7181 y Fm(k)6621 7151 y Fo(\()p Fn(\022)5 b(;)30 b(E)10 b Fo(\))p Fj(k)-201 7409 y Fo(that)65 b(Lemma)c(3.1)k(requires.)94 b(De\014ne)64 b(\002)2626 7439 y Fm(k)2769 7409 y Fo(:=)57 b Fj(f)p Fn(\022)63 b Fj(2)58 b Fk(T)3654 7254 y Fe(\014)3654 7364 y(\014)3738 7337 y Fl(1)p 3735 7367 78 8 v 3735 7472 a Fm(k)3862 7409 y Fo(ln)31 b Fj(k)p Fn(M)4312 7439 y Fm(k)4396 7409 y Fo(\()p Fn(\022)5 b(;)30 b(E)10 b Fo(\))p Fj(k)41 b(\000)f Fn(\015)10 b Fo(\()p Fn(E)g Fo(\))5557 7254 y Fe(\014)5557 7364 y(\014)5676 7409 y Fj(\025)5940 7337 y Fl(1)p 5895 7367 161 8 v 5895 7472 a Fm(k)5972 7434 y Fh(\033)6075 7409 y Fj(g)65 b Fo(and)g(\002)6731 7313 y Fl(\(2\))6731 7466 y Fm(L)6977 7409 y Fo(:=)-201 7635 y Fj([)-80 7665 y Fm(k)t Fi(\025)p Fm(L)212 7635 y Fo(\002)354 7665 y Fm(k)438 7635 y Fo(.)94 b(Lemma)61 b(3.2)j(then)h(pro)-5 b(vides)65 b(a)g(scale)f Fn(L)3394 7662 y Fl(2)3473 7635 y Fo(\()p Fn(\033)l(;)30 b(!)7 b(;)30 b(")p Fo(\),)64 b(suc)-5 b(h)65 b(that)f(for)h(an)-5 b(y)65 b Fn(L)57 b Fj(\025)g Fn(L)6145 7662 y Fl(2)6224 7635 y Fo(\()p Fn(\033)l(;)30 b(!)7 b(;)30 b(")p Fo(\))64 b(one)-201 7861 y(has)-201 8086 y(a\))c Fj(j)p Fo(\002)214 7991 y Fl(\(2\))214 8144 y Fm(L)402 8086 y Fj(j)50 b(\024)g Fn(C)825 8113 y Fm(\033)919 8086 y Fn(e)1004 8020 y Fi(\000)p Fm(L)1210 7974 y Fd(1)p Fa(\000)p Fd(2)p Fh(\033)1518 8086 y Fo(,)60 b(and)-201 8312 y(b\))828 8570 y Fj(8)51 b Fn(\022)56 b Fj(62)50 b Fo(\002)1434 8474 y Fl(\(2\))1434 8627 y Fm(L)1622 8570 y Fn(;)30 b Fj(8)51 b Fn(E)10 b(;)91 b(k)56 b Fj(\025)50 b Fn(L)g Fo(=)-30 b Fj(\))49 b(k)p Fn(M)3269 8600 y Fm(k)3354 8570 y Fo(\()p Fn(\022)5 b(;)30 b(E)10 b Fo(\))p Fj(k)51 b(\024)f Fn(e)4230 8495 y Fm(k)t Fl(\()p Fm(\015)8 b Fl(\()p Fm(E)f Fl(\)+)p Fm(k)4850 8448 y Fa(\000)p Fh(\033)5028 8495 y Fl(\))5142 8570 y Fj(\024)50 b Fn(e)5419 8495 y Fm(k)t Fl(\()p Fm(\015)8 b Fl(\()p Fm(E)f Fl(\)+)p Fm(")p Fl(\))6090 8570 y Fn(:)705 b Fo(\(22\))-201 8939 y(This)74 b(means)d(that)h(Lemma)e(3.1)i(applies) i(for)e Fn(L)f Fj(\025)g Fn(L)3611 8966 y Fl(2)3690 8939 y Fo(\()p Fn(\033)l(;)30 b(!)7 b(;)30 b(")p Fo(\))71 b(and)i(the)g(set)f(of)h(phases)f(\002)6310 8844 y Fl(\(2\))6310 8997 y Fm(L)6498 8939 y Fo(.)118 b(De\014ne)-201 9209 y(\002)-59 9237 y Fm(L)45 9209 y Fo(\()p Fn(y)7 b Fo(\))49 b(:=)h(\002)717 9113 y Fl(\(1\))717 9266 y Fm(L)905 9209 y Fo(\()p Fn(y)7 b Fo(\))39 b Fj([)i Fo(\002)1486 9113 y Fl(\(2\))1486 9266 y Fm(L)1674 9209 y Fo(.)80 b(Th)-5 b(us)61 b(one)f(has)h(for)f Fn(L)g 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b(!)7 b(;)30 b(";)g(s)p Fo(\))75 b(=)-201 10587 y(max)o(\()p Fn(L)333 10614 y Fl(1)411 10587 y Fo(\()p Fn(\013)q(;)30 b(!)7 b(;)30 b(")p Fo(\))p Fn(;)g(L)1242 10614 y Fl(3)1319 10587 y Fo(\()p Fn(\033)l(;)g(\013)q(;)g(!)7 b(;)30 b(";)g(s)p Fo(\)\),)77 b(but)d(with)i(the)f(rate)g Fn(\015)4078 10614 y Fl(0)4157 10587 y Fn(=)p Fo(2)g(instead)f(of)h Fn(\015)10 b Fo(\()p Fn(E)g Fo(\))51 b Fj(\000)e Fn(")p Fo(.)124 b(This)76 b(in)f(turn)-201 10813 y(implies)60 b(Lemma)d(2.1.)5679 b Fb(\003)81 11133 y Fo(W)-15 b(e)61 b(no)-5 b(w)60 b(come)f(bac)-5 b(k)60 b(to)h(Lemma)c(3.1)i(and)i(Lemma) c(3.2.)3439 11631 y(8)p eop %%Page: 9 9 9 8 bop 81 1158 a Fo(Lemma)58 b(3.1)h(is)i(a)g(sort)f(of)g(in)-5 b(termediate)59 b(statemen)-5 b(t)58 b(b)5 b(et)-5 b(w)g(een)60 b(Lemma)d(2.1)j(and)g(Lemma)d(4)k(of)f([21)o(].)-201 1384 y(The)76 b(di\013erence)g(b)5 b(et)-5 b(w)g(een)76 b(our)g(Lemma)c(3.1)j(and)h(Lemma)c(4)k(of)f([21])g(is)h(slim)f(but)h (crucial.)127 b(T)-15 b(o)76 b(get)f(a)-201 1610 y(con)-5 b(trol)77 b(on)g(the)g(dep)5 b(endency)76 b(on)h Fn(\022)5 b Fo(,)81 b(the)c(idea)g(is)h(to)e(a)-5 b(v)g(oid)78 b(the)e(Borel-Can)-5 b(telli)78 b(Lemma)c(that)i(leads)-201 1836 y(to)c(a)h(set)f(\002)g(of)g(full)h(measure)e(in)i([19)o(,)f(20,)g (21,)g(22,)g(13],)i(and)f(to)f(\014nd)g(for)h(a)f(giv)-5 b(en)73 b(scale)f Fn(L)h Fo(a)f(suitable)-201 2093 y(non-resonan)-5 b(t)61 b(set)g(of)h Fn(\022)67 b Fo(with)c(a)e(measure)f(close)i(to)g (one.)84 b(This)62 b(will)i(precisely)f(b)5 b(e)61 b(the)h(set)f(\002) 6378 1998 y Fl(\(1\))6378 2151 y Fm(L)6566 2093 y Fo(\()p Fn(y)7 b Fo(\))61 b(\(see)-201 2363 y(\(18\))53 b(and)g(\(19\)\).)77 b(More)54 b(tec)-5 b(hnically)54 b(w)-5 b(e)53 b(set)g(\002)3090 2267 y Fl(\(1\))3090 2420 y Fm(L)3278 2363 y Fo(\()p Fn(y)7 b Fo(\))53 b(is)g(de\014ned)h(so)f(that)g(the)g(conclusion)h(of) f(Lemma)c(13)-201 2589 y(in)j([21)o(])f(holds)g(for)g(an)-5 b(y)51 b(scale)g(large)g(enough,)h(uniformly)e(b)5 b(oth)51 b(in)h Fn(x)f Fo(suc)-5 b(h)50 b(that)5337 2517 y Fm(L)p Fl(+1)p 5337 2547 277 8 v 5440 2651 a(2)5683 2589 y Fj(\024)g(j)p Fn(x)21 b Fj(\000)g Fn(y)7 b Fj(j)50 b Fn(<)g(L)6727 2523 y Fm(\013)6876 2589 y Fo(and)-201 2863 y(in)71 b Fn(\022)g Fj(62)c Fo(\002)507 2767 y Fl(\(1\))507 2921 y Fm(L)695 2863 y Fo(\()p Fn(y)7 b Fo(\).)107 b(As)70 b(one)g(can)g(see)f(from)f(its)i(de\014nition)h(\(18\),)g(the)e(set)h (of)f(resonances)g(\002)6209 2767 y Fl(\(1\))6209 2921 y Fm(L)6397 2863 y Fo(\()p Fn(y)7 b Fo(\))69 b(where)-201 3089 y(Lemma)49 b(13)k(of)f([21)o(])h(fails)f(do)5 b(es)53 b(dep)5 b(end)52 b(on)h Fn(y)7 b Fo(.)77 b(The)53 b(crucial)g(p)5 b(oin)-5 b(t)53 b(for)f(our)h(pro)5 b(of)53 b(of)f(Strong)g(Dynamical) -201 3315 y(Lo)5 b(calization)61 b(\(see)f(e.g.)79 b(Eq.)h(\(15\)\))60 b(is)h(that)e(its)i Ff(me)-9 b(asur)g(e)59 b Fo(do)5 b(es)60 b(not)g(dep)5 b(end)60 b(on)h Fn(y)7 b Fo(.)81 3541 y(Lemma)75 b(3.2)i(tak)-5 b(es)77 b(care)g(of)h(the)f(\(non\))g (dep)5 b(endency)77 b(on)h(the)f(energy)h Fn(E)88 b Fo(of)78 b(the)f(constan)-5 b(t)76 b Fn(L)6866 3475 y Fi(\003)7023 3541 y Fo(of)-201 3767 y(Lemma)49 b(2.1.)76 b(Actually)53 b(the)f(pro)5 b(of)52 b(of)f(Dynamical)g(Lo)5 b(calization)52 b(in)h([22)o(,)f(13])f(could)i(only)f(handle)g(v)-10 b(alues)-201 3992 y(of)42 b Fn(\025)50 b Fj(\025)g Fo(15)42 b(\(and)g(not)g Fn(\025)51 b(>)f Fo(2\),)45 b(in)e(part,)i(b)5 b(ecause)41 b(of)h(the)g(lac)-5 b(k)42 b(of)g(information)f(on)h(the)g (rate)g(of)f(con)-5 b(v)g(ergence)-201 4218 y(in)69 b(\(16\).)104 b(The)69 b(missing)e(information)h(is)g(actually)h(pro)-5 b(vided)69 b(b)-5 b(y)69 b(the)f(com)-5 b(bination)67 b(of)h(the)h(t)-5 b(w)g(o)68 b(recen)-5 b(t)-201 4444 y(w)g(orks)61 b([3)o(])f(and)h([16)o(],)f(whic)-5 b(h)61 b(establish)g(a)f(large)h(deviation)g(theorem)e(in)i(the)f(con)-5 b(text)60 b(of)g(quasi-p)5 b(erio)g(dic)-201 4670 y(p)g(oten)-5 b(tials)76 b([3],)j(and)d(the)g(con)-5 b(tin)g(uit)g(y)76 b(of)g(the)g(Ly)-5 b(apuno)g(v)75 b(exp)5 b(onen)-5 b(t)76 b Fn(\015)10 b Fo(\()p Fn(E)g Fo(\))76 b([16].)127 b(Indeed,)79 b(the)c(reader)-201 4896 y(will)58 b(easily)e(con)-5 b(vince)55 b(himself)g(that)g(Lemma)d(3.2)j(ab)5 b(o)-5 b(v)g(e)55 b(follo)-5 b(ws)56 b(directly)g(form)e(Lemma)e(1.1)j(in)i ([3)o(])e(and)-201 5121 y(Lemma)61 b(4.2)j(in)i([16)o(].)92 b(Ho)-5 b(w)g(ev)g(er,)66 b(the)e(Diophan)-5 b(tine)65 b(condition)g(that)f(is)h(used)g(in)g([16)o(])f(is)h(stronger)g(than) -201 5347 y(the)52 b(one)g(w)-5 b(e)52 b(allo)-5 b(w)g(ed)53 b(ourselv)-5 b(es)51 b(in)i(the)f(presen)-5 b(t)51 b(pap)5 b(er)52 b(\(see)f(\(2\)\).)77 b(Ho)-5 b(w)g(ev)g(er,)53 b(unlik)-5 b(e)53 b(in)f([16],)h(w)-5 b(e)52 b(do)g(not)-201 5573 y(need)66 b(sharp)g(estimates)e(on)i(the)g(sp)5 b(eed)65 b(of)h(con)-5 b(v)g(ergence)65 b(of)4080 5502 y Fl(1)p 4077 5532 78 8 v 4077 5636 a Fm(k)4204 5427 y Fe(R)4290 5637 y Fc(T)4417 5573 y Fo(ln)31 b Fj(k)p Fn(M)4867 5603 y Fm(k)4951 5573 y Fo(\()p Fn(\022)5 b(;)30 b(E)10 b Fo(\))p Fj(k)p Fo(d)p Fn(\022)72 b Fo(to)-5 b(w)g(ard)66 b Fn(\015)10 b Fo(\()p Fn(E)g Fo(\))67 b(\(but)-201 5799 y(just)k(the)h(uniformit)-5 b(y)71 b(in)h Fn(E)10 b Fo(\).)115 b(That)72 b(means)e(one)i(can)g(adapt)f(the)h(pro)5 b(of)71 b(of)h(Lemma)c(4.2)k(of)f([16)o(],)j(and)-201 6025 y(get,)60 b(under)g(our)h(assumption)e(\(2\),)g(that,)h(for)g(an) -5 b(y)60 b(\014xed)h Fn(")50 b(>)g Fo(0)60 b(and)h Fn(E)10 b Fo(,)1937 6269 y Fe(\014)1937 6379 y(\014)1937 6488 y(\014)1937 6597 y(\014)2023 6410 y Fo(1)p 2018 6491 101 8 v 2018 6658 a Fn(k)2169 6286 y Fe(Z)2270 6698 y Fc(T)2397 6533 y Fo(ln)31 b Fj(k)p Fn(M)2847 6563 y Fm(k)2931 6533 y Fo(\()p Fn(\022)5 b(;)30 b(E)10 b Fo(\))p Fj(k)p Fo(d)p Fn(\022)46 b Fj(\000)40 b Fn(\015)10 b Fo(\()p Fn(E)g Fo(\))4283 6269 y Fe(\014)4283 6379 y(\014)4283 6488 y(\014)4283 6597 y(\014)4395 6533 y Fj(\024)4682 6410 y Fn(C)4812 6437 y Fm(")p 4606 6491 355 8 v 4606 6658 a Fn(k)4707 6605 y Fl(1)p Fi(\000)p Fm(")4981 6533 y Fn(;)1814 b Fo(\(23\))-201 7041 y(where)59 b Fn(C)453 7068 y Fm(")584 7041 y Fo(is)g(a)f(constan)-5 b(t)57 b(uniform)g(in)i Fn(E)10 b Fo(.)80 b(Again)59 b(getting)f(\(23\))g(do)5 b(es)58 b(not)g(require)g(an)-5 b(y)59 b(argumen)-5 b(ts)56 b(but)-201 7267 y(the)62 b(one)f(already)h(con)-5 b(tained)62 b(in)g([16)o(].)84 b(The)62 b(estimate)d(\(23\),)i(together)g(with)h (large)h(deviation)f(Theorem)-201 7493 y(\(Lemma)57 b(1.1)j(in)h([3)o (]\))f(is)h(then)f(clearly)h(su\016cien)-5 b(t)60 b(to)g(get)g(Lemma)d (3.2.)-201 8066 y Fp(References)-110 8472 y Fo([1])90 b(M.)79 b(Aizenman,)j(G.M.)c(Graf,)83 b Ff(L)-9 b(o)g(c)g(alization)81 b(b)-9 b(ounds)82 b(for)g(an)g(ele)-9 b(ctr)g(on)81 b(gas)p Fo(,)i(J.)c(Ph)-5 b(ys.)78 b(A:)h(Math.)173 8698 y(Gen.,)59 b Fg(31)p Fo(,)g(6783)i(\(1998\).)-110 9073 y([2])90 b(Ju.M.)74 b(Berezanskii,)80 b Ff(Exp)-9 b(ansions)81 b(in)e(Eigenfunctions)g(of)g(selfadjoint)g(Op)-9 b(er)g(ators)p Fo(,)78 b(Amer.)d(Mat.)173 9299 y(So)5 b(c,)60 b(Pro)-5 b(vidence,)60 b(RI)g(\(1968\).)-110 9674 y([3])90 b(J.)58 b(Bourgain,)i(M.)f(Goldstein,)g Ff(On)k(nonp)-9 b(erturb)g(ative)62 b(lo)-9 b(c)g(alization)62 b(with)i(quasi-p)-9 b(erio)g(dic)61 b(p)-9 b(otential)p Fo(,)173 9900 y(Preprin)k(t)61 b(\(1999\).)-110 10275 y([4])90 b(H.L.)62 b(Cycon,)i(R.G.)d(F)-15 b(ro)5 b(ese,)63 b(W.)f(Kirsc)-5 b(h,)63 b(B.)g(Simon,)f Ff(Schr\177)-93 b(odinger)66 b(Op)-9 b(er)g(ators)p Fo(,)61 b(Springer-V)-15 b(erlag)173 10501 y(\(1987\).)-110 10876 y([5])90 b(D.)j(Damanik,)101 b(P)-15 b(.)93 b(Stollman,)101 b Ff(Multi-sc)-9 b(ale)93 b(analysis)j(implies)f(str)-9 b(ong)95 b(dynamic)-9 b(al)95 b(lo)-9 b(c)g(alization)p Fo(,)173 11102 y(Preprin)k(t)61 b(\(1999\),)e(or)i(see)f(mp-arc)e(99-461.)3439 11631 y(9)p eop %%Page: 10 10 10 9 bop -110 1158 a Fo([6])90 b(S.)66 b(De)h(Bi)-5 b(\022)-86 b(evre,)68 b(G.)e(F)-15 b(orni,)68 b Ff(T)-14 b(r)-9 b(ansp)g(ort)70 b(pr)-9 b(op)g(erties)69 b(of)i(kicke)-9 b(d)68 b(and)j(quasi-p)-9 b(erio)g(dic)68 b(Hamiltonians)p Fo(,)173 1384 y(J.)59 b(Stat.)h(Ph)-5 b(ys.)59 b Fg(90)p Fo(,)h(1201-1223)f(\(1998\).)-110 1760 y([7])90 b(S.)42 b(De)g(Bi)-5 b(\022)-86 b(evre,)45 b(F.)d(Germinet,)i Ff(Dynamic)-9 b(al)47 b(L)-9 b(o)g(c)g(alization)47 b(for)g(the)h(r)-9 b(andom)48 b(dimer)f(mo)-9 b(del)p Fo(,)43 b(to)f(app)5 b(ear)173 1985 y(in)61 b(J.)f(Stat.)f(Ph)-5 b(ys.)60 b(\(2000\),)f(or)h(see)g(mp-arc)f(99-250.)-110 2361 y([8])90 b(R.)61 b(Del)h(Rio,)g(S.)g(Jitomirsk)-10 b(a)-5 b(y)g(a,)60 b(Y.)i(Last,)e(B.)i(Simon,)e Ff(What)66 b(is)h(L)-9 b(o)g(c)g (alization?)p Fo(,)60 b(Ph)-5 b(ys.)61 b(Rev.)g(Lett.)173 2586 y(75,)f(117-119)g(\(1995\).)-110 2962 y([9])90 b(R.)g(Del)g(Rio,) 98 b(S.)89 b(Jitomirsk)-10 b(a)-5 b(y)g(a,)96 b(Y.)90 b(Last,)96 b(B.)90 b(Simon,)96 b Ff(Op)-9 b(er)g(ators)91 b(with)h(singular)f(c)-9 b(ontinuous)173 3187 y(sp)g(e)g(ctrum)59 b(IV:)j(Hausdor\013)e(dimensions,)h(r)-9 b(ank)61 b(one)g(p)-9 b(ertub)g(ations)60 b(and)h(lo)-9 b(c)g(alization)p Fo(,)55 b(J.)h(d'Analyse)173 3413 y(Math.)j Fg(69)p Fo(,)h(153-200)f(\(1996\).) -201 3789 y([10])90 b(A.)49 b(v)-5 b(on)49 b(Dreifus,)i(A.)e(Klein,)j Ff(A)i(new)h(pr)-9 b(o)g(of)53 b(of)h(lo)-9 b(c)g(alization)53 b(in)i(the)g(A)-5 b(nderson)55 b(tight)g(binding)e(mo)-9 b(del)p Fo(,)173 4014 y(Comm)k(un.)57 b(Math.)j(Ph)-5 b(ys.)59 b Fg(124)p Fo(,)g(285-299)h(\(1989\).)-201 4390 y([11])90 b(L.H.)76 b(Eliasson)g Ff(Discr)-9 b(ete)79 b(one-dimensional)g(quasi-p)-9 b(erio)g(dic)76 b(Schr\177)-93 b(odinger)78 b(op)-9 b(er)g(ators)79 b(with)h(pur)-9 b(e)173 4615 y(p)g(oint)65 b(sp)-9 b(e)g(ctrum)p Fo(,)58 b(Acta)j(Math.)e Fg(179)p Fo(,)g(153-196)h(\(1997\).)-201 4991 y([12])90 b(J.)55 b(F)-15 b(r\177)-91 b(ohlic)-5 b(h,)56 b(T.)g(Sp)5 b(encer,)56 b Ff(A)-5 b(bsenc)c(e)60 b(of)g(di\013usion)g(with)g(A)-5 b(nderson)61 b(tight)g(binding)e(mo)-9 b(del)59 b(for)h(lar)-9 b(ge)173 5217 y(disor)g(der)64 b(or)g(low)g(ener)-9 b(gy)p Fo(,)59 b(Comm)-5 b(un.)58 b(Math.)h(Ph)-5 b(ys.)60 b Fg(88)p Fo(,)f(151-184)g(\(1983\).)-201 5592 y([13])90 b(F.)62 b(Germinet,)f Ff(Dynamic)-9 b(al)66 b(L)-9 b(o)g(c)g(alization)66 b(II)h(with)g(an)h(Applic)-9 b(ation)65 b(to)j(the)f(A)-5 b(lmost)68 b(Mathieu)e(Op-)173 5818 y(er)-9 b(ator)p Fo(,)59 b(J.)h(Stat)f(Ph)-5 b(ys.)60 b Fg(95)p Fo(,)f(273-286)h(\(1999\).)-201 6193 y([14])90 b(F.)51 b(Germinet,)g(S.)g(De)h(Bi)-5 b(\022)-86 b(evre,)53 b Ff(Dynamic)-9 b(al)56 b(L)-9 b(o)g(c)g(alization)55 b(for)h(Discr)-9 b(ete)56 b(and)h(Continuous)g(R)-9 b(andom)173 6419 y(Schr\177)-93 b(odinger)63 b(Op)-9 b(er)g(ators)p Fo(,)59 b(Comm)-5 b(un.)57 b(Math.)i(Ph)-5 b(ys.)60 b Fg(194)p Fo(,)f(323-341)g(\(1998\).)-201 6794 y([15])90 b(F.)53 b(Germinet,)g(A.)i(Klein,)g Ff(Str)-9 b(ong)59 b(Dynamic)-9 b(al)58 b(L)-9 b(o)g(c)g(alization)57 b(for)i(r)-9 b(andom)58 b(op)-9 b(er)g(ators)58 b(and)h(applic)-9 b(a-)173 7020 y(tion)65 b(to)h(classic)-9 b(al)63 b(waves)h(in)h (inhomo)-9 b(gene)g(ous)63 b(me)-9 b(dia)p Fo(,)59 b(in)i(preparation.) -201 7395 y([16])90 b(M.)107 b(Goldstein,)119 b(W.)108 b(Sc)-5 b(hlag,)119 b Ff(H\177)-93 b(older)108 b(c)-9 b(ontinuity)108 b(of)h(the)f(inte)-9 b(gr)g(ate)g(d)108 b(density)h(of)f(states)i(for)173 7621 y(quasip)-9 b(erio)g(dic)93 b(Schr\177)-93 b(odinger)93 b(e)-9 b(quations)95 b(and)h(aver)-9 b(ages)94 b(of)h(shifts)i(of)e(subharmonic)f(functions)p Fo(,)173 7847 y(Preprin)-5 b(t)61 b(\(1999\).)-201 8222 y([17])90 b(A.)g(Y.)f(Gordon,)96 b(S.)90 b(Jitomirsk)-10 b(a)-5 b(y)g(a,)94 b(Y.)c(Last,)95 b(B.)90 b(Simon,)95 b Ff(Duality)d(and)g(singular)e(c)-9 b(ontinuous)173 8448 y(sp)g(e)g(ctrum)63 b(in)i(the)g(almost)g(Mathieu)f(e)-9 b(quation)p Fo(,)59 b(Acta.)h(Math.)g Fg(178)p Fo(,)e(169-183)i (\(1997\).)-201 8823 y([18])90 b(I.)99 b(Guarneri,)109 b Ff(Sp)-9 b(e)g(ctr)g(al)99 b(pr)-9 b(op)g(erties)99 b(of)i(quantum)f(di\013usion)g(on)h(discr)-9 b(ete)100 b(lattic)-9 b(es)p Fo(,)108 b(Europh)-5 b(ys.)173 9049 y(Lett.)56 b(10)h(\(1989\),)g(95-100;)g Ff(On)62 b(an)g(estimate)g(c)-9 b(onc)g(erning)60 b(quantum)h(di\013usion)g(in)h(the)g(pr)-9 b(esenc)g(e)61 b(of)173 9275 y(a)k(fr)-9 b(actal)63 b(sp)-9 b(e)g(ctrum)p Fo(,)59 b(Europh)-5 b(ys.)59 b(Lett.)g Fg(21)h Fo(\(1993\),)f(729-733.)-201 9650 y([19])90 b(S.)66 b(Jitomirsk)-10 b(a)-5 b(y)g(a,)66 b Ff(A)-5 b(nderson)72 b(lo)-9 b(c)g(alization)69 b(for)h(the)h(A)-5 b(lmost)71 b(Mathieu)f(Equation:)95 b(A)71 b(Non)g(p)-9 b(ertu-)173 9876 y(b)g(ative)64 b(Pr)-9 b(o)g(of)p Fo(,)59 b(Comm)-5 b(un.)57 b(Math.)j(Ph)-5 b(ys.)59 b Fg(165)p Fo(,)g(49-57)h(\(1994\).) -201 10251 y([20])90 b(S.)56 b(Jitomirsk)-10 b(a)-5 b(y)g(a,)56 b Ff(A)-5 b(nderson)62 b(lo)-9 b(c)g(alization)60 b(for)h(the)h(A)-5 b(lmost)62 b(Mathieu)f(Equation:)81 b(II.)62 b(Point)g(Sp)-9 b(e)g(c-)173 10477 y(trum)64 b(for)h Fn(\025)50 b(>)g Fo(2,)60 b(Comm)-5 b(un.)58 b(Math.)h(Ph)-5 b(ys.)59 b Fg(168)p Fo(,)g(563-570)h(\(1995\).)-201 10852 y([21])90 b(S.)66 b(Jitomirsk)-10 b(a)-5 b(y)g(a,)65 b Ff(Metal-Insulator)k(tr)-9 b(ansition)71 b(for)e(the)h(A)-5 b(lmost)71 b(Mathieu)e(Op)-9 b(er)g(ator)p Fo(,)66 b(to)f(app)5 b(ear)173 11078 y(in)61 b(Ann.)g(Math.)3393 11631 y(10)p eop %%Page: 11 11 11 10 bop -201 1158 a Fo([22])90 b(S.)76 b(Jitomirsk)-10 b(a)-5 b(y)g(a,)78 b(Y.)f(Last)e Ff(A)-5 b(nderson)80 b(L)-9 b(o)g(c)g(alization)78 b(for)i(the)f(A)-5 b(lmost)80 b(Mathieu)f(Equation,)k(III.)173 1384 y(Semi-Uniform)57 b(L)-9 b(o)g(c)g(alization,)57 b(Continuity)i(of)f(Gaps,)h(and)g(Me)-9 b(asur)g(e)56 b(of)i(the)g(sp)-9 b(e)g(ctrum)p Fo(,)53 b(Comm)-5 b(un.)173 1610 y(Math.)59 b(Ph)-5 b(ys.)60 b Fg(195)p Fo(,)f(1-14)h(\(1998\).)-201 1985 y([23])90 b(A.)100 b(Klein,)110 b(M.)99 b(Seifert)h Ff(Gener)-9 b(alize)g(d)99 b(eigenfunctions)h(for)h(waves)g(in)h(inhomo)-9 b(gene)g(ous)99 b(me)-9 b(dia)p Fo(,)173 2211 y(Preprin)k(t)61 b(\(2000\).)-201 2586 y([24])90 b(Y.)79 b(Last,)j Ff(Quantum)f (Dynamics)i(and)f(De)-9 b(c)g(omp)g(osition)81 b(of)h(Singular)e (Continuous)j(Sp)-9 b(e)g(ctrum)p Fo(,)81 b(J.)173 2812 y(F)-15 b(unct.)59 b(Anal)j Fg(142)d Fo(\(1996\),)g(406-445.)-201 3187 y([25])90 b(F.)64 b(Piec)-5 b(hon,)65 b Ff(A)-5 b(nomalous)69 b(Di\013usion)f(Pr)-9 b(op)g(erties)68 b(of)h(Wave)f(Packets)h(on)g(Quasip)-9 b(erio)g(dic)66 b(Chains)p Fo(,)173 3413 y(Ph)-5 b(ys.)59 b(Rev.)h(Lett.)f Fg(76)p Fo(,)g Fn(n)1969 3347 y Fl(o)2048 3413 y Fo(23,)h(4372-4375)g (\(1996\).)-201 3789 y([26])90 b(B.)43 b(Simon,)j Ff(Schr\177)-93 b(odinger)48 b(semi-gr)-9 b(oups)p Fo(,)45 b(Bull.)f(Amer.)f(Math.)f (So)5 b(c.)44 b Fg(V)-17 b(ol.7)p Fo(,)45 b Fn(n)5544 3723 y Fl(o)5623 3789 y Fo(3,)i(p.447-526)c(\(1982\).)-201 4164 y([27])90 b(B.)60 b(Simon,)f Ff(A)-5 b(bsenc)c(e)64 b(of)h(b)-9 b(al)9 b(listic)63 b(motion)p Fo(,)d(Comm)-5 b(un.)58 b(Math.)h(Ph)-5 b(ys.)60 b Fg(134)p Fo(,)e(209-212)i (\(1990\).)3393 11631 y(11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0001270643991--