Content-Type: multipart/mixed; boundary="-------------0306231107549" This is a multi-part message in MIME format. ---------------0306231107549 Content-Type: text/plain; name="03-296.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-296.comments" Email: marchett@if.usp.br , wreszins@fma.if.usp.br AMS-Code: 47A10, 47A25, 47A35, 47A55, 81Q10 ---------------0306231107549 Content-Type: text/plain; name="03-296.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-296.keywords" Off-diagonal Jacobi matrices, spectral transition, sparse perturbation, Transfer matrices, Prufer variables, uniformly distributed sequences ---------------0306231107549 Content-Type: application/postscript; name="stjacobi2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="stjacobi2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: stjacobi2.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips stjacobi2.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.13:1654 %%BeginProcSet: texc.pro %! 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y Fx(\))1857 2365 y Fp(n)p Fn(\025)p Fr(0)2027 2336 y Fx(is)e(required)h(to)e(b)s(e)h(of)f(the)h (form)1259 2636 y Ft(p)1308 2651 y Fp(n)1382 2636 y Fx(=)1486 2466 y Fs(\()1608 2568 y Fx(1)22 b Fq(\000)g Ft(r)1822 2583 y Fp(j)1942 2568 y Fx(if)90 b Ft(n)27 b Fx(=)h Ft(a)2329 2583 y Fp(j)2393 2568 y Fq(2)g Ft(A)1608 2712 y Fx(1)285 b(if)90 b Ft(n)39 b(=)-61 b Fq(2)29 b Ft(A)2645 2636 y(;)947 b Fx(\(1.2\))111 2915 y(with)1285 3060 y Ft(r)1329 3075 y Fp(j)1365 3060 y Fx(\(0\))28 b(=)f(0)195 b(and)g Ft(r)2261 3075 y Fp(j)2298 3060 y Fx(\(1\))27 b(=)h(1)17 b Ft(;)972 b Fx(\(1.3\))111 3249 y(where)34 b Ft(A)28 b Fx(=)f Fq(f)p Ft(a)698 3264 y Fp(j)735 3249 y Fq(g)785 3281 y Fp(j)t Fn(\025)p Fr(1)944 3249 y Fx(is)33 b(a)f(set)i(of)e (strictly)i(p)s(ositiv)m(e)f(in)m(tegers)h(suc)m(h)g(that)1277 3471 y Ft(a)1328 3486 y Fp(j)1386 3471 y Fq(\000)23 b Ft(a)1537 3486 y Fp(j)t Fn(\000)p Fr(1)1692 3471 y Fq(\025)28 b Fx(2)33 b Ft(;)212 b(j)33 b Fx(=)28 b(2)p Ft(;)17 b Fx(3)p Ft(;)g(:)g(:)g(:)31 b(;)965 b Fx(\(1.4\))111 3692 y(and)1583 3837 y(lim)1564 3897 y Fp(j)t Fn(!1)1764 3769 y Ft(a)1815 3784 y Fp(j)t Fr(+1)p 1764 3814 178 4 v 1809 3905 a Ft(a)1860 3920 y Fp(j)1979 3837 y Fx(=)28 b Ft(\014)33 b(>)28 b Fx(1)17 b Ft(;)1251 b Fx(\(1.5\))111 4055 y(where)34 b Ft(\014)k Fx(is)33 b(the)g(\\sparseness)i(parameter".)258 4199 y(A)d(concrete)i(example)g(satisfying)g(\(1.3\))e(is)h(giv)m(en)g (b)m(y)1566 4421 y Ft(r)1610 4436 y Fp(j)1674 4421 y Fx(=)2043 4354 y Ft(\024)p 1788 4398 568 4 v 1788 4490 a Fx(1)22 b(+)g(\(1)g Fq(\000)g Ft(\024)p Fx(\))17 b Ft(\020)2319 4505 y Fp(j)3619 4421 y Fx(\(1.6\))111 4672 y(with)33 b Ft(\020)376 4687 y Fp(j)440 4672 y Fq(\025)28 b Fx(0.)44 b(If)32 b Ft(\020)805 4687 y Fp(j)869 4672 y Fq(!)27 b(1)32 b Fx(as)h Ft(j)h Fq(!)27 b(1)p Fx(,)32 b(then)1736 4894 y Ft(r)1780 4909 y Fp(j)1816 4894 y Fx(\()p Ft(\024)p Fx(\))c Fq(!)f Fx(0)17 b Ft(;)1423 b Fx(\(1.7\))111 5116 y(for)32 b(all)h Ft(\024)28 b Fq(2)g Fx([0)p Ft(;)17 b Fx(1\))31 b(and)i Ft(r)1046 5131 y Fp(j)1083 5116 y Fx(\(1\))27 b(=)g Ft(\024)p Fx(.)44 b(If)33 b Ft(\020)1606 5131 y Fp(j)1669 5116 y Fx(=)28 b(0)k(for)g(all)h Ft(j)6 b Fx(,)32 b(then)1744 5338 y Ft(r)1788 5353 y Fp(j)1824 5338 y Fx(\()p Ft(\024)p Fx(\))c(=)f Ft(\024)17 b(:)1432 b Fx(\(1.8\))1941 5587 y(2)p eop %%Page: 3 3 3 2 bop 111 312 a Fx(In)31 b(b)s(oth)f(cases)i(\(either)f Ft(\020)1057 327 y Fp(j)1120 312 y Fq(!)d(1)i Fx(or)g Ft(\020)1538 327 y Fp(j)1601 312 y Fx(=)e(0\),)i(as)h Ft(\024)f Fx(v)-5 b(aries)31 b(in)g(the)f(in)m(terv)-5 b(al)31 b(\(0)p Ft(;)17 b Fx(1\),)30 b Ft(J)40 b Fx(in)m(terp)s(olates) 111 456 y(con)m(tin)m(uously)e(t)m(w)m(o)e(opp)s(osite)g(situations:)50 b Ft(J)45 b Fx(has)36 b(purely)h(a.c.)52 b(sp)s(ectrum)37 b(at)e Ft(\024)e Fx(=)g(0)i(and)h(purely)111 601 y(p)s(oin)m(t)42 b(sp)s(ectrum)i(at)e Ft(\024)i Fx(=)g(1.)72 b(Note)42 b(that,)i(in)f(b)s(oth)f(cases,)k Ft(p)2500 616 y Fp(n)2547 601 y Fx(\(1\))d(=)h(0)e(for)g(all)g Ft(n)i Fq(2)g Ft(A)e Fx(and)h Ft(J)111 745 y Fx(b)s(ecomes)c(a)d(direct)i(sum)g(of)e (\014nite)i(matrices)g(with)g(dense)g(p.p.)58 b(sp)s(ectrum)38 b(if)f(\(1.5\))f(holds.)58 b(This)111 890 y(is)30 b(Ho)m(wland's)i (remark)e([H],)h(whic)m(h)g(sho)m(ws)g(o\013{diagonal)e(Jacobi)h (matrices)h(as)f(natural)g(examples)111 1034 y(in)j(whic)m(h)h(a)e(sp)s (ectral)i(transition)f(migh)m(t)g(b)s(e)g(exp)s(ected.)258 1178 y(W)-8 b(e)38 b(shall)h(call)f(the)g(class)i(of)d(Jacobi)i (matrices)g(satisfying)g(\(1.2\)-\(1.5\))e(Ho)m(wland's)i(class)g Fm(H)q Fx(,)111 1323 y(and)d(the)h(subset)g(of)f(those)h(satisfying)g (\(1.7\),)f Fm(H)1943 1338 y Fr(0)1982 1323 y Fx(.)54 b(Note)36 b(that)g Fm(H)q Fq(n)p Fm(H)2746 1338 y Fr(0)2819 1323 y Fx(includes)i(those)f(in)f(whic)m(h)111 1467 y(\(1.8\))c(holds.) 258 1612 y(Let)g Ft(J)486 1627 y Fr(0)558 1612 y Fx(b)s(e)h(the)g(free) g(Jacobi)g(matrix)1364 2133 y Ft(J)1418 2148 y Fr(0)1485 2133 y Fx(=)1589 1754 y Fs(0)1589 1929 y(B)1589 1989 y(B)1589 2048 y(B)1589 2108 y(B)1589 2168 y(B)1589 2228 y(B)1589 2292 y(@)1718 1836 y Fx(0)82 b(1)h(0)g(0)g Fq(\001)17 b(\001)g(\001)1718 1980 y Fx(1)82 b(0)h(1)g(0)g Fq(\001)17 b(\001)g(\001)1718 2125 y Fx(0)82 b(1)h(0)g(1)g Fq(\001)17 b(\001)g(\001)1718 2269 y Fx(0)82 b(0)h(1)g(0)g Fq(\001)17 b(\001)g(\001)1728 2372 y Fx(.)1728 2405 y(.)1728 2438 y(.)1860 2372 y(.)1860 2405 y(.)1860 2438 y(.)1992 2372 y(.)1992 2405 y(.)1992 2438 y(.)2124 2372 y(.)2124 2405 y(.)2124 2438 y(.)2251 2380 y(.)2289 2405 y(.)2328 2430 y(.)2403 1754 y Fs(1)2403 1929 y(C)2403 1989 y(C)2403 2048 y(C)2403 2108 y(C)2403 2168 y(C)2403 2228 y(C)2403 2292 y(A)2540 2133 y Ft(:)111 2664 y Fx(The)40 b(sp)s(ectrum)h Ft(\033)21 b Fx(\()p Ft(J)919 2679 y Fr(0)958 2664 y Fx(\))39 b(=)g Ft(\033)1205 2679 y Fp(ac)1294 2664 y Fx(\()p Ft(J)1386 2679 y Fr(0)1426 2664 y Fx(\))g(=)g([)p Fq(\000)p Fx(2)p Ft(;)17 b Fx(2])39 b(is)h(purely)g(absolutely)h(con)m (tin)m(uum.)65 b(If)39 b Ft(J)49 b Fq(2)39 b Fm(H)3754 2679 y Fr(0)3793 2664 y Fx(,)111 2809 y Ft(Q)f Fx(=)f Ft(J)e Fq(\000)27 b Ft(J)586 2824 y Fr(0)663 2809 y Fx(is)39 b(a)f(compact)h(p)s(erturbation)g(of)f Ft(J)1999 2824 y Fr(0)2076 2809 y Fx(and)h(th)m(us,)h(b)m(y)g(W)-8 b(eyl's)40 b(in)m(v)-5 b(ariance)39 b(theorem,)111 2953 y(the)33 b(essen)m(tial)i(sp)s(ectrum)f(of)f Ft(J)42 b Fx(remains)34 b(the)f(same)g(as)g(that)g(of)f Ft(J)2575 2968 y Fr(0)2615 2953 y Fx(,)h Ft(\033)2730 2968 y Fr(ess)2821 2953 y Fx(\()p Ft(J)9 b Fx(\))28 b(=)g Ft(\033)3147 2968 y Fr(ac)3219 2953 y Fx(\()p Ft(J)3311 2968 y Fr(0)3350 2953 y Fx(\))g(=)g([)p Fq(\000)p Fx(2)p Ft(;)17 b Fx(2].)111 3098 y(The)34 b(essen)m(tial)g (sp)s(ectrum)g(of)e Ft(J)42 b Fx(in)33 b(this)g(sub)s(class)h(is)f (describ)s(ed)i(as)e(follo)m(ws.)111 3397 y Fl(Theorem)k(1.1)49 b Fk(L)-5 b(et)31 b Ft(J)37 b Fq(2)28 b Fm(H)1216 3412 y Fr(0)1286 3397 y Fk(.)43 b(Then)30 b Ft(\033)1664 3412 y Fr(ess)1756 3397 y Fx(\()p Ft(J)9 b Fx(\))30 b Fk(is)h(pur)-5 b(ely)31 b(a.c.)43 b(if)2637 3272 y Fn(1)2601 3302 y Fs(X)2608 3514 y Fp(k)r Fr(=1)2761 3397 y Fq(j)p Ft(r)2833 3412 y Fp(k)2875 3397 y Fq(j)2903 3348 y Fr(2)2970 3397 y Ft(<)28 b Fq(1)i Fk(and)g(pur)-5 b(ely)31 b(s.c.)111 3695 y(if)243 3571 y Fn(1)206 3601 y Fs(X)213 3813 y Fp(k)r Fr(=1)366 3695 y Fq(j)p Ft(r)438 3710 y Fp(k)481 3695 y Fq(j)508 3647 y Fr(2)575 3695 y Fx(=)d Fq(1)34 b Fk(and)h(\\sp)-5 b(arseness)33 b(p)-5 b(ar)g(ameter")34 b Ft(\014)f Fx(=)28 b Fq(1)p Fk(.)258 4008 y Fx(Theorem)39 b(1.1)f(ma)m(y)h(b)s(e)f(pro)m(v)m(en)h(along)f(the)h(lines)g(of)f (Theorem)h(1)p Ft(:)p Fx(7)f(of)f(Kiselev-Last-Simon)111 4153 y([KLS].)44 b(It)32 b(has)h(the)g(follo)m(wing)111 4457 y Fl(Corollary)j(1.2)49 b Fk(L)-5 b(et)35 b Ft(J)44 b Fk(b)-5 b(e)35 b(as)f(in)h(The)-5 b(or)g(em)34 b(1.1.)44 b(If)2179 4332 y Fn(1)2142 4362 y Fs(X)2150 4575 y Fp(k)r Fr(=1)2303 4457 y Fq(j)o Ft(r)2374 4472 y Fp(k)2417 4457 y Fq(j)2445 4409 y Fr(2)2512 4457 y Ft(<)27 b Fq(1)p Fk(,)35 b(then)1186 4867 y Ft(\033)1241 4882 y Fr(ess)1333 4867 y Fx(\()p Ft(J)9 b Fx(\))28 b(=)1603 4696 y Fs(\()1731 4799 y Ft(\033)1786 4814 y Fr(ac)1857 4799 y Fx(\()p Ft(J)9 b Fx(\))89 b(if)h(0)27 b Fq(\024)h Ft(\024)g(<)g Fx(1)1725 4943 y Ft(\033)1780 4958 y Fr(pp)1863 4943 y Fx(\()p Ft(J)9 b Fx(\))83 b(if)181 b Ft(\024)27 b Fx(=)h(1)2718 4867 y Ft(:)1941 5587 y Fx(3)p eop %%Page: 4 4 4 3 bop 111 375 a Fk(If)250 251 y Fn(1)213 281 y Fs(X)221 493 y Fp(k)r Fr(=1)374 375 y Fq(j)p Ft(r)446 390 y Fp(k)488 375 y Fq(j)516 327 y Fr(2)583 375 y Fx(=)28 b Fq(1)p Fk(,)34 b(then)1175 844 y Ft(\033)1230 859 y Fr(ess)1321 844 y Fx(\()p Ft(J)9 b Fx(\))28 b(=)1591 610 y Fs(8)1591 700 y(>)1591 729 y(<)1591 909 y(>)1591 939 y(:)1727 704 y Ft(\033)1782 719 y Fr(ac)1854 704 y Fx(\()p Ft(J)9 b Fx(\))89 b(if)179 b Ft(\024)28 b Fx(=)g(0)1731 848 y Ft(\033)1786 863 y Fr(sc)1850 848 y Fx(\()p Ft(J)9 b Fx(\))93 b(if)c(0)28 b Ft(<)f(\024)h(<)g Fx(1)1722 992 y Ft(\033)1777 1007 y Fr(pp)1860 992 y Fx(\()p Ft(J)9 b Fx(\))83 b(if)179 b Ft(\024)28 b Fx(=)g(1)2729 844 y Ft(:)111 1205 y Fk(Henc)-5 b(e,)45 b(the)e(sp)-5 b(e)g(ctr)g(al)42 b(tr)-5 b(ansition)43 b(in)f(the)h(sub)-5 b(class)42 b Fm(H)2181 1220 y Fr(0)2263 1205 y Fk(o)-5 b(c)g(curs)43 b(only)f(at)h(the)g(end)g(p)-5 b(oints)42 b(of)h(the)111 1350 y(r)-5 b(ange)34 b(of)70 b(the)35 b(\\c)-5 b(oupling)34 b(c)-5 b(onstants",)34 b(at)h(le)-5 b(ast)35 b(under)f(the)h(sp)-5 b(arseness)34 b(c)-5 b(ondition)33 b(\(1.5\).)258 1593 y Fx(Corollary)f(1.2)f(sho)m(ws)j(that,)e(in)g(order)h(to)e(o)s(ccur)i (a)f(transition)g(at)g(a)f(non)m(trivial)i(v)-5 b(alue)33 b(of)e Ft(\024)p Fx(,)i Ft(J)111 1737 y Fx(has)39 b(to)e(b)s(e)i(so)f (that)g(\(1.7\))g(do)s(es)g(not)g(hold)g(\(i.e.)61 b Ft(r)2028 1752 y Fp(n)2112 1737 y Fo(9)37 b Fx(0\))g(and)i(it)f(is)g (lik)m(ely)i(to)e(b)s(e)g(from)g(s.c.)61 b(to)111 1882 y(\(dense\))34 b(p.p.,)f(with)h(a.c.)43 b(sp)s(ectrum)34 b(just)f(for)f Ft(\024)c Fx(=)g(0.)43 b(Our)32 b(main)h(result)h(is)f (as)g(follo)m(ws)111 2125 y Fl(Theorem)k(1.3)49 b Fk(L)-5 b(et)36 b Ft(J)i Fq(2)30 b Fm(H)q Fq(n)p Fm(H)1364 2140 y Fr(0)1437 2125 y Fk(with)36 b 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Fp(j)760 3160 y Fx(for)32 b(some)h Ft(j)h Fq(\025)28 b Fx(1,)862 3484 y Ft(T)14 b Fx(\()p Ft(n;)j(n)22 b Fq(\000)h Fx(1;)17 b Ft(\025)p Fx(\))27 b(=)g Ft(T)1628 3499 y Fn(\000)1704 3484 y Fx(\()p Ft(a)1793 3499 y Fp(j)1830 3484 y Fx(;)17 b Ft(\025)p Fx(\))27 b(:=)2126 3284 y Fs(0)2126 3463 y(@)2362 3340 y Ft(\025)p 2265 3384 252 4 v 2265 3476 a Fx(1)22 b Fq(\000)h Ft(r)2480 3491 y Fp(j)2677 3340 y Fq(\000)p Ft(\025)p 2619 3384 V 2619 3476 a Fx(1)f Fq(\000)h Ft(r)2834 3491 y Fp(j)2366 3605 y Fx(1)305 b(0)2922 3284 y Fs(1)2922 3463 y(A)3042 3484 y Ft(:)550 b Fx(\(2.6\))111 3806 y(Let)28 b Ft(\025)f Fx(=)h(2)17 b(cos)f Ft(')28 b Fx(where)g Ft(')g Fq(2)g Fx(\(0)p Ft(;)17 b(\031)t Fx(\))26 b(\(see)j(Theorem)g(2.1)e(for)f(a)h (pro)s(of)g(that)g(this)h(parameterization)111 3950 y(is)33 b(exhaustiv)m(e\),)i(and)1523 4158 y Ft(U)j Fx(=)1731 3987 y Fs( )1851 4089 y Fx(0)135 b(sin)18 b Ft(')1851 4234 y Fx(1)83 b Fq(\000)17 b Fx(cos)g Ft(')2329 3987 y Fs(!)3619 4158 y Fx(\(2.7\))111 4403 y(b)s(e)33 b(the)g(nonsingular)g (matrix)g(with)1493 4730 y Ft(U)1569 4689 y Fn(\000)p Fr(1)1691 4730 y Fx(=)1795 4560 y Fs( )1915 4662 y Fx(cot)17 b Ft(')83 b Fx(1)1918 4806 y(csc)18 b Ft(')85 b Fx(0)2299 4560 y Fs(!)2411 4730 y Ft(:)111 4981 y Fx(The)34 b(matrix)f Ft(U)43 b Fx(has)33 b(a)f(remark)-5 b(able)33 b(prop)s(ert)m(y)h(of)e (transforming)h Ft(T)2639 4996 y Fr(0)2678 4981 y Fx(\()p Ft(\025)p Fx(\))g(in)m(to)f(a)h(pure)g(rotation)1044 5263 y Ft(R)17 b Fx(\()p Ft(')p Fx(\))28 b(=)1406 5093 y Fs( )1568 5195 y Fx(cos)17 b Ft(')130 b Fx(sin)18 b Ft(')1527 5340 y Fq(\000)f Fx(sin)g Ft(')83 b Fx(cos)17 b Ft(')2157 5093 y Fs(!)2263 5263 y Fx(=)28 b Ft(U)10 b(T)2500 5278 y Fr(0)2540 5263 y Fx(\()p Ft(\025)p Fx(\))p Ft(U)2749 5222 y Fn(\000)p Fr(1)2860 5263 y Ft(:)732 b Fx(\(2.8\))1941 5587 y(6)p eop %%Page: 7 7 7 6 bop 111 312 a Fx(It)32 b(will)g(b)s(e)f(most)h(con)m(v)m(enien)m(t) j(to)c(in)m(tro)s(duce,)h(for)f(eac)m(h)i Ft(a)2245 327 y Fp(j)2309 312 y Fq(2)28 b Ft(A)p Fx(,)k(the)g(matrices)h Ft(P)3155 327 y Fr(+)p Fn(\000)3268 312 y Fx(\()p Ft(a)3357 327 y Fp(j)3394 312 y Fx(\))e(giv)m(en)i(b)m(y)1314 541 y Ft(T)1385 499 y Fp(U)1371 565 y Fr(+)p Fn(\000)1502 541 y Fx(\()p Ft(a)1591 556 y Fp(j)1628 541 y Fx(\))27 b(=)h Ft(R)18 b Fx(\()o Ft(')p Fx(\))f Ft(P)2108 556 y Fr(+)p Fn(\000)2222 541 y Fx(\()p Ft(a)2311 556 y Fp(j)2347 541 y Fx(\))p Ft(R)h Fx(\()p Ft(')p Fx(\))1002 b(\(2.9\))111 769 y(where)49 b Ft(T)479 733 y Fp(U)465 794 y Fr(+)p Fn(\000)595 769 y Fx(\()p Ft(a)684 784 y Fp(j)721 769 y Fx(\))k(=)g Ft(U)10 b(T)1074 784 y Fr(+)1150 769 y Fx(\()p Ft(a)1239 784 y Fp(j)1276 769 y Fx(\))16 b Ft(T)1387 784 y Fn(\000)1463 769 y Fx(\()p Ft(a)1552 784 y Fp(j)1589 769 y Fx(\))g Ft(U)1719 733 y Fn(\000)p Fr(1)1814 769 y Fx(.)88 b(F)-8 b(rom)47 b(\(2.3\))g(-)g(\(2.9\))g(and,)k(since)e(a)e (pro)s(duct)g(of)111 914 y(rotations)33 b(is)g(a)f(rotation,)g(w)m(e)i (ha)m(v)m(e)228 1143 y Ft(U)10 b(T)k Fx(\()p Ft(n)p Fx(;)j Ft(\025)p Fx(\))p Ft(U)686 1102 y Fn(\000)p Fr(1)808 1143 y Fx(=)28 b Ft(R)18 b Fx(\(\()p Ft(n)k Fq(\000)h Ft(a)1311 1158 y Fp(k)1353 1143 y Fx(\))p Ft(')p Fx(\))17 b Ft(P)1573 1158 y Fr(+)p Fn(\000)1687 1143 y Fx(\()p Ft(a)1776 1158 y Fp(k)1818 1143 y Fx(\))p Ft(R)h Fx(\(\()p Ft(a)2075 1158 y Fp(k)2140 1143 y Fq(\000)23 b Ft(a)2291 1158 y Fp(k)r Fn(\000)p Fr(1)2424 1143 y Fx(\))p Ft(')p Fx(\))16 b Fq(\001)h(\001)g(\001)e Ft(P)2776 1158 y Fr(+)p Fn(\000)2890 1143 y Fx(\()p Ft(a)2979 1158 y Fr(1)3018 1143 y Fx(\))p Ft(R)j Fx(\(\()p Ft(a)3275 1158 y Fr(1)3314 1143 y Fx(\))p Ft(')p Fx(\))117 b(\(2.10\))111 1372 y(whic)m(h)34 b(will)g(b)s(e)e(useful)i(in)f(Section)g(3.)258 1516 y(According)g(to)f(\(1.2\))g(and)h(\(1.8\),)f(in)h(the)g(equations)h (ab)s(o)m(v)m(e)1713 1745 y Ft(r)1757 1760 y Fp(j)1821 1745 y Fx(=)28 b(1)21 b Fq(\000)i Ft(p)2144 1760 y Fp(a)2181 1770 y Fi(j)3571 1745 y Fx(\(2.11\))111 1974 y(where)399 1893 y Fs(\000)444 1974 y Ft(p)493 1989 y Fp(a)530 1999 y Fi(j)567 1893 y Fs(\001)613 2013 y Fp(j)t Fn(\025)p Fr(1)777 1974 y Fq(\021)37 b Fx(0)h(refers)h(to)f(p.)60 b(p.)h(case,)40 b(and)2090 1893 y Fs(\000)2136 1974 y Ft(p)2185 1989 y Fp(a)2222 1999 y Fi(j)2259 1893 y Fs(\001)2305 2013 y Fp(j)t Fn(\025)p Fr(1)2469 1974 y Fq(\021)d Fx(1)h(to)g(the)g (unp)s(erturb)s(ed)i(\(a.)59 b(c.\))111 2118 y(case.)45 b(W)-8 b(e)33 b(ha)m(v)m(e)111 2353 y Fl(Theorem)k(2.1)49 b Fk(If)35 b Ft(J)h Fq(2)28 b Fm(H)36 b Fk(and)e Fx(0)28 b Fq(\024)g Ft(r)1605 2368 y Fp(j)1669 2353 y Fq(\024)g Fx(1)35 b Fk(for)f(al)5 b(l)35 b Ft(j)e Fq(2)c Fo(N)8 b Fk(,)41 b(i.)k(e.)f Ft(p)2730 2368 y Fp(a)2767 2378 y Fi(j)2832 2353 y Fq(2)28 b Fx([0)p Ft(;)17 b Fx(1])p Fk(,)34 b(then)1625 2582 y Ft(\033)t Fx(\()p Ft(J)9 b Fx(\))28 b(=)f([)p Fq(\000)p Fx(2)p Ft(;)17 b Fx(2])52 b Ft(;)111 2811 y Fk(wher)-5 b(e)34 b Ft(\033)t Fx(\()p Ft(A)p Fx(\))h Fk(denotes)f(the)h(sp)-5 b(e)g(ctrum)35 b(of)g Ft(A)p Fk(.)111 3046 y(Pr)-5 b(o)g(of.)61 b Fx(Let)39 b Ft(J)47 b Fq(2)38 b Fm(H)h Fx(with)g(\()p Ft(p)1258 3061 y Fp(n)1305 3046 y Fx(\))1343 3075 y Fn(\025)p Fr(0)1476 3046 y Fx(so)f(that)h(0)e Fq(\024)h Ft(p)2069 3061 y Fp(a)2106 3071 y Fi(j)2181 3046 y Fq(\024)h Fx(1,)g Fq(8)p Ft(j)6 b Fx(.)62 b(T)-8 b(o)39 b(pro)m(v)m(e)g Ft(\033)t Fx(\()p Ft(J)9 b Fx(\))38 b Fq(\022)g Fx([)q Fq(\000)p Fx(2)p Ft(;)17 b Fx(2])o(,)40 b(let)111 3190 y Ft(u)27 b Fx(=)h(\()p Ft(u)392 3205 y Fp(n)438 3190 y Fx(\))476 3220 y Fp(n)p Fn(\025)p Fr(0)641 3190 y Fq(2)g Ft(l)764 3205 y Fr(2)820 3190 y Fx(\()p Fo(N)9 b Fx(\))d(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)1194 3498 y(\()p Ft(u;)17 b(J)9 b(u)p Fx(\))27 b(=)1656 3373 y Fn(1)1619 3403 y Fs(X)1625 3612 y Fp(n)p Fr(=0)1780 3498 y Fx(\()p Ft(u)1874 3513 y Fp(n)1937 3498 y Ft(u)1993 3513 y Fp(n)p Fr(+1)2129 3498 y Fx(\))17 b Ft(h)2240 3513 y Fp(n)2304 3327 y Fs( )2469 3430 y Ft(u)2525 3445 y Fp(n)2424 3574 y Ft(u)2480 3589 y Fp(n)p Fr(+1)2658 3327 y Fs(!)3571 3498 y Fx(\(2.12\))111 3784 y(where)1552 3967 y Ft(h)1608 3982 y Fp(n)1683 3967 y Fx(=)1786 3797 y Fs( )1930 3899 y Fx(0)107 b Ft(p)2135 3914 y Fp(n)1907 4043 y Ft(p)1956 4058 y Fp(n)2109 4043 y Fx(0)2223 3797 y Fs(!)2335 3967 y Ft(:)111 4215 y Fx(Inserting)1520 4360 y Ft(h)1576 4375 y Fp(n)1651 4360 y Fx(=)28 b Ft(\025)1812 4319 y Fr(+)1812 4384 y Fp(n)1887 4360 y Ft(P)1964 4319 y Fr(+)1950 4384 y Fp(n)2045 4360 y Fx(+)22 b Ft(\025)2200 4319 y Fn(\000)2200 4384 y Fp(n)2275 4360 y Ft(P)2352 4319 y Fn(\000)2338 4384 y Fp(n)111 4552 y Fx(in)m(to)45 b(\(2.12\),)h(where)f Ft(P)1014 4515 y Fn(\006)1000 4576 y Fp(n)1117 4552 y Fx(pro)5 b(jects)45 b(on)m(to)f(the)h (eigendirection)h(asso)s(ciated)f(to)f(the)g(eingen)m(v)-5 b(alue)111 4696 y Ft(\025)168 4660 y Fn(\006)168 4721 y Fp(n)255 4696 y Fx(=)27 b Fq(\006)p Ft(p)484 4711 y Fp(n)532 4696 y Fx(,)32 b(w)m(e)i(ha)m(v)m(e)1515 4871 y Fq(\000)p Fx(2)p 1658 4790 57 4 v 17 w Ft(\025)27 b Fq(\024)1857 4804 y Fx(\()p Ft(u;)17 b(J)9 b(u)p Fx(\))p 1857 4848 294 4 v 1889 4939 a(\()p Ft(u;)17 b(u)p Fx(\))2189 4871 y Fq(\024)28 b Fx(2)p 2360 4790 57 4 v 17 w Ft(\025)111 5109 y Fx(where)p 393 5028 V 34 w Ft(\025)g Fx(=)f(sup)728 5133 y Fp(n)792 5109 y Ft(\025)849 5073 y Fr(+)849 5134 y Fp(n)935 5109 y Fx(=)h(sup)1186 5133 y Fp(n)1249 5109 y Ft(p)1298 5124 y Fp(n)1373 5109 y Fx(=)f(1.)44 b(Hence)1335 5338 y Ft(\033)21 b Fx(\()p Ft(J)9 b Fx(\))27 b Fq(\022)1683 5257 y Fs(\002)1724 5338 y Fq(\000)p Fx(2)p 1867 5257 V 17 w Ft(\025;)17 b Fx(2)p 2034 5257 V 17 w Ft(\025)2090 5257 y Fs(\003)2159 5338 y Fx(=)28 b([)p Fq(\000)p Fx(2)p Ft(;)17 b Fx(2])33 b Ft(:)975 b Fx(\(2.13\))1941 5587 y(7)p eop %%Page: 8 8 8 7 bop 111 312 a Fx(Note)41 b(that,)h(b)s(ecause)g Ft(J)50 b Fx(is)41 b(a)g(symmetric)h(o\013{diagonal)e(matrix,)j(the)e(sp)s (ectrum)h(is)g(symmetric)111 456 y(and)33 b(inf)419 471 y Fp(n)482 456 y Ft(\025)539 420 y Fn(\000)539 481 y Fp(n)626 456 y Fx(=)28 b Fq(\000)17 b Fx(sup)971 480 y Fp(n)1034 456 y Ft(\025)1091 420 y Fr(+)1091 481 y Fp(n)1178 456 y Fx(=)27 b Fq(\000)p 1358 375 57 4 v Ft(\025)q Fx(.)43 b(In)33 b(order)g(to)f(pro)m(v)m(e)i(the)f(opp)s(osite)g (inclusion)1625 700 y Ft(\033)21 b Fx(\()p Ft(J)9 b Fx(\))27 b Fq(\023)i Fx([)p Fq(\000)p Fx(2)p Ft(;)17 b Fx(2])33 b Ft(;)1265 b Fx(\(2.14\))111 944 y(w)m(e)41 b(use)g(W)-8 b(eyl's)41 b(criterion)g(\(see)f(Theorem)i(VI)s(I)p Ft(:)p Fx(12)d(of)h([S1]\):)97 b(if)40 b Ft(A)g Fx(is)g(a)g(b)s(ounded)g (self{adjoin)m(t)111 1089 y(op)s(erator)35 b(in)i(separable)g(Hilb)s (ert)f(space)h Fj(H)29 b Fx(,)37 b Ft(\025)f Fx(is)g(a)g(p)s(oin)m(t)g (of)g(the)g(sp)s(ectrum)h Ft(\033)t Fx(\()p Ft(A)p Fx(\))f(of)g Ft(A)g Fx(if)g(and)111 1233 y(only)d(if)g(there)g(exist)h(a)e(sequence) j(\()p Ft( )1481 1248 y Fp(n)1529 1233 y Fx(\))1567 1262 y Fp(n)p Fn(\025)p Fr(0)1736 1233 y Fx(in)e Fj(H)c Fx(,)k(with)g Fq(k)p Ft( )2371 1248 y Fp(n)2418 1233 y Fq(k)28 b Fx(=)f(1,)33 b(suc)m(h)h(that)1500 1477 y(lim)1476 1537 y Fp(n)p Fn(!1)1677 1477 y Fq(k)p Fx(\()p Ft(A)22 b Fq(\000)g Ft(\025)p Fx(\))17 b Ft( )2134 1492 y Fp(n)2181 1477 y Fq(k)28 b Fx(=)f(0)17 b Ft(:)1116 b Fx(\(2.15\))258 1735 y(Let)25 b Ft(\025)j Fx(=)g(2)17 b(cos)f Ft(')p Fx(,)27 b Ft(')h Fq(2)g Fx([0)p Ft(;)17 b(\031)t Fx(],)27 b(and)f(de\014ne)h Ft( )1911 1750 y Fp(n)1986 1735 y Fx(=)g(\(1)p Ft(=)2225 1663 y Fq(p)p 2308 1663 59 4 v 72 x Ft(n)p Fx(\))16 b(\()p Ft(e)2503 1699 y Fp(i')2578 1735 y Ft(;)h(e)2667 1699 y Fp(i)p Fr(2)p Fp(')2776 1735 y Ft(;)g(:)g(:)g(:)f(;)h(e)3040 1699 y Fp(in')3157 1735 y Ft(;)g Fx(0)p Ft(;)g(:)g(:)g(:)o Fx(\))o(.)42 b(Clearly)111 1880 y Ft( )174 1895 y Fp(n)249 1880 y Fq(2)28 b Ft(l)372 1895 y Fr(2)428 1880 y Fx(\()p Fo(N)9 b Fx(\))39 b(and)32 b(w)m(e)i(claim)f(that)1518 2149 y Fq(k)p Fx(\()p Ft(J)e Fq(\000)23 b Ft(\025)p Fx(\))16 b Ft( )1965 2164 y Fp(n)2012 2149 y Fq(k)28 b(\024)g Ft(c)2247 2082 y Fx(ln)17 b Ft(n)p 2247 2126 157 4 v 2254 2146 a Fq(p)p 2337 2146 59 4 v 72 x Ft(n)3571 2149 y Fx(\(2.16\))111 2424 y(holds)37 b(with)h Ft(c)c Fx(=)g Ft(c)p Fx(\()p Ft(\014)6 b Fx(\))36 b(indep)s(enden)m(t)j(of)d Ft(n)h Fx(for)f(an)m(y)h Ft(J)44 b Fq(2)35 b Fm(H)q Fx(.)55 b(In)36 b(order)h(to)f(pro)m(v)m(e)i(\(2.16\),)f(notice)111 2569 y(that)30 b(\()p Ft(J)h Fq(\000)22 b Ft(\025)p Fx(\))17 b Ft( )717 2584 y Fp(n)794 2569 y Fx(consists)31 b(of)e(the)h(action)g (on)g Ft( )1909 2584 y Fp(n)1986 2569 y Fx(of)f(a)g(sum)i(of)e(lo)s (cal)g(matrices,)j(eac)m(h)e(b)s(ounded)h(in)111 2713 y(norm)25 b(b)m(y)h(one.)42 b(Tw)m(o)26 b(of)e(them)i(in)m(v)m(olv)m(e) i(the)d(extremes)i Ft(e)2186 2677 y Fp(i')2286 2713 y Fx(and)e Ft(e)2513 2677 y Fp(in')2630 2713 y Fx(,)i(and)e(there)h(are)f Ft(O)19 b Fx(\(ln)e Ft(n)p Fx(\))25 b(non{)111 2858 y(diagonal)37 b(matrices,)j(eac)m(h)e(acting)g(just)g(on)f(t)m(w)m(o)h(successiv)m(e) j(comp)s(onen)m(ts)e(of)e Ft( )3156 2873 y Fp(n)3204 2858 y Fx(.)58 b(The)38 b Ft(O)19 b Fx(\(ln)e Ft(n)p Fx(\))111 3002 y(comes)31 b(from)f(the)g(sequence)i(\()p Ft(a)1279 3017 y Fp(j)1316 3002 y Fx(\))1354 3034 y Fp(j)t Fn(\025)p Fr(1)1510 3002 y Fx(satisfying)f(the)f(sparseness)i (condition)e(\(1.5\),)g Ft(a)3306 3017 y Fp(j)3371 3002 y Fq(\030)e Ft(\014)3537 2966 y Fp(j)3573 3002 y Fx(,)i(with)111 3147 y(at)i(most)h Ft(r)j Fx(p)s(oin)m(ts)d Ft(a)893 3162 y Fp(j)962 3147 y Fx(within)h([1)p Ft(;)17 b(n)p Fx(])o(,)33 b(where)h Ft(r)h Fx(is)e(suc)m(h)h(that)f Ft(\014)2482 3110 y Fp(r)2547 3147 y Fq(\024)28 b Ft(n)33 b Fx(or,)f(equiv)-5 b(alen)m(tly)d(,)1757 3419 y Ft(r)30 b Fq(\024)1947 3352 y Fx(ln)17 b Ft(n)p 1946 3396 159 4 v 1946 3487 a Fx(ln)g Ft(\014)2147 3419 y(:)111 3689 y Fx(This)34 b(pro)m(v)m(es)g(\(2.16\))e(and)h(therefore)g(\(2.14\),)f (concluding)i(the)f(pro)s(of)e(of)h(Theorem)i(2.1.)3745 3834 y Fh(2)111 4203 y Fy(3)161 b(Pr)t(\177)-85 b(ufer)54 b(V)-13 b(ariables)111 4446 y Fx(W)-8 b(e)36 b(in)m(tro)s(duce)g(the)g (so)g(called)g(\(mo)s(di\014ed\))g(Pr)s(\177)-51 b(ufer)35 b(v)-5 b(ariables)36 b(for)e(the)i(o\013{diagonal)f(Jacobi)g(ma-)111 4591 y(trices)j(in)g(consideration.)58 b(F)-8 b(or)36 b Ft(\025)g Fq(2)f Fo(C)63 b Fx(and)38 b Ft(n)d Fq(2)h Fo(N)9 b Fx(,)44 b(let)38 b Ft(T)14 b Fx(\()p Ft(n;)j(n)25 b Fq(\000)h Fx(1;)17 b Ft(\025)p Fx(\))36 b(b)s(e)h(the)h(2)25 b Fq(\002)h Fx(2)36 b(transfer)111 4798 y(matrix)g(giv)m(en)g(b)m(y)g (\(2.2\))f(and)g(note)g(that,)h(if)f Fl(u)1869 4813 y Fp(n)1948 4798 y Fx(=)2056 4628 y Fs( )2221 4730 y Ft(u)2277 4745 y Fp(n)2176 4874 y Ft(u)2232 4889 y Fp(n)p Fn(\000)p Fr(1)2410 4628 y Fs(!)2524 4798 y Fx(is)h(a)f(v)m(ector)h(in)f Fo(C)3186 4762 y Fr(2)3267 4798 y Fx(comp)s(osed)h(b)m(y)111 5006 y(the)d(comp)s(onen)m(ts)h(of)e(a)h Ft(l)1038 5021 y Fr(2)1094 5006 y Fx(\()p Fo(N)9 b Fx(\))d({solution)32 b Ft(u)c Fx(=)f(\()p Ft(u)1940 5021 y Fr(0)1979 5006 y Ft(;)17 b(u)2079 5021 y Fr(1)2118 5006 y Ft(;)g(:)g(:)g(:)o Fx(\))32 b(of)g(\()p Ft(J)g Fq(\000)22 b Ft(\025)p Fx(\))17 b Ft(u)27 b Fx(=)g(0,)33 b(then)1448 5250 y Fl(u)1510 5265 y Fp(n)p Fr(+1)1675 5250 y Fx(=)28 b Ft(T)14 b Fx(\()p Ft(n;)j(n)22 b Fq(\000)h Fx(1;)17 b Ft(\025)p Fx(\))g Fl(u)2437 5265 y Fp(n)1941 5587 y Fx(8)p eop %%Page: 9 9 9 8 bop 111 386 a Fx(holds)33 b(for)f Ft(n)c Fx(=)g(0)p Ft(;)17 b Fx(1)p Ft(;)g(:)g(:)g(:)n Fx(,)33 b(with)g Fl(u)1348 401 y Fr(0)1416 386 y Fx(=)1519 216 y Fs( )1667 318 y Ft(u)1723 333 y Fr(0)1639 463 y Ft(u)1695 478 y Fn(\000)p Fr(1)1831 216 y Fs(!)1942 386 y Fx(satisfying)h(some)f (initial)g(data.)258 587 y(W)-8 b(riting)1615 732 y Ft(T)1686 691 y Fp(k)1672 756 y Fr(0)1756 732 y Fx(=)28 b Ft(T)1917 747 y Fr(0)1973 732 y Ft(T)2030 747 y Fr(0)2086 732 y Fq(\001)17 b(\001)g(\001)e Ft(T)2276 747 y Fr(0)1860 784 y Fs(|)p 1905 784 139 12 v 139 w({z)p 2134 784 V 139 w(})1959 871 y Fp(k)r Fn(\000)p Fr(times)111 1020 y Fx(and)1317 1164 y Ft(T)1374 1179 y Fr(+)p Fp(;)p Fn(\000)1507 1164 y Fx(\()p Ft(a)p Fx(;)i Ft(\025)p Fx(\))28 b(=)f Ft(T)1923 1179 y Fr(+)1982 1164 y Fx(\()p Ft(a)p Fx(;)17 b Ft(\025)p Fx(\))g Ft(T)2284 1179 y Fn(\000)2343 1164 y Fx(\()p Ft(a)p Fx(;)g Ft(\025)p Fx(\))g Ft(;)111 1362 y Fx(where)34 b Ft(T)450 1377 y Fr(0)522 1362 y Fx(and)f Ft(T)769 1377 y Fn(\006)861 1362 y Fx(is)g(giv)m(en)g(b)m(y)h (\(2.4\)-\(2.6\),)d(equation)i(\(2.3\))f(can)h(b)s(e)g(written)g(as)396 1607 y Ft(T)d Fx(\()p Ft(n)p Fx(;)17 b Ft(\025)p Fx(\))27 b(=)h Ft(T)920 1561 y Fp(n)p Fn(\000)p Fp(a)1055 1573 y Fi(k)1093 1561 y Fn(\000)p Fr(1)906 1631 y(0)1204 1607 y Ft(T)1261 1622 y Fr(+)p Fp(;)p Fn(\000)1395 1607 y Fx(\()p Ft(a)1484 1622 y Fp(k)1526 1607 y Fx(\))17 b Ft(T)1652 1554 y Fp(a)1689 1566 y Fi(k)1727 1554 y Fn(\000)p Fp(a)1819 1566 y Fi(k)q Fg(\000)p Ff(1)1936 1554 y Fn(\000)p Fr(2)1638 1631 y(0)2047 1607 y Ft(T)2104 1622 y Fr(+)p Fp(;)p Fn(\000)2238 1607 y Fx(\()p Ft(a)2327 1622 y Fp(k)r Fn(\000)p Fr(1)2460 1607 y Fx(\))33 b Fq(\001)17 b(\001)g(\001)31 b Ft(T)2737 1622 y Fr(+)p Fp(;)p Fn(\000)2871 1607 y Fx(\()p Ft(a)2960 1622 y Fr(1)3000 1607 y Fx(\))17 b Ft(T)3126 1563 y Fp(a)3163 1572 y Ff(1)3197 1563 y Fn(\000)p Fr(1)3112 1631 y(0)3308 1607 y Ft(;)284 b Fx(\(3.1\))111 1851 y(from)35 b(whic)m(h)i(equation)f(\(2.10\))e(can)i(b)s(e)f (obtained)h(follo)m(wing)f(the)h(indications)g(of)f(Section)h(2.)51 b(In-)111 1995 y(serting)1040 2197 y Ft(R)q Fx(\()p Fq(\000)p Ft(')p Fx(\))p Ft(U)10 b(T)1465 2212 y Fr(+)1525 2197 y Fx(\()p Ft(a)p Fx(\))p Ft(U)1728 2156 y Fn(\000)p Fr(1)1851 2197 y Fx(=)1954 2027 y Fs( )2312 2129 y Ft(p)2361 2144 y Fp(a)2722 2129 y Fx(0)2075 2273 y(\(1)22 b Fq(\000)g Ft(p)2332 2288 y Fp(a)2374 2273 y Fx(\))16 b(cot)h Ft(')83 b Fx(1)2812 2027 y Fs(!)111 2453 y Fx(and)921 2636 y Ft(U)10 b(T)1054 2651 y Fn(\000)1114 2636 y Fx(\()p Ft(a)p Fx(\))p Ft(U)1317 2595 y Fn(\000)p Fr(1)1412 2636 y Ft(R)q Fx(\()p Fq(\000)p Ft(')p Fx(\))28 b(=)1836 2465 y Fs( )2263 2568 y Fx(1)459 b(0)1956 2712 y(\(1)p Ft(=p)2141 2727 y Fp(a)2205 2712 y Fq(\000)22 b Fx(1\))16 b(cot)h Ft(')83 b Fx(1)p Ft(=p)2848 2727 y Fp(a)2931 2465 y Fs(!)111 2897 y Fx(in)m(to)33 b(\(2.9\))f(giv)m(es)1327 3105 y Ft(P)1390 3120 y Fr(+)p Fn(\000)1503 3105 y Fx(\()p Ft(a)p Fx(\))c(=)1762 2934 y Fs( )2002 3037 y Ft(p)2051 3052 y Fp(a)2366 3037 y Fx(0)1882 3181 y Ft(\013)1944 3196 y Fp(a)2002 3181 y Fx(cot)17 b Ft(')83 b Fx(1)p Ft(=p)2443 3196 y Fp(a)2526 2934 y Fs(!)3619 3105 y Fx(\(3.2\))111 3361 y(where)1673 3505 y Ft(\013)1735 3520 y Fp(a)1804 3505 y Fx(=)1939 3438 y(1)p 1918 3482 91 4 v 1918 3574 a Ft(p)1967 3589 y Fp(a)2041 3505 y Fq(\000)22 b Ft(p)2189 3520 y Fp(a)2231 3505 y Ft(:)258 3730 y Fx(W)-8 b(e)31 b(no)m(w)g(use)h(equation)g(\(2.10\),)f(where)h Ft(R)g Fx(and)f Ft(P)2134 3745 y Fr(+)p Fp(;)p Fn(\000)2298 3730 y Fx(are)g(giv)m(en)h(b)m(y)g(\(2.8\))e(and)h(\(3.2\),)g(resp)s (ec-)111 3874 y(tiv)m(ely)-8 b(,)34 b(to)f(de\014ne)g(the)g(Pr)s(\177) -51 b(ufer)33 b(v)-5 b(ariables)33 b(\()p Ft(R)1786 3889 y Fp(k)1829 3874 y Ft(;)17 b(\022)1918 3889 y Fp(k)1961 3874 y Fx(\))1998 3903 y Fp(k)r Fn(\025)p Fr(0)2155 3874 y Fx(as)33 b(follo)m(ws.)44 b(Let)1336 4119 y Fl(v)1395 4134 y Fp(k)1465 4119 y Fx(=)28 b(\()p Ft(R)1681 4134 y Fp(k)r Fn(\000)p Fr(1)1830 4119 y Fx(cos)17 b Ft(\022)2022 4134 y Fp(k)2066 4119 y Ft(;)g(R)2184 4134 y Fp(k)r Fn(\000)p Fr(1)2333 4119 y Fx(sin)g Ft(\022)2514 4134 y Fp(k)2557 4119 y Fx(\))1024 b(\(3.3\))111 4363 y(and)1407 4506 y Fs(e)1404 4507 y Fl(v)1463 4522 y Fp(k)1534 4507 y Fx(=)1637 4396 y Fs(\020)1697 4507 y Ft(R)1771 4522 y Fp(k)1830 4507 y Fx(cos)1982 4481 y Fs(e)1977 4507 y Ft(\022)2022 4522 y Fp(k)2066 4507 y Ft(;)17 b(R)2184 4522 y Fp(k)2243 4507 y Fx(sin)2384 4481 y Fs(e)2379 4507 y Ft(\022)2424 4522 y Fp(k)2467 4396 y Fs(\021)3619 4507 y Fx(\(3.4\))111 4705 y(b)s(e)33 b(giv)m(en)h(b)m(y)1391 4850 y Fl(v)1450 4865 y Fp(k)1521 4850 y Fx(=)27 b Ft(R)18 b Fx(\(\()p Ft(a)1843 4865 y Fp(k)1908 4850 y Fq(\000)k Ft(a)2058 4865 y Fp(k)r Fn(\000)p Fr(1)2191 4850 y Fx(\))p Ft(')p Fx(\))2351 4849 y Fs(e)2348 4850 y Fl(v)2407 4865 y Fp(k)r Fn(\000)p Fr(1)3619 4850 y Fx(\(3.5\))111 5048 y(and)1589 5192 y Fs(e)1586 5193 y Fl(v)1645 5208 y Fp(k)1716 5193 y Fx(=)27 b Ft(P)1882 5208 y Fr(+)p Fn(\000)2013 5193 y Fx(\()p Ft(a)2102 5208 y Fp(k)2145 5193 y Fx(\))16 b Fl(v)2258 5208 y Fp(k)2318 5193 y Ft(;)1274 b Fx(\(3.6\))1941 5587 y(9)p eop %%Page: 10 10 10 9 bop 111 312 a Fx(with)41 b(initial)g(condition)1070 311 y Fs(e)1067 312 y Fl(v)1126 327 y Fr(0)1207 312 y Fx(=)g Fl(v)1383 327 y Fr(0)1464 312 y Fx(=)g(\(cos)17 b Ft(\022)1811 327 y Fr(0)1851 312 y Ft(;)g Fx(sin)g Ft(\022)2076 327 y Fr(0)2116 312 y Fx(\).)67 b(With)41 b(these)h(notation)e(equation)i(\(2.10\))111 456 y(reads)1491 600 y Fs(e)1489 601 y Fl(v)1548 616 y Fp(k)1618 601 y Fx(=)28 b Ft(U)10 b(T)31 b Fx(\()p Ft(a)1975 616 y Fp(k)2018 601 y Fx(;)17 b Ft(\025)p Fx(\))f Ft(U)2249 560 y Fn(\000)p Fr(1)2344 601 y Fl(v)2403 616 y Fr(0)111 799 y Fx(and)33 b(w)m(e)g(ha)m(v)m(e)531 1098 y Ft(R)606 1057 y Fr(2)605 1122 y Fp(k)676 1098 y Fx(=)28 b Fq(k)o Fl(v)888 1113 y Fp(k)r Fr(+1)1021 1098 y Fq(k)1071 1049 y Fr(2)1138 1098 y Fx(=)g Fq(k)1294 1097 y Fs(e)1292 1098 y Fl(v)1351 1113 y Fp(k)1394 1098 y Fq(k)1443 1049 y Fr(2)1510 1098 y Fx(=)g Ft(R)1689 1057 y Fr(2)1688 1122 y Fp(k)r Fn(\000)p Fr(1)1838 924 y Fs(\015)1838 983 y(\015)1838 1043 y(\015)1838 1103 y(\015)1838 1163 y(\015)1893 927 y( )2134 1030 y Ft(p)2183 1045 y Fp(a)2220 1057 y Fi(k)2554 1030 y Fx(0)2013 1174 y Ft(\013)2075 1189 y Fp(a)2112 1201 y Fi(k)2172 1174 y Fx(cot)16 b Ft(')83 b Fx(1)p Ft(=p)2612 1189 y Fp(a)2649 1201 y Fi(k)2733 927 y Fs(!)17 b( )2949 1030 y Fx(cos)g Ft(\022)3141 1045 y Fp(k)2954 1174 y Fx(sin)h Ft(\022)3136 1189 y Fp(k)3226 927 y Fs(!)3305 924 y(\015)3305 983 y(\015)3305 1043 y(\015)3305 1103 y(\015)3305 1163 y(\015)3360 950 y Fr(2)111 1399 y Fx(and)1126 1544 y Ft(\022)1171 1559 y Fp(k)r Fr(+1)1331 1544 y Fx(=)28 b(arg)17 b(\()p Ft(P)1689 1559 y Fr(+)p Fn(\000)1803 1544 y Fx(\()p Ft(a)1892 1559 y Fp(k)1935 1544 y Fx(\))p Fl(v)2032 1559 y Fp(k)2075 1544 y Fx(\))22 b(+)g(\()p Ft(a)2322 1559 y Fp(k)r Fr(+1)2477 1544 y Fq(\000)h Ft(a)2628 1559 y Fp(k)2670 1544 y Fx(\))17 b Ft(')111 1742 y Fx(whic)m(h)34 b(yield)891 1864 y Ft(R)966 1827 y Fr(2)965 1890 y Fp(k)p 846 1908 208 4 v 846 1999 a Ft(R)921 1965 y Fr(2)920 2027 y Fp(k)r Fn(\000)p Fr(1)1091 1931 y Fx(=)27 b Ft(p)1243 1890 y Fr(2)1243 1956 y Fp(a)1280 1968 y Fi(k)1340 1931 y Fx(cos)1470 1890 y Fr(2)1526 1931 y Ft(\022)1571 1946 y Fp(k)1636 1931 y Fx(+)1734 1791 y Fs(\022)1808 1931 y Ft(\013)1870 1946 y Fp(a)1907 1958 y Fi(k)1966 1931 y Fx(cot)17 b Ft(')g Fx(cos)g Ft(\022)2386 1946 y Fp(k)2451 1931 y Fx(+)2599 1864 y(1)p 2559 1908 129 4 v 2559 1999 a Ft(p)2608 2014 y Fp(a)2645 2026 y Fi(k)2714 1931 y Fx(sin)g Ft(\022)2895 1946 y Fp(k)2939 1791 y Fs(\023)3012 1813 y Fr(2)3068 1931 y Ft(:)524 b Fx(\(3.7\))111 2165 y(and)792 2318 y Ft(\022)837 2333 y Fp(k)r Fr(+1)998 2318 y Fx(=)27 b(tan)1242 2277 y Fn(\000)p Fr(1)1353 2178 y Fs(\022)1476 2251 y Fx(1)p 1436 2295 V 1436 2387 a Ft(p)1485 2358 y Fr(2)1485 2411 y Fp(a)1522 2423 y Fi(k)1592 2318 y Fx(tan)16 b Ft(\022)1794 2333 y Fp(k)1859 2318 y Fx(+)1967 2251 y Ft(\013)2029 2266 y Fp(a)2066 2278 y Fi(k)p 1967 2295 142 4 v 1974 2387 a Ft(p)2023 2402 y Fp(a)2060 2414 y Fi(k)2136 2318 y Fx(cot)g Ft(')2346 2178 y Fs(\023)2442 2318 y Fx(+)22 b(\()p Ft(a)2629 2333 y Fp(k)r Fr(+1)2784 2318 y Fq(\000)g Ft(a)2934 2333 y Fp(k)2977 2318 y Fx(\))17 b Ft(')g(:)479 b Fx(\(3.8\))258 2564 y(The)33 b(Euclidean)h(norm)f(of)f Ft(U)10 b(T)31 b Fx(\()p Ft(n)p Fx(;)17 b Ft(\025)p Fx(\))f Ft(U)1766 2528 y Fn(\000)p Fr(1)1861 2564 y Fx(,)33 b(giv)m(en)h(b)m(y)877 2724 y Fs(\015)877 2783 y(\015)932 2808 y Ft(U)10 b(T)31 b Fx(\()p Ft(n)p Fx(;)17 b Ft(\025)p Fx(\))f Ft(U)1423 2767 y Fn(\000)p Fr(1)1518 2808 y Fl(v)1577 2823 y Fr(0)1617 2724 y Fs(\015)1617 2783 y(\015)1672 2750 y Fr(2)1739 2808 y Fx(=)1843 2724 y Fs(\015)1843 2783 y(\015)1898 2808 y Ft(U)10 b(T)31 b Fx(\()p Ft(a)2151 2823 y Fp(N)2219 2808 y Fx(;)17 b Ft(\025)p Fx(\))f Ft(U)2450 2767 y Fn(\000)p Fr(1)2545 2808 y Fl(v)2604 2823 y Fr(0)2643 2724 y Fs(\015)2643 2783 y(\015)2699 2750 y Fr(2)2782 2808 y Fx(=)28 b Ft(R)2961 2767 y Fr(2)2960 2833 y Fp(N)3027 2808 y Ft(;)565 b Fx(\(3.9\))111 3052 y(for)32 b(an)m(y)h Ft(n)g 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Ft(;)17 b(\022)1895 4573 y Fp(k)1938 4558 y Fx(\))22 b(+)g(\()p Ft(a)2185 4573 y Fp(k)r Fr(+1)2340 4558 y Fq(\000)g Ft(a)2490 4573 y Fp(k)2533 4558 y Fx(\))17 b Ft(')919 b Fx(\(3.12\))111 4802 y(where)1002 4955 y Ft(g)t Fx(\()p Ft(p;)17 b(\022)s Fx(\))27 b(=)g(tan)1541 4914 y Fn(\000)p Fr(1)1652 4814 y Fs(\022)1755 4887 y Fx(1)p 1736 4932 V 1736 5023 a Ft(p)1785 4994 y Fr(2)1851 4955 y Fx(\()o(tan)17 b Ft(\022)25 b Fx(+)d(cot)17 b Ft(')p Fx(\))22 b Fq(\000)h Fx(cot)16 b Ft(')2795 4814 y Fs(\023)2902 4955 y Ft(:)642 b Fx(\(3.13\))1917 5587 y(10)p eop %%Page: 11 11 11 10 bop 111 312 a Fy(4)161 b(The)54 b(Main)g(Lemma)111 555 y Fx(In)31 b(this)g(section)g(w)m(e)h(pro)m(v)m(e)f(the)g(main)f (Lemma)h(connecting)h(uniform)e(distribution)i(of)e(the)g(Pr)s(\177)-51 b(ufer)111 699 y(angles)40 b(with)g(the)g(sp)s(ectral)g(transition.)65 b(W)-8 b(e)40 b(use)g(the)g(p)s(o)m(w)m(erful)g(theorems)h(of)e(Y.)h (Last)f(and)h(B.)111 844 y(Simon)h([LS])f(relating)g(the)g(nature)g(of) 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3733 y Fx(card)17 b(\()p Fq(f)o Ft(k)31 b Fx(:)d Ft(\022)1600 3748 y Fp(k)1671 3733 y Fq(2)g Ft(I)8 b(;)33 b Fx(1)27 b Fq(\024)i Ft(k)h Fq(\024)e Ft(N)10 b Fq(g)q Fx(\))p 1130 3778 1291 4 v 1731 3869 a Ft(N)2458 3801 y Fx(=)28 b Fq(j)p Ft(I)8 b Fq(j)27 b(\021)h Ft(b)23 b Fq(\000)f Ft(a)604 b Fx(\(4.5\))111 4055 y(where)34 b(card)17 b(\()p Ft(S)6 b Fx(\),)32 b(the)h(cardinalit) m(y)h(of)e(the)h(set)g Ft(S)6 b Fx(,)33 b(is)g(the)g(n)m(um)m(b)s(er)h (of)e(elemen)m(ts)j(in)e Ft(S)6 b Fx(.)258 4200 y(F)-8 b(or)30 b(simplicit)m(y)-8 b(,)34 b(and)e(without)g(loss)g(of)f (generalit)m(y)-8 b(,)32 b(w)m(e)h(\014x)f(here)g(and)g(henceforth)g Ft(p)3412 4215 y Fp(a)3449 4227 y Fi(k)3519 4200 y Fx(=)c Ft(p)j Fx(for)111 4344 y(all)i Ft(k)s Fx(,)f(0)c Ft(<)f(p)h(<)g Fx(1,)k(and)h(tak)m(e)g Ft(A)28 b Fx(=)f(\()p Ft(a)1523 4359 y Fp(k)1566 4344 y Fx(\))1604 4373 y Fp(k)r Fn(\025)p Fr(1)1769 4344 y Fx(so)33 b(that)1648 4588 y Ft(a)1699 4603 y Fp(k)1764 4588 y Fq(\000)23 b Ft(a)1915 4603 y Fp(k)r Fn(\000)p Fr(1)2076 4588 y Fx(=)k Ft(\014)2240 4547 y Fp(k)3619 4588 y Fx(\(4.6\))111 4832 y(i.)44 b(e.,)33 b Ft(a)390 4847 y Fp(k)460 4832 y Fx(=)28 b Ft(a)615 4847 y Fp(k)658 4832 y Fx(\()p Ft(\014)6 b Fx(\))27 b(=)g Ft(\014)986 4796 y Fp(k)r Fr(+1)1119 4832 y Ft(=)p Fx(\()p Ft(\014)g Fq(\000)c Fx(1\),)32 b Ft(\014)i(>)27 b Fx(1.)43 b(In)33 b(this)g(case)h(\(3.10\))e(and)g(\(3.12\))g(read)1229 5164 y Ft(R)1304 5123 y Fr(2)1303 5189 y Fp(N)1398 5164 y Fx(=)1501 4994 y Fs( )1610 5097 y Fx(1)p 1590 5141 89 4 v 1590 5233 a Ft(p)1639 5204 y Fr(2)1705 5164 y Fx(exp)1871 4994 y Fs(\()1981 5097 y Fx(1)p 1961 5141 V 1961 5233 a Ft(N)2117 5040 y Fp(N)2076 5070 y Fs(X)2084 5282 y Fp(k)r Fr(=1)2237 5164 y Ft(f)27 b Fx(\()p Ft(\022)2395 5179 y Fp(k)2438 5164 y Fx(\))2476 4994 y Fs(\)!)2635 5011 y Fp(N)3619 5164 y Fx(\(4.7\))1917 5587 y(11)p eop %%Page: 12 12 12 11 bop 111 312 a Fx(with)1560 456 y Ft(\022)1605 471 y Fp(k)1676 456 y Fx(=)27 b Ft(g)t Fx(\()p Ft(\022)1913 471 y Fp(k)r Fn(\000)p 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2352 y Ft(')28 b Fq(2)g Fx(\(0)p Ft(;)17 b(\031)t Fx(\))o Fk(,)35 b(then)g(the)f(c)-5 b(onclusions)34 b(of)h(The)-5 b(or)g(em)33 b(1.3)i(hold.)111 2580 y(Pr)-5 b(o)g(of.)58 b Fx(W)-8 b(e)38 b(m)m(ust)g(v)m(erify)h (that)f(the)g(conditions)g(on)f Ft(p)p Fx(,)i Ft(\014)k Fx(and)38 b Ft(\025)e Fx(=)g(2)17 b(cos)f Ft(')38 b Fx(whic)m(h)h (de\014ne)f(\(1.9\))111 2725 y(and)33 b(\(1.10\))f(imply)h(\(4.3\))f (and)h(\(4.4\),)f(resp)s(ectiv)m(ely)-8 b(.)258 2869 y(By)32 b(assumption)i(\(4.9\))d(the)i(large)f Ft(N)10 b Fx({b)s(eha)m(vior)33 b(of)e Ft(R)2283 2884 y Fp(N)2383 2869 y Fx(is)h(go)m(v)m(erned)i(b)m(y)f(\(4.7\))f(with)g(the)h(sum)111 3014 y(in)39 b(the)h(exp)s(onen)m(t)g(substituted)h(b)m(y)f(the)f(in)m (tegral)g(\(4.9\).)62 b(In)39 b(order)g(to)g(emplo)m(y)h(Theorem)g(4.1) f(it)111 3158 y(is)c(necessary)i(to)e(ascertain)g(that)f Ft(f)46 b Fx(is)35 b(a)f(Riemann{in)m(tegrable)i(p)s(erio)s(dic)f (function)g(of)f(p)s(erio)s(d)h Ft(\031)t Fx(,)111 3303 y(and)i(compute)h(its)g(in)m(tegral.)57 b(By)37 b(simple)h (trigonometric)g(iden)m(tities)h(\(3.11\))d(ma)m(y)i(b)s(e)f(written)h (in)111 3447 y(the)33 b(form)1279 3592 y Ft(f)11 b Fx(\()p Ft(\022)s Fx(\))27 b(=)h(ln)17 b(\()p Ft(a)22 b Fx(+)g Ft(b)17 b Fx(cos)g(2)p Ft(\022)25 b Fx(+)d Ft(c)17 b Fx(sin)g(2)p Ft(\022)s Fx(\))919 b(\(4.10\))111 3781 y(where)1333 3925 y(2)p Ft(a)28 b Fx(=)1564 3845 y Fs(\000)1610 3925 y Fx(1)22 b Fq(\000)g Ft(p)1829 3884 y Fr(2)1869 3845 y Fs(\001)1914 3867 y Fr(2)1971 3925 y Fx(cot)2101 3884 y Fr(2)2157 3925 y Ft(')g Fx(+)g(1)g(+)g Ft(p)2559 3884 y Fr(4)1337 4115 y Fx(2)p Ft(b)28 b Fx(=)1559 4034 y Fs(\000)1604 4115 y Fx(1)22 b Fq(\000)h Ft(p)1824 4073 y Fr(2)1863 4034 y Fs(\001)1909 4056 y Fr(2)1965 4115 y Fx(cot)2095 4073 y Fr(2)2151 4115 y Ft(')f Fq(\000)h Fx(1)f(+)g Ft(p)2555 4073 y Fr(4)111 4304 y Fx(and)1568 4448 y Ft(c)28 b Fx(=)1742 4368 y Fs(\000)1787 4448 y Fx(1)22 b Fq(\000)h Ft(p)2007 4407 y Fr(2)2046 4368 y Fs(\001)2108 4448 y Fx(cot)17 b 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Fv(3)f(of)h([KLS],)h(it)g(is)f(enough)111 5338 y(to)28 b(hold)f(for)g(t)n(w)n(o)g(distinct)h(v)-5 b(alues.)1917 5587 y Fx(12)p eop %%Page: 13 13 13 12 bop 111 312 a Fx(since)1518 456 y(0)28 b Ft(<)f(b)1739 415 y Fr(2)1801 456 y Fx(+)22 b Ft(c)1941 415 y Fr(2)2008 456 y Fx(=)28 b Ft(a)2163 415 y Fr(2)2225 456 y Fq(\000)22 b Ft(p)2373 415 y Fr(4)3571 456 y Fx(\(4.12\))111 655 y(if)31 b(0)d Ft(<)f(p)h(<)f Fx(1,)32 b(and)f(this)h(implies)h(that)e Ft(f)11 b Fx(\()p Ft(\022)s Fx(\))31 b(is)g(Riemann{in)m(tegrable.)45 b(F)-8 b(rom)31 b([PBM],)h Fl(29)f Fx(p.)44 b(546)111 799 y(and)33 b(\(4.12\),)f(w)m(e)h(ha)m(v)m(e)852 1017 y(1)p 847 1061 59 4 v 847 1152 a Ft(\031)932 948 y Fs(Z)1032 975 y Fp(\031)988 1174 y Fr(0)1096 1084 y Ft(f)11 b Fx(\()p Ft(\022)s Fx(\))17 b Ft(d\022)85 b Fx(=)1675 1017 y(1)p 1646 1061 108 4 v 1646 1152 a(2)p Ft(\031)1780 948 y Fs(Z)1880 975 y Fr(2)p Fp(\031)1835 1174 y Fr(0)1978 1084 y Fx(ln)17 b(\()p Ft(a)23 b Fx(+)f Ft(b)17 b Fx(cos)g Ft(x)23 b Fx(+)f Ft(c)17 b Fx(sin)g Ft(x)p Fx(\))33 b Ft(dx)1477 1364 y Fx(=)83 b(ln)1734 1224 y Fs(\022)1817 1297 y Ft(a)22 b Fx(+)1988 1215 y Fq(p)p 2071 1215 497 4 v 82 x Ft(a)2122 1268 y Fr(2)2184 1297 y Fq(\000)h Ft(b)2325 1268 y Fr(2)2387 1297 y Fq(\000)f Ft(c)2528 1268 y Fr(2)p 1817 1342 751 4 v 2168 1433 a Fx(2)2578 1224 y Fs(\023)1477 1639 y Fx(=)83 b(ln)1734 1498 y Fs(\022)1817 1571 y Fx(\(1)22 b Fq(\000)g Ft(p)2074 1535 y Fr(2)2114 1571 y Fx(\))2152 1535 y Fr(2)2208 1571 y Fx(csc)2333 1535 y Fr(2)2389 1571 y Ft(')p 1817 1616 636 4 v 2110 1707 a Fx(4)2485 1639 y(+)g Ft(p)2632 1598 y Fr(2)2671 1498 y Fs(\023)111 1916 y Fx(whic)m(h,)34 b(together)f(with)g(\(4.7\))f (and)h(\(4.9\),)f(yields)440 2214 y Ft(C)510 2229 y Fr(1)582 2074 y Fs(\022)666 2147 y Fx(\(1)21 b Fq(\000)i Ft(p)923 2111 y Fr(2)962 2147 y Fx(\))1000 2111 y Fr(2)1056 2147 y Fx(csc)1182 2111 y Fr(2)1238 2147 y Ft(')p 666 2191 V 915 2283 a Fx(4)p Ft(p)1013 2254 y Fr(2)1334 2214 y Fx(+)f(1)1481 2074 y Fs(\023)1554 2094 y Fp(N)1649 2214 y Fq(\024)28 b Ft(R)1829 2173 y Fr(2)1828 2239 y Fp(N)1923 2214 y Fq(\024)g Ft(C)2098 2229 y Fr(2)2154 2074 y Fs(\022)2237 2147 y Fx(\(1)22 b Fq(\000)h Ft(p)2495 2111 y Fr(2)2534 2147 y Fx(\))2572 2111 y Fr(2)2628 2147 y Fx(csc)2753 2111 y Fr(2)2809 2147 y Ft(')p 2237 2191 V 2487 2283 a Fx(4)p Ft(p)2585 2254 y Fr(2)2905 2214 y Fx(+)f(1)3052 2074 y Fs(\023)3126 2094 y Fp(N)3571 2214 y Fx(\(4.13\))111 2486 y(for)32 b(some)i(p)s(ositiv)m(e)g(constan)m(ts)g Ft(C)1368 2501 y Fr(1)1439 2486 y Fx(and)f Ft(C)1699 2501 y Fr(2)1771 2486 y Fx(uniformly)g(in)g Ft(N)10 b Fx(.)44 b(Consequen)m(tly)-8 b(,)36 b(using)888 2635 y Fs(X)880 2846 y Fp(m)p Fn(\025)p Fp(n)1057 2730 y Fq(k)p Ft(T)30 b Fx(\()p Ft(m)p Fx(;)17 b Ft(\025)p Fx(\))p Fq(k)1506 2682 y Fn(\000)p Fr(2)1506 2759 y Fp(U)1683 2730 y Fq(\024)1850 2635 y Fs(X)1844 2848 y Fp(k)r Fn(\025)p Fp(N)2017 2730 y Fq(k)p Ft(T)30 b Fx(\()p Ft(a)2243 2745 y Fp(k)2286 2730 y Fx(;)17 b Ft(\025)p Fx(\))o Fq(k)2474 2682 y Fn(\000)p Fr(2)2474 2759 y Fp(U)2713 2635 y Fs(X)2585 2846 y Fp(a)2622 2858 y Fi(k)2661 2846 y Fn(\024)p Fp(m)f Fx(\(1)22 b Fq(\000)h Ft(p)2171 4941 y Fr(2)2210 4982 y Fx(\))2248 4941 y Fr(2)2304 4982 y Fx(csc)2429 4941 y Fr(2)2485 4982 y Ft(')17 b(:)978 b Fx(\(4.15\))111 5181 y(In)30 b(view)g(of)e(\(4.3\))h(and)g(\(4.2\),)g Ft(\025)f Fq(2)g Ft(\033)t Fx(\()p Ft(J)9 b Fx(\))28 b Fq(\022)g Fx([)p Fq(\000)p Fx(2)p Ft(;)17 b Fx(2])29 b(b)s(elongs)g(to)g(the)g(essen)m (tial)i(supp)s(ort)f(of)e(the)i(s.c.)111 5325 y(sp)s(ectrum)k(of)e Ft(J)42 b Fx(pro)m(vided)34 b(\(4.15\))e(holds)h(whic)m(h,)h(in)f(view) h(of)e(4)17 b(csc)2629 5289 y Fn(\000)p Fr(2)2740 5325 y Ft(')44 b Fx(=)27 b(4)22 b Fq(\000)h Ft(\025)3179 5289 y Fr(2)3218 5325 y Fx(,)33 b(yields)h(\(1.9\).)1917 5587 y(13)p eop %%Page: 14 14 14 13 bop 258 312 a Fx(No)m(w,)33 b(assuming)h(that)e(\(4.14\))g (holds,)h(in)g(view)h(of)e(\(4.4\),)g Ft(\025)c Fq(2)g Ft(\033)t Fx(\()p Ft(J)9 b Fx(\))32 b(is)h(an)g(eigen)m(v)-5 b(alue)34 b(of)e Ft(J)42 b Fx(if)801 601 y Ft(S)33 b Fx(=)1039 476 y Fn(1)1002 506 y Fs(X)998 718 y Fp(N)7 b Fr(=1)1167 601 y Ft(\014)1228 560 y Fp(N)1295 601 y Ft(R)1370 560 y Fr(2)1369 626 y Fp(N)1453 601 y Ft(S)1519 560 y Fr(2)1513 626 y Fp(N)1608 601 y Fq(\024)28 b Ft(C)1790 536 y Fg(0)1875 476 y Fn(1)1838 506 y Fs(X)1833 718 y Fp(N)7 b Fr(=1)2003 460 y Fs(\022)2415 533 y Fx(4)p Ft(p)2513 497 y Fr(2)2552 533 y Ft(\014)2613 497 y Fr(3)p 2087 578 894 4 v 2087 669 a Fx(\(1)22 b Fq(\000)g Ft(p)2344 640 y Fr(2)2384 669 y Fx(\))2422 640 y Fr(2)2477 669 y Fx(csc)2603 640 y Fr(2)2659 669 y Ft(')g Fx(+)g(4)p Ft(p)2941 640 y Fr(2)2990 460 y Fs(\023)3063 481 y Fp(N)111 893 y Fx(con)m(v)m(erges,)35 b(i.e.,)1478 999 y Ft(p)1527 963 y Fr(2)p 1336 1044 375 4 v 1336 1135 a Fx(\(1)21 b Fq(\000)i Ft(p)1593 1106 y Fr(2)1632 1135 y Fx(\))1670 1106 y Fr(2)1720 1067 y Fx(\()p Ft(\014)1819 1025 y Fr(3)1880 1067 y Fq(\000)f Fx(1\)\(4)g Fq(\000)h Ft(\025)2332 1025 y Fr(2)2371 1067 y Fx(\))28 b Ft(<)f Fx(1)982 b(\(4.16\))111 1293 y(whic)m(h)34 b(yields)h(\(1.10\))c(and)i(concludes)i(the)e(pro)s (of)e(of)h(Theorem)i(4.2.)3745 1437 y Fh(2)111 1679 y Fl(Remark)j(4.3)227 1864 y Fk(1.)48 b(The)27 b(c)-5 b(onditions)26 b(\(4.15\))g(for)h(diver)-5 b(genc)g(e)26 b(of)h Ft(S)2068 1879 y Fp(N)2163 1864 y Fk(and)g(\(4.16\))f(for)h(c)-5 b(onver)g(genc)g(e)26 b(of)h Ft(S)33 b Fk(wil)5 b(l)27 b(not)355 2008 y(change)i(if)g Ft(J)39 b Fk(b)-5 b(elongs)28 b(to)i(the)g(class)f Fm(H)q Ft(=)p Fm(H)1910 2023 y Fr(0)1977 2008 y Fk(with)h Fx(lim)2319 2023 y Fp(k)r Fn(!1)2520 2008 y Ft(a)2571 2023 y Fp(k)2613 2008 y Ft(=a)2713 2023 y Fp(k)r Fn(\000)p Fr(1)2874 2008 y Fx(=)d Ft(\014)36 b Fk(and)29 b Fx(lim)3387 2023 y Fp(k)r Fn(!1)3588 2008 y Ft(p)3637 2023 y Fp(a)3674 2035 y Fi(k)3744 2008 y Fx(=)355 2153 y Ft(p)p Fk(.)111 2521 y Fy(5)161 b(Uniform)54 b(Distribution)g(of)g(Pr)t(\177)-85 b(ufer)53 b(Angles)111 2764 y Fx(W)-8 b(e)49 b(shall)h(no)m(w)f(establish)h(\(4.9\))f(for)f (\()p Ft(\022)1675 2779 y Fp(k)1718 2764 y Fx(\))1756 2793 y Fp(k)r 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Ft(g)t Fx(\()p Ft(p;)17 b(\022)s Fx(\))26 b(=)1482 4192 y Fs(\()1603 4294 y Fq(\000)p Ft(\031)t(=)p Fx(2)83 b(if)90 b Fq(\000)p Ft(\031)t(=)p Fx(2)28 b Ft(<)g(\022)i(<)e(')22 b Fq(\000)h Ft(\031)t(=)p Fx(2)1603 4438 y Ft(\031)t(=)p Fx(2)160 b(if)90 b Ft(')22 b Fq(\000)h Ft(\031)t(=)p Fx(2)k Ft(<)h(\022)j(<)c(\031)t(=)p Fx(2)111 4660 y(and)33 b(\(4.8\))f(is)h(simply)h(giv)m(en)g(b)m(y)1634 4805 y Ft(\022)1679 4820 y Fp(k)1750 4805 y Fx(=)27 b Fq(\006)1940 4737 y Ft(\031)p 1940 4782 59 4 v 1945 4873 a Fx(2)2032 4805 y(+)22 b Ft(\014)2191 4763 y Fp(k)2233 4805 y Ft(')1322 b Fx(\(5.1\))111 4999 y(as)33 b(long)f(as)1565 5144 y(tan)17 b Ft(\022)1768 5159 y Fp(k)1833 5144 y Fx(+)22 b(cot)17 b Ft(')28 b Fq(6)p Fx(=)f(0)17 b Ft(;)1253 b Fx(\(5.2\))111 5338 y(i.)44 b(e.,)33 b(as)g(long)f(as)h Ft(\022)835 5353 y Fp(k)878 5338 y Fx(\()p Ft(')p Fx(\))39 b Ft(=)-61 b Fq(2)28 b(f)p Ft(')22 b Fx(+)g(\(2)p Ft(m)g Fx(+)g(1\))p Ft(\031)t(=)p Fx(2)p Ft(;)33 b(m)28 b Fq(2)g Fo(N)9 b Fq(g)33 b(\021)c Fm(M)p Fx(\()p Ft(')p Fx(\).)1917 5587 y(14)p eop %%Page: 15 15 15 14 bop 258 312 a Fx(Eac)m(h)29 b(p)s(oin)m(t)h(of)e Fm(M)h Fx(is)g(a)g(linear)g(function)h(of)e Ft(')h Fx(with)h(slop)s(e)f (1)g(.)42 b(Using)30 b(the)f(fact)g(that)g Ft(g)3492 276 y Fn(0)3514 312 y Fx(\()p Ft(\022)s Fx(\))f Fq(\025)g Fx(0)111 456 y(\(see)34 b(\(5.7\))e(b)s(elo)m(w\),)h(it)g(follo)m(ws)g (from)1394 617 y Ft(@)5 b(\022)1495 632 y Fp(k)p 1394 662 146 4 v 1406 753 a Ft(@)g(')1577 685 y Fx(=)27 b Ft(g)1731 643 y Fn(0)1754 685 y Fx(\()p Ft(\022)1837 700 y Fp(k)r Fn(\000)p Fr(1)1970 685 y Fx(\))2035 617 y Ft(@)5 b(\022)2136 632 y Fp(k)r Fn(\000)p Fr(1)p 2035 662 236 4 v 2092 753 a Ft(@)g(')2302 685 y Fx(+)22 b Ft(\014)2461 643 y Fp(k)2520 685 y Ft(;)111 921 y Fx(and)h(an)g (inductiv)m(e)i(argumen)m(t)f(that)e Ft(@)5 b(\022)1559 936 y Fp(k)1603 921 y Ft(=@)g(')29 b Fq(\025)f Ft(\014)1967 885 y Fp(k)2032 921 y Fx(and)23 b Ft(\022)2257 936 y Fp(k)2323 921 y Fx(is)g(a)g(monotone)g(increasing)h(function)f(of)111 1066 y Ft(')h Fx(.)41 b(Hence,)28 b(as)c Ft(')h Fx(v)-5 b(aries)25 b(o)m(v)m(er)g(\(0)p Ft(;)17 b(\031)t Fx(\))o(,)26 b Ft(\022)1569 1081 y Fp(k)1613 1066 y Fx(\()p Ft(')p Fx(\))e(in)m(tercepts)i(eac)m(h)f(linear)g(function)g Ft(')5 b Fx(+)g(\(2)p Ft(m)g Fx(+)g(1\))p Ft(\031)t(=)p Fx(2)25 b(of)111 1210 y Fm(M)j Fx(in)g(at)g(most)g(one)h(p)s(oin)m(t.) 42 b(F)-8 b(or)27 b(eac)m(h)i Ft(k)s Fx(,)g(\012)1726 1225 y Fp(k)1797 1210 y Fx(=)e Fq(f)p Ft(')h Fq(2)g Fx(\(0)p Ft(;)17 b(\031)t Fx(\))27 b(:)g Ft(\022)2490 1225 y Fp(k)2533 1210 y Fx(\()p Ft(')p Fx(\))h Fq(2)g Fm(M)p Fx(\()p Ft(')p Fx(\))p Fq(g)f Fx(is)i(a)f(den)m(umerable)111 1355 y(\(\014nite\))33 b(set)f(and)g(\(5.1\))g(holds)g(for)g(all)f Ft(k)g Fq(2)d Fo(N)47 b Fx(and)32 b(all)g Ft(')27 b Fq(2)h Fx(\(0)p Ft(;)17 b(\031)t Fx(\))32 b(except)h(for)e(a)h(den)m(umerable)i(set)111 1499 y(\012)28 b(=)313 1424 y Fs(S)396 1528 y Fp(k)455 1499 y Fx(\012)525 1514 y Fp(k)568 1499 y Fx(.)258 1644 y(No)m(w,)33 b(giv)m(en)g(a)g(sequence)i(of)d(n)m(um)m(b)s(ers)j(\()p Ft(u)1846 1659 y Fp(k)1888 1644 y Fx(\))1926 1673 y Fp(k)r Fn(\025)p Fr(1)2059 1644 y Fx(let)d(\()p Ft(S)2297 1659 y Fp(n)2344 1644 y Fx(\))2382 1673 y Fp(n)p Fn(\025)p Fr(1)2552 1644 y Fx(b)s(e)h(de\014ned)h(b)m(y)1603 1914 y Ft(S)1663 1929 y Fp(n)1738 1914 y Fx(=)1856 1846 y(1)p 1851 1891 59 4 v 1851 1982 a Ft(n)1987 1789 y Fp(n)1936 1819 y Fs(X)1944 2031 y Fp(k)r Fr(=1)2097 1914 y Ft(e)2142 1873 y Fr(2)p Fp(iu)2242 1885 y Fi(k)2301 1914 y Ft(:)111 2183 y Fx(According)29 b(to)g(W)-8 b(eyl)29 b(criterion,)h(the)f (sequence)i(\()p Ft(u)2000 2198 y Fp(k)2042 2183 y Fx(\))2080 2212 y Fp(k)r Fn(\025)p Fr(1)2241 2183 y Fx(is)e(u.d.)43 b(mo)s(d)28 b Ft(\031)33 b Fx(if)28 b(and)h(only)f(if)53 b(lim)3409 2243 y Fp(n)p Fn(!1)3609 2183 y Ft(S)3669 2198 y Fp(n)3744 2183 y Fx(=)111 2336 y(0)17 b Ft(:)p Fx(If)33 b Ft(u)358 2351 y Fp(k)429 2336 y Fx(=)28 b Ft(u)589 2351 y Fp(k)632 2336 y Fx(\()p Ft(x)p Fx(\))33 b(is)h(de\014ned)g(for)f(eac)m(h)h Ft(x)g Fx(lying)g(in)f(some)h(in)m (terv)-5 b(al)34 b([)p Ft(a;)17 b(b)p Fx(])29 b Fq(\032)g Fo(R)5 b Fx(,)40 b(then)34 b Ft(S)3385 2351 y Fp(n)3460 2336 y Fx(=)29 b Ft(S)3625 2351 y Fp(n)3689 2336 y Fx(\()o Ft(x)p Fx(\))111 2548 y(is)f(de\014ned)h(for)d(eac)m(h)j Ft(x)f Fq(2)g Fx([)p Ft(a;)17 b(b)p Fx(].)42 b(Denoting)27 b(b)m(y)h Fq(k)p Ft(f)11 b Fq(k)2034 2577 y Fp(L)2082 2586 y Ff(2)2116 2577 y Fr(\()q([)o Fp(a;b)p Fr(]\))2329 2548 y Fx(=)2433 2407 y Fs(\022)2506 2412 y(Z)2606 2438 y Fp(b)2562 2638 y(a)2657 2548 y Fq(j)o Ft(f)g Fx(\()p Ft(x)p Fx(\))p Fq(j)2902 2499 y Fr(2)2958 2548 y Ft(dx)3064 2407 y Fs(\023)3138 2419 y Fr(1)p Fp(=)p Fr(2)3275 2548 y Fx(the)28 b Ft(L)3504 2563 y Fr(2)3571 2548 y Fx(norm,)111 2725 y(the)33 b(general)g(metric)h(result)f(is)g(as)g(follo)m(ws)g (\(see)h(Theorem)g(4)p Ft(:)p Fx(2)e(of)g([KN],)h(for)f(a)g(pro)s(of)7 b(\))111 2931 y Fl(Theorem)37 b(5.1)49 b Fk(If)35 b(the)f(series)1637 2997 y Fn(1)1601 3027 y Fs(X)1606 3236 y Fp(n)p Fr(=1)1776 3054 y Fx(1)p 1771 3099 V 1771 3190 a Ft(n)1856 3122 y Fq(k)p Ft(S)1966 3137 y Fp(n)2013 3122 y Fq(k)2062 3073 y Fr(2)2062 3151 y Fp(L)2110 3160 y Ff(2)2145 3151 y Fr(\([)p Fp(a;b)p Fr(]\))111 3366 y Fk(c)-5 b(onver)g(ges)37 b(then)h(the)g(se)-5 b(quenc)g(e)38 b Fq(f)p Ft(u)1440 3381 y Fp(k)1482 3366 y Fq(g)g Fk(is)g(u.d.)55 b(mo)-5 b(d)37 b Ft(\031)42 b Fk(for)c(almost)g(al)5 b(l)38 b Ft(x)c Fq(2)g Fx([)p Ft(a;)17 b(b)p Fx(])q Fk(,)39 b(i.)54 b(e.)h Fq(f)p Ft(u)3620 3381 y Fp(k)3662 3366 y Fq(g)38 b Fk(is)111 3510 y(u.d.)45 b(mo)-5 b(d)34 b Ft(\031)39 b Fk(for)c(every)f Ft(x)i Fk(lying)e(in)h Fx([)p Ft(a;)17 b(b)p Fx(])35 b Fk(ap)-5 b(art)35 b(fr)-5 b(om)34 b(a)h(set)g(with)g(L) -5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e)34 b Fx(0)p Fk(.)258 3715 y Fx(Returning)42 b(to)g(the)h(sequence)i(\()p Ft(\022)1533 3730 y Fp(k)1576 3715 y Fx(\()p Ft(\014)6 b Fx(\)\))1751 3745 y Fp(k)r Fn(\025)p Fr(1)1926 3715 y Fx(in)42 b(the)h(limit)g Ft(p)h Fq(!)g Fx(0,)h(except)e(b)m(y)h(the)e(set)i(\012)e(of)111 3860 y(v)-5 b(alues)29 b Ft(')f Fx(of)g(Leb)s(esgue)h(measure)h(0)e (that)f(\(5.1\))h(do)g(not)g(hold,)h Ft(\022)2486 3875 y Fp(k)2557 3860 y Fx(=)f Fq(\006)p Ft(\031)t(=)p Fx(2)13 b(+)g Ft(\014)3058 3824 y Fp(k)3100 3860 y Ft(')28 b Fx(is)g(con)m(tin)m(uously)111 4004 y(di\013eren)m(tiable)34 b(with)g(resp)s(ect)g(to)e Ft(\014)38 b Fx(and)1189 4207 y Ft(\022)1237 4166 y Fn(0)1234 4232 y Fp(k)1299 4207 y Fq(\000)23 b Ft(\022)1447 4166 y Fn(0)1444 4232 y Fp(m)1538 4207 y Fx(=)28 b Ft(\014)1703 4166 y Fp(k)r Fn(\000)p Fr(1)1852 4127 y Fs(\000)1898 4207 y Ft(k)d Fq(\000)e Ft(\014)2135 4166 y Fn(\000)p Fp(k)r Fr(+)p Fp(m)2349 4207 y Ft(m)2434 4127 y Fs(\001)2497 4207 y Ft(')k Fq(\025)h Fx(1)877 b(\(5.3\))111 4410 y(increases)48 b(monotonously)e(for)f(an)m (y)h Ft(k)53 b(>)c(m)p Fx(.)82 b(These)47 b(are)f(su\016cien)m(t)h(to)f (apply)g(the)f(follo)m(wing)111 4555 y(p)s(o)m(w)m(erful)34 b(result.)111 4760 y Fl(Theorem)j(5.2)h(\(Koksma's)f(Metric)f(Theorem)h (\))49 b Fk(L)-5 b(et)45 b Fx(\()p Ft(u)2523 4775 y Fp(k)2565 4760 y Fx(\()p Ft(x)p Fx(\)\))2735 4789 y Fp(k)r Fn(\025)p Fr(1)2913 4760 y Fk(b)-5 b(e)45 b(a)g(se)-5 b(quenc)g(e)44 b(of)h(r)-5 b(e)g(al)111 4905 y(value)g(d)38 b(c)-5 b(ontinuously)37 b(di\013er)-5 b(entiable)37 b(functions)h(de\014ne)-5 b(d)36 b(in)i(an)f(interval)h Fx([)p Ft(a;)17 b(b)p Fx(])p Fk(.)54 b(If)37 b(for)h(any)g(two)111 5049 y(p)-5 b(ositive)41 b(inte)-5 b(gers)41 b Ft(k)j Fq(6)p Fx(=)c Ft(m)p Fk(,)j Ft(u)1263 5013 y Fn(0)1263 5075 y Fp(k)1333 5049 y Fq(\000)27 b Ft(u)1493 5013 y Fn(0)1493 5074 y Fp(m)1601 5049 y Fk(is)41 b(a)h(monotone)e(function)i(of)f Ft(x)h Fk(such)g(that)g Fq(j)o Ft(u)3388 5013 y Fn(0)3388 5075 y Fp(k)3453 5049 y Fq(\000)22 b Ft(u)3608 5013 y Fn(0)3608 5074 y Fp(m)3674 5049 y Fq(j)40 b(\025)111 5194 y Ft(K)35 b(>)28 b Fx(0)33 b Fk(holds)g(uniformly)h(in)f Ft(k)s Fk(,)h Ft(m)h Fk(and)e Ft(x)28 b Fq(2)g Fx([)p Ft(a;)17 b(b)p Fx(])p Fk(,)35 b(then)e Fx(\()p Ft(u)2392 5209 y Fp(k)2435 5194 y Fx(\))2472 5223 y Fp(k)r Fn(\025)p Fr(1)2639 5194 y Fk(is)h(u.d.)44 b(mo)-5 b(d)33 b Ft(\031)38 b Fk(for)c(almost)f(al)5 b(l)111 5338 y Ft(x)28 b Fq(2)g Fx([)p Ft(a;)17 b(b)p Fx(])p Fk(.)1917 5587 y Fx(15)p eop %%Page: 16 16 16 15 bop 258 312 a Fx(Theorem)40 b(5.2)f(cannot)g(b)s(e)g(emplo)m(y)m (ed)i(for)e Ft(p)g Fq(2)g Fx(\(0)p Ft(;)17 b Fx(1\))38 b(in)h(the)h(presen)m(t)g(form.)63 b(Ho)m(w)m(ev)m(er,)43 b(for)111 456 y(con)m(v)m(enience,)j(the)41 b(pro)s(of)f(of)g(Koksma's) i(theorem)f(will)g(b)s(e)g(included)i(in)d(order)h(to)f(iden)m(tify)i (the)111 601 y(problems)34 b(and)f(the)g(w)m(a)m(ys)h(to)e(circum)m(v)m (en)m(t)k(them.)111 745 y Fk(Pr)-5 b(o)g(of.)43 b Fx(By)33 b(de\014nition,)676 1000 y Fq(k)p Ft(S)786 1015 y Fp(n)833 1000 y Fq(k)883 952 y Fr(2)883 1029 y Fp(L)931 1038 y Ff(2)966 1029 y Fr(\([)p Fp(a;b)p Fr(]\))1235 1000 y Fx(=)1429 933 y(1)p 1405 977 98 4 v 1405 1069 a Ft(n)1463 1040 y Fr(2)1613 876 y Fp(n)1562 906 y Fs(X)1529 1118 y Fp(k)r(;m)p Fr(=1)1756 865 y Fs(Z)1856 891 y Fp(b)1812 1090 y(a)1907 1000 y Ft(e)1952 959 y Fr(2)p Fp(i)p Fr(\()q Fp(u)2080 971 y Fi(k)2118 959 y Fr(\()p Fp(x)p Fr(\))p Fn(\000)p Fp(u)2308 967 y Fi(m)2367 959 y Fr(\()p Fp(x)p Fr(\)\))2509 1000 y Ft(dx)1234 1354 y Fq(\024)1405 1287 y(j)o Ft(b)23 b Fq(\000)g Ft(a)p Fq(j)p 1405 1332 270 4 v 1510 1423 a Ft(n)1707 1354 y Fx(+)1839 1287 y(2)p 1815 1332 98 4 v 1815 1423 a Ft(n)1873 1394 y Fr(2)1989 1230 y Fp(n)1939 1260 y Fs(X)1946 1472 y Fp(k)r Fr(=2)2111 1230 y Fp(k)r Fn(\000)p Fr(1)2104 1260 y Fs(X)2099 1469 y Fp(m)p Fr(=1)2268 1210 y Fs(\014)2268 1270 y(\014)2268 1330 y(\014)2268 1389 y(\014)2302 1219 y(Z)2401 1245 y Fp(b)2357 1444 y(a)2452 1354 y Ft(e)2497 1313 y Fr(2)p Fp(i)p Fr(\()q Fp(u)2625 1325 y Fi(k)2663 1313 y Fr(\()p Fp(x)p Fr(\))p Fn(\000)p Fp(u)2853 1321 y Fi(m)2912 1313 y Fr(\()p Fp(x)p Fr(\)\))3055 1354 y Ft(dx)3161 1210 y Fs(\014)3161 1270 y(\014)3161 1330 y(\014)3161 1389 y(\014)3227 1354 y Ft(:)111 1624 y Fx(Under)41 b(the)g(h)m(yp)s (othesis,)k(the)c(second)h(mean)f(v)-5 b(alue)40 b(theorem)i(holds)f (and)f(there)i(exist)f Ft(c)g Fx(,)h(with)111 1768 y Ft(a)28 b(<)g(c)f(<)h(b)p Fx(,)33 b(suc)m(h)h(that)494 1871 y Fs(\014)494 1931 y(\014)494 1991 y(\014)494 2051 y(\014)527 1880 y(Z)626 1906 y Fp(b)582 2106 y(a)677 2016 y Ft(e)722 1975 y Fr(2)p Fp(i)p Fr(\()q Fp(u)850 1987 y Fi(k)888 1975 y Fr(\()p Fp(x)p Fr(\))p Fn(\000)p Fp(u)1078 1983 y Fi(m)1137 1975 y Fr(\()p Fp(x)p Fr(\)\))1280 2016 y Ft(dx)1386 1871 y Fs(\014)1386 1931 y(\014)1386 1991 y(\014)1386 2051 y(\014)1503 2016 y Fx(=)1663 1871 y Fs(\014)1663 1931 y(\014)1663 1991 y(\014)1663 2051 y(\014)1723 1948 y Fx(1)p 1706 1993 83 4 v 1706 2084 a(2)p Ft(i)1815 1880 y Fs(Z)1914 1906 y Fp(b)1870 2106 y(a)1975 1948 y Ft(de)2071 1912 y Fr(2)p Fp(i)p Fr(\()q Fp(u)2199 1924 y Fi(k)2237 1912 y Fr(\()p Fp(x)p Fr(\))p Fn(\000)p Fp(u)2427 1920 y Fi(m)2486 1912 y Fr(\()p Fp(x)p Fr(\)\))p 1975 1993 637 4 v 1991 2084 a Ft(u)2047 2050 y Fn(0)2047 2112 y Fp(k)2089 2084 y Fx(\()p Ft(x)p Fx(\))23 b Fq(\000)g Ft(u)2399 2055 y Fn(0)2399 2109 y Fp(m)2465 2084 y Fx(\()p Ft(x)p Fx(\))2622 1871 y Fs(\014)2622 1931 y(\014)2622 1991 y(\014)2622 2051 y(\014)1503 2289 y Fx(=)1663 2145 y Fs(\014)1663 2205 y(\014)1663 2265 y(\014)1663 2324 y(\014)1723 2222 y Fx(1)p 1706 2267 83 4 v 1706 2358 a(2)p Ft(i)1815 2149 y Fs(\022)2172 2222 y Fx(1)p 1898 2267 597 4 v 1898 2358 a Ft(u)1954 2323 y Fn(0)1954 2386 y Fp(k)1996 2358 y Fx(\()p Ft(a)p Fx(\))f Fq(\000)h Ft(u)2301 2329 y Fn(0)2301 2382 y Fp(m)2367 2358 y Fx(\()p Ft(a)p Fx(\))2521 2154 y Fs(Z)2620 2180 y Fp(c)2576 2379 y(a)2672 2289 y Ft(de)2768 2248 y Fr(2)p Fp(i)p Fr(\()p Fp(u)2895 2260 y Fi(k)2933 2248 y Fr(\()p Fp(x)p Fr(\))p Fn(\000)p Fp(u)3123 2256 y Fi(m)3182 2248 y Fr(\()p Fp(x)p Fr(\)\))3335 2289 y Fx(+)1673 2574 y(+)2049 2507 y(1)p 1785 2551 577 4 v 1785 2643 a Ft(u)1841 2608 y Fn(0)1841 2670 y Fp(k)1883 2643 y Fx(\()p Ft(b)p Fx(\))g Fq(\000)g Ft(u)2179 2614 y Fn(0)2179 2667 y Fp(m)2245 2643 y Fx(\()p Ft(b)p Fx(\))2389 2439 y Fs(Z)2488 2465 y Fp(b)2444 2664 y(c)2539 2574 y Ft(de)2635 2533 y Fr(2)p Fp(i)p Fr(\()q Fp(u)2763 2545 y Fi(k)2801 2533 y Fr(\()p Fp(x)p Fr(\))p Fn(\000)p Fp(u)2991 2541 y Fi(m)3050 2533 y Fr(\()p Fp(x)p Fr(\)\))3176 2434 y Fs(\023)3249 2430 y(\014)3249 2490 y(\014)3249 2549 y(\014)3249 2609 y(\014)1502 2835 y Fq(\024)1974 2768 y Fx(1)p 1673 2812 652 4 v 1673 2903 a Fq(j)p Ft(u)1757 2869 y Fn(0)1757 2931 y Fp(k)1799 2903 y Fx(\()p Ft(a)p Fx(\))f Fq(\000)h Ft(u)2104 2875 y Fn(0)2104 2928 y Fp(m)2170 2903 y Fx(\()p Ft(a)p Fx(\))p Fq(j)2357 2835 y Fx(+)2756 2768 y(1)p 2464 2812 633 4 v 2464 2903 a Fq(j)p Ft(u)2548 2869 y Fn(0)2548 2931 y Fp(k)2590 2903 y Fx(\()p Ft(b)p Fx(\))g Fq(\000)f Ft(u)2885 2875 y Fn(0)2885 2928 y Fp(m)2952 2903 y Fx(\()p Ft(b)p Fx(\))p Fq(j)3123 2835 y Ft(:)469 b Fx(\(5.4\))111 3081 y(By)44 b(ordering)g Fq(f)p Ft(u)777 3045 y Fn(0)777 3107 y Fp(k)819 3081 y Fx(\()p Ft(x)p Fx(\))p Fq(g)1000 3032 y Fp(n)1000 3111 y(k)r Fr(=1)1177 3081 y Fx(suc)m(h)h(that)e Ft(u)1686 3045 y Fn(0)1686 3105 y Fr(1)1725 3081 y Fx(\()p Ft(x)p Fx(\))k Ft(<)f(u)2081 3045 y Fn(0)2081 3105 y Fr(2)2120 3081 y Fx(\()p Ft(x)p Fx(\))h Ft(<)f Fq(\001)17 b(\001)g(\001)45 b Ft(<)h(u)2761 3045 y Fn(0)2761 3105 y Fp(n)2808 3081 y Fx(\()p Ft(x)p Fx(\))e(holds)g(for)f(all)h Ft(x)p Fx(,)i(the)111 3225 y(di\013erence)34 b(of)f(an)m(y)g(t)m(w)m(o) g(consecutiv)m(e)i(n)m(um)m(b)s(ers)1936 3141 y Fs(\014)1936 3200 y(\014)1969 3225 y Ft(u)2025 3189 y Fn(0)2025 3251 y Fp(k)2067 3225 y Fx(\()p Ft(x)p Fx(\))23 b Fq(\000)f Ft(u)2376 3189 y Fn(0)2376 3251 y Fp(k)r Fn(\000)p Fr(1)2509 3225 y Fx(\()p Ft(x)p Fx(\))2640 3141 y Fs(\014)2640 3200 y(\014)2701 3225 y Ft(>)27 b(K)40 b Fx(and)33 b(w)m(e)g(ha)m(v)m (e)1019 3389 y Fp(k)r Fn(\000)p Fr(1)1011 3419 y Fs(X)1007 3628 y Fp(m)p Fr(=1)1492 3446 y Fx(1)p 1186 3491 661 4 v 1186 3582 a Fq(j)p Ft(u)1270 3547 y Fn(0)1270 3610 y Fp(k)1312 3582 y Fx(\()p Ft(x)p Fx(\))23 b Fq(\000)f Ft(u)1621 3553 y Fn(0)1621 3606 y Fp(m)1687 3582 y Fx(\()p Ft(x)p Fx(\))p Fq(j)1884 3513 y Ft(<)27 b Fx(2)2108 3389 y Fp(n)2057 3419 y Fs(X)2053 3628 y Fp(m)p Fr(=1)2295 3446 y Fx(1)p 2232 3491 176 4 v 2232 3582 a Ft(mK)2445 3513 y(<)2579 3446 y Fx(2)p 2559 3491 91 4 v 2559 3582 a Ft(K)2675 3513 y Fx(ln)17 b(3)p Ft(n)g(:)111 3780 y Fx(Putting)33 b(all)g(together,)g(yields)1263 4014 y Fq(k)p Ft(S)1373 4029 y Fp(n)1420 4014 y Fq(k)1469 3966 y Fr(2)1469 4043 y Fp(L)1517 4052 y Ff(2)1552 4043 y Fr(\([)p Fp(a;b)p Fr(]\))1765 4014 y Ft(<)1879 3947 y Fq(j)o Ft(b)23 b Fq(\000)g Ft(a)p Fq(j)p 1879 3991 270 4 v 1984 4082 a Ft(n)2180 4014 y Fx(+)2338 3947 y(4)p 2288 3991 149 4 v 2288 4082 a Ft(nK)2463 4014 y Fx(ln)17 b(3)p Ft(n)111 4223 y Fx(whic)m(h)34 b(com)m(bined)h(with)e(Theorem)h (5.1)e(concludes)i(the)f(pro)s(of)f(of)g(the)h(theorem.)3745 4367 y Fh(2)111 4579 y Fl(Remark)k(5.3)227 4760 y Fk(1.)48 b(Sinc)-5 b(e)35 b Ft(a)666 4775 y Fp(k)732 4760 y Fq(\000)23 b Ft(a)883 4775 y Fp(k)r Fn(\000)p Fr(1)1052 4760 y Fk(is)35 b(an)g(inte)-5 b(ger)36 b(numb)-5 b(er)35 b(\(the)g(distanc)-5 b(e)35 b(b)-5 b(etwe)g(en)35 b(two)h(nonzer)-5 b(o)35 b(elements)355 4905 y(of)i Ft(J)d Fq(\000)24 b Ft(J)715 4920 y Fr(0)792 4905 y Fk(along)37 b(the)h(se)-5 b(c)g(ond)36 b(diagonal\))h Ft(\014)6 b Fk(,)37 b(under)h(the)g(p)-5 b(ar)g(ameterization)36 b(\(4.6\),)i(has)f(to)355 5049 y(b)-5 b(e)38 b(an)f(inte)-5 b(ger.)53 b(Henc)-5 b(e,)38 b(Koksma's)f(the)-5 b(or)g(em,)38 b(which)e(holds)h(almost)h(everywher) -5 b(e,)37 b(c)-5 b(annot)355 5194 y(ensur)g(e)35 b(that)g Fx(\()p Ft(\022)945 5209 y Fp(k)988 5194 y Fx(\()p Ft(\014)6 b Fx(\)\))1162 5223 y Fp(k)1240 5194 y Fk(is)34 b(u.d.)45 b(mo)-5 b(d)34 b Ft(\031)k Fk(for)d Ft(\014)e Fx(=)28 b(2)p Ft(;)17 b Fx(3)p Ft(;)g(:)g(:)g(:)n Fk(.)45 b(The)34 b(p)-5 b(ar)g(ameter)34 b(dep)-5 b(endenc)g(e)33 b(of)355 5338 y Ft(\022)400 5353 y Fp(k)478 5338 y Fk(fr)-5 b(om)34 b(her)-5 b(e)35 b(on)f(wil)5 b(l)35 b(b)-5 b(e)34 b(switche)-5 b(d)34 b(fr)-5 b(om)35 b Ft(\014)40 b Fk(to)35 b Ft(')p Fk(.)1917 5587 y Fx(16)p eop %%Page: 17 17 17 16 bop 227 312 a Fk(2.)48 b(T)-7 b(o)45 b(apply)h(The)-5 b(or)g(em)45 b(5.2,)k Ft(\022)1454 276 y Fn(0)1451 338 y Fp(k)1494 312 y Fx(\()p Ft(')p Fx(\))30 b Fq(\000)h Ft(\022)1820 276 y Fn(0)1817 336 y Fp(m)1884 312 y Fx(\()p Ft(')p Fx(\))46 b Fk(has)g(to)g(b)-5 b(e)46 b(a)g(monotone)g(function)g (for)g(e)-5 b(ach)355 456 y(interval)35 b Fx([)p Ft(a;)17 b(b)p Fx(])36 b Fk(indep)-5 b(endent)34 b(of)h Ft(k)k Fk(and)34 b Ft(m)p Fk(.)47 b(But)36 b(this)f(is)g(not)g(true)h(for)g (the)f(se)-5 b(quenc)g(e)35 b(\(4.8\))355 601 y(for)j Fx(0)d Ft(<)f(p)g(<)g Fx(1)p Fk(.)56 b(As)38 b(we)h(shal)5 b(l)37 b(se)-5 b(e,)39 b(the)g(numb)-5 b(er)38 b(of)g(monotone)f (intervals)h(of)g Ft(\022)3417 564 y Fn(0)3414 627 y Fp(k)3482 601 y Fq(\000)26 b Ft(\022)3633 564 y Fn(0)3630 625 y Fp(m)3735 601 y Fk(in)355 745 y Ft(I)36 b Fq(\032)28 b Fx(\(0)p Ft(;)17 b(\031)t Fx(\))o Fk(,)32 b(with)f Ft(k)f(<)e(m)j Fk(\014xe)-5 b(d,)31 b(is)g(pr)-5 b(op)g(ortional)31 b(to)g Ft(\014)2408 709 y Fp(k)2481 745 y Fk(and)g(this)g(numb)-5 b(er)30 b(is)h(c)-5 b(omp)g(ensate)g(d)355 890 y(by)32 b(the)g(c)-5 b(onstant)31 b Ft(K)39 b Fk(of)32 b(The)-5 b(or)g(em)30 b(5.2)i(which,)f(in)g(our)h(c)-5 b(ase,)32 b(has)f(the)h(same)f(dep)-5 b(endenc)g(e)30 b(as)355 1034 y(in)35 b(\(5.3\).)258 1270 y Fx(W)-8 b(e)32 b(no)m(w)i(need)f (some)h(prop)s(erties)f(of)f(\(3.13\).)111 1505 y Fl(Prop)s(osition)k (5.4)49 b Fk(The)36 b(function)h Ft(g)j Fk(given)c(by)i(\(3.13\))e (maps)g Fx([)p Fq(\000)p Ft(\031)t(=)p Fx(2)p Ft(;)17 b(\031)t(=)p Fx(2])36 b Fk(into)h(itself)g(and)f(has)111 1649 y(thr)-5 b(e)g(e)29 b(\014xe)-5 b(d)28 b(p)-5 b(oint)29 b(solutions:)41 b Ft(\022)1295 1613 y Fp(s)1292 1674 y Fn(\006)1379 1649 y Fx(=)28 b Fq(\006)1570 1582 y Ft(\031)p 1570 1627 59 4 v 1575 1718 a Fx(2)1668 1649 y Fk(ar)-5 b(e)29 b(two)g(stable)f(solutions)g(and)h Ft(\022)2903 1613 y Fp(u)2976 1649 y Fx(=)e Ft(')9 b Fq(\000)3248 1582 y Ft(\031)p 3248 1627 V 3253 1718 a Fx(2)3317 1649 y Fk(,)30 b Ft(')e Fq(2)g Fx(\(0)p Ft(;)17 b(\031)t Fx(\))p Fk(,)111 1805 y(is)34 b(unstable.)45 b Ft(g)38 b Fk(is)c(monotone)f (incr)-5 b(e)g(asing)33 b(with)i Ft(g)1995 1769 y Fn(0)2017 1805 y Fx(\()p Ft(\022)2103 1769 y Fp(s)2100 1830 y Fn(\006)2160 1805 y Fx(\))27 b(=)h Ft(p)2378 1769 y Fr(2)2417 1805 y Fk(,)35 b Ft(g)2533 1769 y Fn(0)2555 1805 y Fx(\()p Ft(\022)2641 1769 y Fp(u)2687 1805 y Fx(\))27 b(=)h(1)p Ft(=p)3003 1769 y Fr(2)3076 1805 y Fk(and)34 b(ther)-5 b(e)35 b(ar)-5 b(e)34 b(two)111 1950 y(in\015e)-5 b(ction)35 b(p)-5 b(oints)34 b Ft(\022)874 1913 y Fp(i)871 1974 y Fn(\006)965 1950 y Fk(such)h(that)1210 2179 y Ft(g)1261 2138 y Fn(0)1283 2179 y Fx(\()p Ft(\022)1369 2138 y Fp(i)1366 2203 y Fn(\000)1426 2179 y Fx(\))27 b(=)h Ft(c)1637 2194 y Fr(1)1676 2179 y Ft(p)1725 2138 y Fr(2)1792 2179 y Fq(\024)g Ft(g)1948 2138 y Fn(0)1971 2179 y Fx(\()p Ft(\022)s Fx(\))g Fq(\024)2241 2111 y Ft(c)2283 2126 y Fr(2)p 2238 2156 89 4 v 2238 2247 a Ft(p)2287 2218 y Fr(2)2364 2179 y Fx(=)f Ft(g)2518 2138 y Fn(0)2541 2179 y Fx(\()p Ft(\022)2627 2138 y Fp(i)2624 2203 y Fr(+)2683 2179 y Fx(\))898 b(\(5.5\))111 2440 y Fk(holds)39 b(for)h Ft(c)568 2455 y Fr(1\(2\))734 2440 y Fx(=)c(1)26 b(+)f Ft(O)s Fx(\()p Ft(p)1187 2404 y Fr(2)1226 2440 y Fx(\))p Fk(.)59 b(In)39 b(addition,)h Ft(\022)1950 2404 y Fp(i)1947 2464 y Fr(+)2043 2440 y Fq(!)c Ft(\022)2227 2404 y Fp(u)2272 2440 y Fk(,)41 b Ft(\022)2391 2404 y Fp(i)2388 2464 y Fn(\000)2484 2440 y Fq(!)c Ft(')i Fk(\(or)h Ft(')25 b Fq(\000)h Ft(\031)t Fk(\))40 b(as)f Ft(p)e Fq(!)f Fx(0)j Fk(and)111 2584 y Ft(g)162 2548 y Fn(0)185 2584 y Fx(\()p Ft(\022)s Fx(\))27 b Fq(\024)i Ft(O)19 b Fx(\()o(1\))35 b Fk(for)g Ft(\022)c Fq(2)d Ft(I)8 b Fq(n)1139 2503 y Fs(\002)1179 2584 y Ft(\022)1227 2548 y Fp(i)1224 2609 y Fn(\000)1306 2584 y Fq(\000)23 b Ft(p;)17 b(\022)1547 2548 y Fp(i)1544 2609 y Fr(+)1625 2584 y Fx(+)22 b Ft(p)1772 2503 y Fs(\003)1813 2584 y Fk(.)111 2820 y(Pr)-5 b(o)g(of.)43 b Fx(W)-8 b(riting)33 b Ft(g)t Fx(\()p Ft(\022)s Fx(\))27 b(=)g(tan)1215 2784 y Fn(\000)p Fr(1)1326 2820 y Fx(\()p Ft(\020)8 b(=p)1513 2784 y Fr(2)1551 2820 y Fx(\))33 b(where)1355 3049 y Ft(\020)8 b Fx(\()p Ft(\022)s Fx(\))27 b(=)h(tan)16 b Ft(\022)26 b Fx(+)c(\(1)f Fq(\000)i Ft(p)2244 3008 y Fr(2)2283 3049 y Fx(\))17 b(cot)g Ft(';)1043 b Fx(\(5.6\))111 3278 y(it)33 b(follo)m(ws)g(from)1519 3433 y Ft(g)1570 3392 y Fn(0)1593 3433 y Fx(\()p Ft(\022)s Fx(\))27 b(=)1963 3366 y Ft(p)2012 3329 y Fr(2)p 1858 3410 299 4 v 1858 3501 a Ft(p)1907 3472 y Fr(4)1968 3501 y Fx(+)22 b Ft(\020)2117 3472 y Fr(2)2183 3433 y Fx(sec)2308 3392 y Fr(2)2364 3433 y Ft(\022)1210 b Fx(\(5.7\))111 3651 y(that)43 b Ft(g)k Fx(is)d(a)f(monotone)g(increasing)i(function)f(with)f Ft(g)t Fx(\()p Fq(\000)p Ft(\031)t(=)p Fx(2\))i(=)h Fq(\000)p Ft(\031)t(=)p Fx(2)e(and)f Ft(g)t Fx(\()p Ft(\031)t(=)p Fx(2\))h(=)i Ft(\031)t(=)p Fx(2.)111 3796 y(Hence)30 b Ft(g)h Fx(:)d([)p Fq(\000)p Ft(\031)t(=)p Fx(2)p Ft(;)17 b(\031)t(=)p Fx(2])27 b Fq(!)g Fx([)q Fq(\000)p Ft(\031)t(=)p Fx(2)p Ft(;)17 b(\031)t(=)p Fx(2])o(.)42 b(As)29 b Ft(\022)j Fx(go)s(es)d(to)g Fq(\006)p Ft(\031)t(=)p Fx(2,)g(sec)2691 3760 y Fr(2)2747 3796 y Ft(\022)s(=\020)2895 3760 y Fr(2)2961 3796 y Fq(!)e Fx(1)i(and)g Ft(g)3403 3760 y Fn(0)3425 3796 y Fx(\()p Ft(\022)s Fx(\))f Fq(!)f Ft(p)3753 3760 y Fr(2)3793 3796 y Fx(.)111 3940 y(Since)36 b Ft(p)c(<)f Fx(1,)k Ft(\022)g Fx(=)c Fq(\006)p Ft(\031)t(=)p Fx(2)k(are)f(stable)i (\014xed)g(p)s(oin)m(t)f(of)f Ft(g)t Fx(.)50 b(T)-8 b(o)34 b(\014nd)i(the)f(unstable)h(\014xed)g(p)s(oin)m(t)f(w)m(e)111 4085 y(observ)m(e)g(that)1170 4229 y(tan)17 b Ft(g)t Fx(\()p Ft(\022)s Fx(\))k(+)h(cot)17 b Ft(')27 b Fx(=)1994 4162 y(1)p 1974 4206 89 4 v 1974 4297 a Ft(p)2023 4269 y Fr(2)2072 4229 y Fx(\(tan)17 b Ft(\022)25 b Fx(+)d(cot)17 b Ft(')p Fx(\))32 b Ft(;)111 4447 y Fx(together)h(with)g(the)g(\014xed) h(p)s(oin)m(t)f(equation)g Ft(g)t Fx(\()p Ft(\022)s Fx(\))27 b(=)h Ft(\022)35 b Fx(and)e Ft(p)28 b(<)f Fx(1,)32 b(implies)1585 4677 y(tan)16 b Ft(\022)1790 4636 y Fp(u)1858 4677 y Fx(+)22 b(cot)16 b Ft(')28 b Fx(=)f(0)1273 b(\(5.8\))111 4906 y(whic)m(h)39 b(is)f(equiv)-5 b(alen)m(t)39 b(to)e Ft(\022)1138 4870 y Fp(u)1219 4906 y Fx(=)f Ft(')25 b Fq(\000)h Ft(\031)t(=)p Fx(2.)58 b(A)m(t)37 b Ft(\022)i Fx(=)d Ft(\022)2154 4870 y Fp(u)2236 4906 y Fx(equation)j(\(5.6\))d (reads)j Ft(\020)8 b Fx(\()p Ft(\022)3275 4870 y Fp(u)3319 4906 y Fx(\))36 b(=)f Ft(p)3553 4870 y Fr(2)3609 4906 y Fx(cot)17 b Ft(')111 5051 y Fx(whic)m(h)34 b(together)f(with)g(sec) 1122 5014 y Fr(2)1178 5051 y Ft(\022)1226 5014 y Fp(u)1299 5051 y Fx(=)28 b(1)22 b(+)g(tan)1713 5014 y Fr(2)1769 5051 y Ft(\022)1817 5014 y Fp(u)1890 5051 y Fx(=)27 b(1)22 b(+)g(cot)2292 5014 y Fr(2)2348 5051 y Ft(')33 b Fx(and)f(\(5.7\))g (giv)m(es)1703 5300 y Ft(g)1754 5259 y Fn(0)1776 5300 y Fx(\()p Ft(\022)1862 5259 y Fp(u)1907 5300 y Fx(\))c(=)2106 5233 y(1)p 2086 5277 V 2086 5369 a Ft(p)2135 5340 y Fr(2)2201 5300 y Ft(:)1917 5587 y Fx(17)p eop %%Page: 18 18 18 17 bop 258 312 a Fx(A)32 b(direct)i(computation)f(yields)1018 561 y Ft(g)1069 520 y Fn(00)1111 561 y Fx(\()p Ft(\022)s Fx(\))27 b(=)1391 493 y(2)p Ft(p)1489 457 y Fr(2)1545 493 y Fx(sec)1670 457 y Fr(2)1726 493 y Ft(\022)p 1376 538 414 4 v 1376 641 a Fx(\()p Ft(p)1463 612 y Fr(4)1524 641 y Fx(+)22 b Ft(\020)1673 612 y Fr(2)1712 641 y Fx(\))1750 593 y Fr(2)1816 480 y Fs(\002)1857 561 y Fx(\()p Ft(p)1944 520 y Fr(4)2006 561 y Fx(+)g Ft(\020)2155 520 y Fr(2)2193 561 y Fx(\))17 b(tan)f Ft(\022)26 b Fq(\000)c Ft(\020)i Fx(sec)2767 520 y Fr(2)2823 561 y Ft(\022)2871 480 y Fs(\003)3619 561 y Fx(\(5.9\))111 828 y(from)33 b(where)i(w)m(e)g(get)e (the)h(t)m(w)m(o)g(in\015ection)h(p)s(oin)m(ts)f(b)m(y)g(solving)g(\()p Ft(p)2557 792 y Fr(4)2619 828 y Fx(+)23 b Ft(\020)2769 792 y Fr(2)2808 828 y Fx(\))17 b(tan)f Ft(\022)26 b Fq(\000)d Ft(\020)h Fx(sec)3383 792 y Fr(2)3439 828 y Ft(\022)32 b Fx(=)d(0)k(for)111 973 y(tan)17 b Ft(\022)s Fx(:)915 1128 y Ft(\022)963 1086 y Fp(i)960 1152 y Fn(\006)1047 1128 y Fx(=)27 b(tan)1291 1086 y Fn(\000)p Fr(1)1402 987 y Fs(\022)1476 1128 y Fq(\007)17 b Fx(\(cot)f Ft(')p Fx(\))1856 1079 y Fn(\006)p Fr(1)1967 987 y Fs(\022)2040 1128 y Fx(1)22 b Fq(\006)2299 1060 y Ft(p)2348 1024 y Fr(2)p 2221 1105 246 4 v 2221 1196 a Fx(sec)2346 1167 y Fr(2)2402 1196 y Ft(')2498 1128 y Fx(+)g Ft(O)2690 1047 y Fs(\000)2736 1128 y Ft(p)2785 1086 y Fr(4)2824 1047 y Fs(\001)2870 987 y(\023\023)3571 1128 y Fx(\(5.10\))111 1354 y(giv)m(es)38 b Ft(\022)402 1318 y Fp(i)399 1379 y Fr(+)492 1354 y Fx(=)33 b Ft(\022)649 1318 y Fp(u)719 1354 y Fx(+)24 b Ft(O)s Fx(\()p Ft(p)984 1318 y Fr(2)1023 1354 y Fx(\))35 b(and)h Ft(\022)1337 1318 y Fp(i)1334 1379 y Fn(\000)1427 1354 y Fx(=)e Ft(')24 b Fx(+)h Ft(O)s Fx(\()p Ft(p)1891 1318 y Fr(2)1929 1354 y Fx(\),)37 b(if)f Ft(')d Fq(2)h Fx([0)p Ft(;)17 b(\031)t(=)p Fx(2\))35 b(and)h Ft(\022)2912 1318 y Fp(i)2909 1379 y Fn(\000)3002 1354 y Fx(=)d Ft(')25 b Fq(\000)g Ft(\031)j Fx(+)c Ft(O)s Fx(\()p Ft(p)3650 1318 y Fr(2)3689 1354 y Fx(\))36 b(if)111 1498 y Ft(')28 b Fq(2)h Fx([)p Ft(\031)t(=)p Fx(2)p Ft(;)17 b(\031)t Fx(\).)43 b(T)-8 b(o)33 b(sho)m(w)h(that)f Ft(g)1342 1462 y Fn(0)1365 1498 y Fx(\()p Ft(\022)s Fx(\))28 b Fq(\024)h Ft(O)18 b Fx(\(1\))33 b(if)g(w)m(e)h(are)e(out)h(of)g(a)f Ft(p)h Fx({neigh)m(b)s(orho)s(o)s(d)f(of)h Ft(\022)3540 1462 y Fp(i)3537 1523 y Fr(+)3596 1498 y Fx(,)g(it)g(is)111 1643 y(enough)g(to)f(observ)m(e)j(that,)d(using)685 1863 y(tan\()p Ft(\022)912 1821 y Fp(u)980 1863 y Fx(+)22 b Ft(p)p Fx(\))27 b(=)h(tan)16 b Ft(\022)1501 1821 y Fp(u)1569 1863 y Fx(+)22 b Ft(p)17 b Fx(sec)1857 1821 y Fr(2)1913 1863 y Ft(\022)1961 1821 y Fp(u)2029 1863 y Fq(\000)22 b Ft(p)2177 1821 y Fr(2)2233 1863 y Fx(tan)17 b Ft(\022)2439 1821 y Fp(u)2517 1863 y Fx(sec)2642 1821 y Fr(2)2698 1863 y Ft(\022)2746 1821 y Fp(u)2814 1863 y Fx(+)22 b Ft(O)3006 1782 y Fs(\000)3051 1863 y Ft(p)3100 1821 y Fr(3)3140 1782 y Fs(\001)3219 1863 y Ft(;)111 2082 y Fx(\(5.8\))32 b(and)h(\(5.7\),)f(w)m(e)i(ha)m(v)m(e)716 2337 y Ft(g)767 2296 y Fn(0)789 2337 y Fx(\()p Ft(\022)875 2296 y Fp(u)943 2337 y Fx(+)22 b Ft(p)p Fx(\))82 b(=)1634 2269 y Ft(p)1683 2233 y Fr(2)1739 2269 y Fx(csc)1864 2233 y Fr(2)1920 2269 y Ft(')17 b Fx(\(1)22 b Fq(\000)g Fx(2)p Ft(p)17 b Fx(cot)f Ft(')22 b Fx(+)g Ft(O)d Fx(\()p Ft(p)2835 2233 y Fr(2)2875 2269 y Fx(\)\))p 1379 2314 1827 4 v 1379 2472 a Ft(p)1428 2443 y Fr(2)1484 2472 y Fx(csc)1610 2443 y Fr(4)1666 2472 y Ft(')1747 2332 y Fs(\022)1819 2472 y Fx(1)j Fq(\000)h Fx(2)p Ft(p)17 b Fx(cot)f Ft(')2325 2405 y Fx(2)22 b(+)g(cot)2624 2369 y Fr(2)2680 2405 y Ft(')p 2325 2449 419 4 v 2325 2541 a Fx(1)g(+)g(cot)2624 2506 y Fr(2)2680 2541 y Ft(')2776 2472 y Fx(+)g Ft(O)d Fx(\()p Ft(p)3055 2443 y Fr(2)3094 2472 y Fx(\))3132 2332 y Fs(\023)1210 2732 y Fx(=)1478 2664 y(1)p 1379 2709 246 4 v 1379 2800 a(csc)1505 2771 y Fr(2)1561 2800 y Ft(')1651 2732 y Fx(\(1)j(+)g Ft(O)d Fx(\()p Ft(p)p Fx(\)\))111 2978 y(and)33 b(the)g(last)g(assertion)g (follo)m(ws)h(from)e(\(5.10\).)258 3122 y(Since)39 b(equation)g (\(5.5\))f(follo)m(ws)i(from)e(a)g(similar)h(calculation,)i(this)e (concludes)i(the)e(pro)s(of)e(of)111 3266 y(Prop)s(osition)c(5.4.)3745 3411 y Fh(2)258 3555 y Fx(The)50 b(function)h Ft(g)60 b Fx(:)d Fo(R)69 b Fq(\000)-17 b(!)57 b Fx([)p Fq(\000)p Ft(\031)t(=)p Fx(2)p Ft(;)17 b(\031)t(=)p Fx(2])49 b(is)i(p)s(erio)s (dic)f(and)g(monotone)g(in)g(eac)m(h)h(in)m(terv)-5 b(al)111 3700 y([)p Fq(\000)p Ft(\031)t(=)p Fx(2)p Ft(;)17 b(\031)t(=)p Fx(2])29 b(+)h Ft(m\031)t Fx(,)47 b Ft(m)h Fq(2)g Fo(Z)p Fx(.)75 b(So)44 b Ft(\022)1563 3715 y Fp(k)1606 3700 y Fx(\()p Ft(')p Fx(\),)j(giv)m(en)e(b)m(y)g(\(4.8\),)h(is)f(a)f (piecewise)i(monotone)f(func-)111 3844 y(tion)31 b(of)f Ft(')p Fx(.)43 b(Note)30 b(that)h(the)g(deriv)-5 b(ativ)m(e)32 b(at)e(righ)m(t)h(coincides)h(with)g(the)f(deriv)-5 b(ativ)m(e)32 b(at)e(left)h(of)f(ev)m(ery)111 3989 y(discon)m(tin)m(uit)m(y)k(p)s (oin)m(t)e(of)f Ft(g)j Fx(and)e(this)g(prop)s(ert)m(y)g(is)g(inherited) g(b)m(y)h Ft(\022)2597 4004 y Fp(k)2640 3989 y Fx(\()p Ft(')p Fx(\).)42 b(Our)32 b(aim)f(is)h(to)f(use)h(The-)111 4133 y(orem)37 b(5.2)e(to)h(all)g(segmen)m(t)i(of)e(monotonicit)m(y)h (of)f Ft(\022)2034 4097 y Fn(0)2031 4159 y Fp(k)2074 4133 y Fx(\()p Ft(')p Fx(\))g(but)g(this)h(can)g(b)s(e)f(accomplished)i (only)f(if)111 4278 y(\(4.8\))32 b(is)h(replaced)h(b)m(y)1499 4422 y Ft( )1562 4437 y Fp(k)1632 4422 y Fx(=)28 b Ft(G)17 b Fx(\()o Ft( )1930 4437 y Fp(k)r Fn(\000)p Fr(1)2064 4422 y Fx(\))22 b(+)g Ft(\014)2283 4381 y Fp(k)2325 4422 y Ft(')17 b(:)1138 b Fx(\(5.11\))111 4610 y(where)34 b Ft(G)28 b Fx(:)f Fo(R)39 b Fq(\000)-16 b(!)27 b Fo(R)76 b Fx(is)33 b(a)g(smo)s(oth)f(function)h(de\014ned)h(in)f(the)g(whole)h (line)f(as)797 4830 y Ft(G)p Fx(\()p Ft(\022)25 b Fx(+)d Ft(m\031)t Fx(\))28 b(=)g Ft(g)t Fx(\()p Ft(\022)s Fx(\))21 b(+)h Ft(m\031)32 b(;)2109 4762 y Fq(\000)p Ft(\031)p 2109 4807 137 4 v 2153 4898 a Fx(2)2283 4830 y Fq(\024)c Ft(\022)j(<)2577 4762 y(\031)p 2577 4807 59 4 v 2582 4898 a Fx(2)2674 4830 y Ft(;)i(m)28 b Fq(2)g Fo(Z)17 b Ft(:)517 b Fx(\(5.12\))111 5049 y(The)30 b(follo)m(wing)f(result)g (sho)m(ws)h(that,)f(as)g(far)f(as)h(\(4.9\))f(is)h(concerned,)i(there)e (is)g(no)g(loss)g(in)g(replacing)111 5194 y Fq(f)p Ft(\022)206 5209 y Fp(k)249 5194 y Fx(\()p Ft(')p Fx(\))p Fq(g)439 5223 y Fp(k)r Fn(\025)p Fr(0)618 5194 y Fx(b)m(y)46 b(the)h(sequence)i (of)c(smo)s(oth)h(functions)h Fq(f)p Ft( )2395 5209 y Fp(k)2438 5194 y Fx(\()p Ft(')p Fx(\))p Fq(g)2628 5223 y Fp(k)r Fn(\025)p Fr(0)2761 5194 y Fx(.)83 b(Note)46 b(that,)k(in)c(view)h(of)111 5338 y(\(4.10\),)32 b Ft(f)43 b Fx(giv)m(en)34 b(b)m(y)g(\(3.11\))d(is)i(a)g(p)s(erio)s(dic)g (function)g(with)g(p)s(erio)s(d)f Ft(\031)t Fx(.)1917 5587 y(18)p eop %%Page: 19 19 19 18 bop 111 312 a Fl(Prop)s(osition)36 b(5.5)49 b Ft(f)11 b Fx(\()p Ft( )1069 327 y Fp(k)1111 312 y Fx(\))28 b(=)g Ft(f)11 b Fx(\()p Ft(\022)1423 327 y Fp(k)1465 312 y Fx(\))35 b Fk(holds)f(for)h(al)5 b(l)34 b Ft(k)d Fq(2)d Fo(N)50 b Fk(and)34 b Ft(f)46 b Fk(p)-5 b(erio)g(dic)34 b(of)g(p)-5 b(erio)g(d)35 b Ft(\031)t Fk(.)111 564 y(Pr)-5 b(o)g(of.)57 b Fx(Prop)s(osition)38 b(5.5)f(follo)m(ws)h(if,)g(for)f (ev)m(ery)i Ft(k)s Fx(,)g Ft( )2163 579 y Fp(k)2242 564 y Fx(=)c Ft(\022)2398 579 y Fp(k)2467 564 y Fx(+)25 b Ft(l)r(\031)42 b Fx(holds)c(for)e(some)j Ft(l)f Fq(2)e Fo(N)9 b Fx(.)63 b(W)-8 b(e)111 709 y(pro)m(v)m(e)34 b(it)f(b)m(y)g(induction.)45 b(F)-8 b(or)31 b Ft(k)g Fx(=)d(1,)k(w)m(e)i(ha)m(v)m(e)1568 953 y Ft( )1631 968 y Fr(1)1699 953 y Fx(=)27 b Ft(G)17 b Fx(\()p Ft( )1997 968 y Fr(0)2037 953 y Fx(\))22 b(+)g Ft(\014)6 b(')17 b(:)111 1197 y Fx(If)25 b Ft( )264 1212 y Fr(0)332 1197 y Fx(=)i Ft(\022)480 1212 y Fr(0)545 1197 y Fx(or)d Ft( )719 1212 y Fr(0)787 1197 y Fx(=)j Ft(\022)935 1212 y Fr(0)982 1197 y Fx(+)6 b Ft(m\031)t Fx(,)27 b(then)e Ft( )1539 1212 y Fr(1)1607 1197 y Fx(=)i Ft(\022)1755 1212 y Fr(1)1802 1197 y Fx(+)6 b Ft(l)r(\031)29 b Fx(b)m(y)d(\(5.12\).)40 b(Let)25 b(us)g(assume)i(that)d Ft( )3324 1212 y Fp(k)3395 1197 y Fx(=)k Ft(\022)3544 1212 y Fp(k)3593 1197 y Fx(+)6 b Ft(m\031)111 1341 y Fx(holds)33 b(for)f Ft(k)f Fx(=)d Ft(n)p Fx(.)43 b(Then)1114 1585 y Ft( )1177 1600 y Fp(n)p Fr(+1)1397 1585 y Fx(=)83 b Ft(G)17 b Fx(\()p Ft( )1751 1600 y Fp(n)1798 1585 y Fx(\))22 b(+)g Ft(\014)2017 1544 y Fp(n)2064 1585 y Ft(')1397 1755 y Fx(=)83 b Ft(G)17 b Fx(\()p Ft(\022)1733 1770 y Fp(n)1802 1755 y Fx(+)22 b Ft(m\031)t Fx(\))h(+)f Ft(\014)2264 1714 y Fp(n)2310 1755 y Ft(')1397 1924 y Fx(=)83 b Ft(g)20 b Fx(\()p Ft(\022)1706 1939 y Fp(n)1753 1924 y Fx(\))i(+)g Ft(l)r(\031)27 b Fx(+)22 b Ft(\014)2183 1883 y Fp(n)2229 1924 y Ft(')28 b Fx(=)f Ft(\022)2469 1939 y Fp(n)p Fr(+1)2629 1924 y Fx(+)22 b Ft(l)r(\031)111 2168 y Fx(In)42 b(the)g(third)g(line)g(w)m(e) g(ha)m(v)m(e)h(used)g(the)f(decomp)s(osition)g Ft(\022)2356 2183 y Fp(n)2447 2168 y Fx(=)g Ft(\022)32 b Fx(+)c Ft(r)s(\031)t Fx(,)43 b Ft(\022)j Fq(2)d Fx([)p Fq(\000)p Ft(\031)t(=)p Fx(2)p Ft(;)17 b(\031)t(=)p Fx(2\))41 b(and)111 2313 y Ft(r)31 b Fx(=)c Ft(l)d Fq(\000)f Ft(m)28 b Fq(2)g Fo(N)9 b Fx(,)39 b(and)32 b(the)h(fact)g Ft(g)t Fx(\()p Ft(\022)s Fx(\))27 b(=)g Ft(g)t Fx(\()p Ft(\022)e Fx(+)d Ft(r)s(\031)t Fx(\))27 b(=)h Ft(g)t Fx(\()p Ft(\022)2302 2328 y Fp(n)2348 2313 y Fx(\).)3745 2457 y Fh(2)258 2601 y Fx(W)-8 b(e)32 b(are)h(no)m(w)g(ready)h(to)e(state)h(and)f(pro)m(v)m (e)i(our)f(result.)111 2854 y Fl(Theorem)k(5.6)49 b Fk(The)31 b(se)-5 b(quenc)g(e)31 b Fq(f)p Ft( )1480 2869 y Fp(k)1523 2854 y Fx(\()p Ft(')p Fx(\))p Fq(g)1712 2883 y Fp(k)r Fn(\025)p Fr(0)1876 2854 y Fk(given)g(by)h(\(5.11\),)f(\(5.12\))f(and)h (\(3.13\))g(is)g(u.d.)44 b(mo)-5 b(d)111 2998 y Ft(\031)30 b Fk(for)d(almost)e(al)5 b(l)26 b Ft(')i Fq(2)g Fx(\(0)p Ft(;)17 b(\031)t Fx(\))26 b Fk(and)f(for)i(al)5 b(l)26 b Ft( )1739 3013 y Fr(0)1806 2998 y Fq(2)i Fx([0)p Ft(;)17 b(\031)t Fx(])p Fk(,)28 b Ft(p)f Fq(2)h Fx(\(0)p Ft(;)17 b Fx(1\))26 b Fk(and)g Ft(\014)32 b Fk(such)26 b(that)h Ft(p)3299 2962 y Fr(2)3338 2998 y Fx(\()p Ft(\014)9 b Fq(\000)s Fx(1\))28 b Ft(>)f(c)3780 3013 y Fr(2)111 3143 y Fk(\()p Ft(c)193 3158 y Fr(2)267 3143 y Fk(as)35 b(in)f(\(5.5\)\))g (holds.)111 3395 y(Pr)-5 b(o)g(of.)46 b Fx(W)-8 b(e)34 b(b)s(egin)g(b)m(y)h(determining)g(the)f(condition)g(under)g(whic)m(h)i Ft( )2750 3359 y Fn(0)2746 3421 y Fp(k)2789 3395 y Fx(\()p Ft(')p Fx(\))22 b Fq(\000)i Ft( )3119 3359 y Fn(0)3115 3420 y Fp(m)3182 3395 y Fx(\()p Ft(')p Fx(\))33 b(is)h(b)s(ounded)111 3540 y(from)f(b)s(elo)m(w)g(uniformly)g(in)g Ft(')28 b Fq(2)g Ft(I)35 b Fq(\032)28 b Fx(\(0)p Ft(;)17 b(\031)t Fx(\).)258 3684 y(If)32 b Ft(p)404 3648 y Fr(2)443 3684 y Ft(\014)i(>)27 b(c)677 3699 y Fr(2)744 3684 y Fx(=)h(1)22 b(+)g Ft(O)d Fx(\()p Ft(p)1198 3648 y Fr(2)1237 3684 y Fx(\),)33 b(then)g(from)f(\(5.11\),)g(\(5.12\))g(and)g(\(5.5\))1268 3928 y Ft( )1335 3887 y Fn(0)1331 3953 y Fp(k)1374 3928 y Fx(\()p Ft(')p Fx(\))84 b(=)g Ft(g)1809 3887 y Fn(0)1831 3928 y Fx(\()p Ft( )1932 3943 y Fp(k)r Fn(\000)p Fr(1)2065 3928 y Fx(\))p Ft( )2170 3887 y Fn(0)2166 3953 y Fp(k)r Fn(\000)p Fr(1)2300 3928 y Fx(\()p Ft(')p Fx(\))22 b(+)g Ft(\014)2621 3887 y Fp(k)1597 4098 y Fq(\024)1771 4030 y Ft(c)1813 4045 y Fr(2)p 1768 4075 89 4 v 1768 4166 a Ft(p)1817 4137 y Fr(2)1866 4098 y Ft( )1933 4056 y Fn(0)1929 4122 y Fp(k)r Fn(\000)p Fr(1)2084 4098 y Fx(+)g Ft(\014)2243 4056 y Fp(k)1597 4350 y Fq(\024)2014 4283 y Fx(1)p 1768 4327 543 4 v 1768 4418 a(1)g Fq(\000)g Ft(c)1980 4433 y Fr(2)2020 4418 y Ft(=)17 b Fx(\()o Ft(p)2172 4390 y Fr(2)2211 4418 y Ft(\014)6 b Fx(\))2320 4350 y Ft(\014)2381 4309 y Fp(k)2440 4350 y Ft(;)1104 b Fx(\(5.13\))111 4620 y(b)m(y)34 b(induction.)44 b(Analogously)-8 b(,)1356 4864 y Ft( )1423 4823 y Fn(0)1419 4889 y Fp(k)1462 4864 y Fx(\()p Ft(')p Fx(\))83 b Fq(\025)h Ft(c)1888 4879 y Fr(1)1927 4864 y Ft(p)1976 4823 y Fr(2)2015 4864 y Ft( )2082 4823 y Fn(0)2078 4889 y Fp(k)r Fn(\000)p Fr(1)2212 4864 y Fx(\()p Ft(')p Fx(\))22 b(+)g Ft(\014)2533 4823 y Fp(k)1685 5061 y Fq(\025)2056 4993 y Fx(1)p 1856 5038 450 4 v 1856 5129 a(1)f Fq(\000)i Ft(c)2068 5144 y Fr(1)2108 5129 y Ft(p)2157 5100 y Fr(2)2196 5129 y Ft(=\014)2315 5061 y(\014)2376 5020 y Fp(k)2435 5061 y Ft(:)1109 b Fx(\(5.14\))1917 5587 y(19)p eop %%Page: 20 20 20 19 bop 111 312 a Fx(whic)m(h)34 b(will)g(b)s(e)e(used)i(later.)44 b(Equations)33 b(\(5.13\))f(and)h(\(5.5\))f(imply)900 556 y Ft( )967 515 y Fn(0)963 580 y Fp(k)1006 556 y Fx(\()p Ft(')p Fx(\))22 b Fq(\000)g Ft( )1334 515 y Fn(0)1330 580 y Fp(k)r Fn(\000)p Fr(1)1463 556 y Fx(\()p Ft(')p Fx(\))84 b(=)g(\()p Ft(g)1936 515 y Fn(0)1958 556 y Fx(\()p Ft( )2059 571 y Fp(k)r Fn(\000)p Fr(1)2192 556 y Fx(\))23 b Fq(\000)f Fx(1\))33 b Ft( )2539 515 y Fn(0)2535 580 y Fp(k)r Fn(\000)p Fr(1)2668 556 y Fx(\()p Ft(')p Fx(\))22 b(+)g Ft(\014)2989 515 y Fp(k)1686 725 y Fq(\025)84 b(\000)1941 644 y Fs(\000)1986 725 y Fx(1)22 b Fq(\000)h Ft(p)2206 684 y Fr(2)2245 644 y Fs(\001)2308 725 y Ft( )2375 684 y Fn(0)2371 750 y Fp(k)r Fn(\000)p Fr(1)2526 725 y Fx(+)f Ft(\014)2685 684 y Fp(k)1686 940 y Fq(\025)1857 872 y Ft(\014)6 b(p)1967 836 y Fr(2)2028 872 y 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b(\(se)-5 b(e)39 b(The)-5 b(or)g(em)37 b Fx(4)p Ft(:)p Fx(4)i Fk(of)g([KN]\))g(in)f(or)-5 b(der)39 b(to)g(c)-5 b(onclude)38 b(that)355 1834 y Fx(\()p Ft( )456 1849 y Fp(k)499 1834 y Fx(\))537 1863 y Fp(k)r Fn(\025)p Fr(0)701 1834 y Fk(is)30 b(u.d.)44 b(mo)-5 b(d)30 b Ft(\031)35 b Fk(for)30 b(almost)h(every)g Ft(')f Fk(in)h(the)g(c)-5 b(omplementary)29 b(r)-5 b(e)g(gion)31 b Ft(p)3388 1797 y Fr(2)3427 1834 y Fx(\()p Ft(\014)19 b Fq(\000)13 b Fx(1\))28 b Ft(<)355 1978 y(c)397 1993 y Fr(2)436 1978 y Fk(.)83 b(Detaile)-5 b(d)47 b(analysis)g(of)g Fx(\()p Ft( )1554 1993 y Fp(k)1597 1978 y Fx(\))1635 2007 y Fp(k)r Fn(\025)p Fr(0)1815 1978 y Fk(in)h(the)f(latter)h(r)-5 b(e)g(gion)47 b(to)-5 b(gether)48 b(with)f(a)h(numeric)-5 b(al)355 2122 y(investigation)34 b(wil)5 b(l)34 b(b)-5 b(e)35 b(pr)-5 b(esente)g(d)34 b(sep)-5 b(ar)g(ately)35 b([AMW].)111 2489 y Fy(6)161 b(Conclusion:)71 b(Op)t(en)53 b(Problems)111 2732 y Fx(Theorem)38 b(1.3)e(sho)m(ws)i(a)f(sp)s(ectral) 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Ft(U)47 b Fx(de\014ne)38 b(an)111 456 y(isomorphism)c(in)f Fo(C)861 420 y Fr(2)907 456 y Fx(,)796 730 y Fq(k)p Ft(A)p Fq(k)968 759 y Fp(U)1055 730 y Fx(=)41 b(sup)1159 817 y Fb(v)q Fn(2)p Fa(C)1292 798 y Ff(2)1359 662 y Fq(k)p Ft(U)10 b(AU)1634 626 y Fn(\000)p Fr(1)1729 662 y Fl(v)r Fq(k)p 1359 707 482 4 v 1519 798 a(k)p Fl(v)r Fq(k)1878 730 y Fx(=)41 b(sup)1981 817 y Fb(v)q Fn(2)p Fa(C)2114 798 y Ff(2)2181 662 y Fq(k)p Ft(U)10 b(A)p Fl(v)r Fq(k)p 2181 707 311 4 v 2218 798 a(k)p Ft(U)g Fl(v)r Fq(k)2529 730 y Fx(=)41 b(sup)2633 817 y Fb(v)q Fn(2)p Fa(C)2766 798 y Ff(2)2833 662 y Fq(k)p Ft(A)p Fl(v)r Fq(k)3066 691 y Fp(U)p 2833 707 293 4 v 2870 798 a Fq(k)o Fl(v)r Fq(k)3030 827 y Fp(U)111 1014 y Fx(and)33 b(the)g(result)g(follo)m(ws)h (from)e(\(A.1\).)3745 1158 y Fh(2)258 1303 y Fx(Prop)s(osition)g(A.1)h (and)g(\(3.9\))f(imply)851 1547 y Fq(k)p Ft(T)e Fx(\()p Ft(n)p Fx(;)17 b Ft(\025)p Fx(\))p Fq(k)28 b(\024)g Ft(C)23 b Fx(sup)1539 1629 y Fp(\022)1572 1638 y Ff(0)1663 1462 y Fs(\015)1663 1522 y(\015)1718 1547 y Ft(U)10 b(T)31 b Fx(\()p Ft(n)p Fx(;)17 b Ft(\025)p Fx(\))f Ft(U)2209 1505 y Fn(\000)p Fr(1)2304 1547 y Fl(v)2363 1562 y Fr(0)2403 1462 y Fs(\015)2403 1522 y(\015)2486 1547 y Fq(\024)28 b Ft(C)c Fx(sup)2724 1629 y Fp(\022)2757 1638 y Ff(0)2848 1547 y Ft(R)2922 1562 y Fp(N)7 b Fr(+1)3595 1547 y Fx(\(A.2\))111 1825 y(and)929 1969 y Fq(k)p Ft(T)31 b Fx(\()p Ft(n)p Fx(;)17 b Ft(\025)p Fx(\))o Fq(k)28 b(\025)g Ft(C)1561 1928 y Fn(\000)p Fr(1)1672 1885 y Fs(\015)1672 1945 y(\015)1727 1969 y Ft(U)10 b(T)31 b Fx(\()p Ft(n)p Fx(;)17 b Ft(\025)p Fx(\))f Ft(U)2218 1928 y Fn(\000)p Fr(1)2313 1969 y Fl(v)2372 1984 y Fr(0)2412 1885 y Fs(\015)2412 1945 y(\015)2495 1969 y Fx(=)28 b Ft(C)2676 1928 y Fn(\000)p Fr(1)2770 1969 y Ft(R)2844 1984 y Fp(N)7 b Fr(+1)3595 1969 y Fx(\(A.3\))111 2168 y(for)38 b Ft(n)g Fx(so)g(that)g Ft(a)755 2183 y Fp(N)7 b Fr(+1)950 2168 y Ft(<)36 b(n)i Fq(\024)f Ft(a)1323 2183 y Fp(N)1429 2168 y Fx(holds)h(for)g(some)h Ft(N)10 b Fx(.)60 b(The)39 b(analog)e(Theorem)j(2)p Ft(:)p Fx(3)d(of)h (Kiselev,)111 2312 y(Last)d(and)g(Simon)g([KLS])g(uses)h(explicitly)g (the)g(fact)e(that)g Ft(T)14 b Fx(,)35 b(and)g(consequen)m(tly)j Ft(T)3277 2327 y Fp(U)3367 2312 y Fx(=)31 b Ft(U)10 b(T)k(U)3697 2276 y Fn(\000)p Fr(1)3793 2312 y Fx(,)111 2457 y(is)33 b(a)g(unimo)s(dal)f(matrix.)44 b(In)33 b(this)h(case,)818 2701 y Fq(k)p Ft(T)925 2716 y Fp(U)984 2701 y Fl(v)1043 2716 y Fr(0)1082 2701 y Fq(k)1132 2652 y Fr(2)1199 2701 y Fx(=)28 b(cos)1433 2660 y Fr(2)1489 2701 y Fx(\()p Ft(\022)1572 2716 y Fr(0)1634 2701 y Fq(\000)23 b Ft(\022)1782 2660 y Fn(\003)1822 2701 y Fx(\))16 b Fq(k)p Ft(T)e Fq(k)2047 2652 y Fr(2)2047 2730 y Fp(U)2128 2701 y Fx(+)22 b(sin)2345 2660 y Fr(2)2401 2701 y Fx(\()p Ft(\022)2484 2716 y Fr(0)2546 2701 y Fq(\000)h Ft(\022)2694 2660 y Fn(\003)2734 2701 y Fx(\))16 b Fq(k)p Ft(T)e Fq(k)2959 2652 y Fn(\000)p Fr(2)2959 2730 y Fp(U)3086 2701 y Ft(;)111 2945 y Fx(where)37 b Ft(\022)444 2909 y Fn(\003)520 2945 y Fx(is)f(the)g(angle)g(in)g (whic)m(h)i(the)e(suprem)m(um)i(is)e(attained,)h(can)f(b)s(e)g(used)h (to)f(reco)m(v)m(er)h(their)111 3089 y(result.)45 b(This)33 b(will)h(not)e(b)s(e)h(necessary)i(in)e(this)g(pap)s(er.)1531 3342 y FD(Ac)m(kno)m(wledgmen)m(t)258 3619 y Fx(W)-8 b(e)37 b(are)g(grateful)g(to)g(R.)g(Killip)g(for)g(p)s(oin)m(ting)g(us) h(out)e(an)h(error)g(in)h(a)f(previous)h(v)m(ersion)h(and)111 3764 y(to)31 b(Prof.)43 b(B.)31 b(Simon)g(for)f(instructiv)m(e)k (commen)m(ts.)44 b(W.)32 b(W.)f(should)h(lik)m(e)g(to)e(thank)i(J.)f (S.)g(Ho)m(wland)111 3908 y(and)i(A.)g(Jo)m(y)m(e)g(for)f(fruitful)h (con)m(v)m(ersations)i(and)e(remarks.)111 4277 y Fy(References)111 4520 y Fx([A])239 b(N.)36 b(I.)g(Akhiezer.)h(\\The)g(classical)g(momen) m(t)g(problem)f(and)g(some)h(related)f(questions)i(in)477 4665 y(analysis",)c(Oliv)m(er)f(&)g(Bo)m(yd)g(\(1965\))111 4892 y([AMW])50 b(R.)c(M.)h(Angelo,)k(D.)46 b(H.)h(U.)g(Marc)m(hetti)h (and)f(W.)f(F.)h(W)-8 b(reszinski.)49 b(\\Comp)s(osition)e(of)477 5037 y(nonlinear)d(map)g(p)s(ossessing)h(an)f(attractiv)m(e)g(and)g(a)f (repulsiv)m(e)j(\014xed)f(p)s(oin)m(ts)f(with)g(an)477 5181 y(exp)s(onen)m(tial)34 b(translation",)e(in)h(Preparation.)1917 5587 y(25)p eop %%Page: 26 26 26 25 bop 111 312 a Fx([C])242 b(Kai)27 b(Lai)g(Ch)m(ung.)i(\\A)e (course)i(in)e(Probabilit)m(y)i(theory",)g(Academic)g(Press)g(-)e(2nd)g (edition)477 456 y(\(1974\))111 684 y([H])239 b(J.)36 b(S.)g(Ho)m(wland.)g(\\Quan)m(tum)g(stabilit)m(y")h(in)f Fk(Schr\177)-50 b(odinger)36 b(op)-5 b(er)g(ators)p Fx(,)36 b(Ed.)g(E.)g(Balslev.)477 828 y(Lecture)d(Notes)h(in)e(Ph)m(ysics)k Fl(403)p Fx(,)c(Springer)i(V)-8 b(erlag)32 b(\(1992\))111 1056 y([K])236 b(Ab)s(el)39 b(Klein.)g(\\Extended)i(States)e(in)g(the)g (Anderson)h(Mo)s(del)f(on)g(the)g(Bethe)h(Lattice",)477 1200 y(Adv.)33 b(Math.)g Fl(133)p Fx(,)g(163-184)e(\(1998\))111 1428 y([KLS])121 b(Alexander)31 b(Kiselev,)g(Y)-8 b(oram)29 b(Last)h(and)f(Barry)h(Simon.)f(\\Mo)s(di\014ed)h(Pr)s(\177)-51 b(ufer)29 b(and)h(EF)m(GP)477 1572 y(transforms)22 b(and)g(the)g(sp)s (ectral)h(analysis)g(of)f(one{dimensional)h(Sc)m(hr\177)-49 b(odinger)23 b(op)s(erators",)477 1716 y(Comm)m(un.)34 b(Math.)f(Ph)m(ys.)h Fl(194)p Fx(,)f(1-45)e(\(1997\))111 1944 y([KN])163 b(L.)28 b(Kuip)s(ers)h(and)f(H.)g(Niederrreiter.)h (\\Uniform)f(Distribution)h(of)e(Sequences"John)k(Wi-)477 2088 y(ley)i(&)g(Sons)g(\(1974\))111 2316 y([LS])197 b(Y)-8 b(oram)42 b(Last)g(and)g(Barry)g(Simon.)h(\\Eigenfunctions,)j (transfer)c(matrices,)k(and)c(abso-)477 2460 y(lutely)30 b(con)m(tin)m(uous)h(sp)s(ectrum)f(of)e(one{dimensional)j(Sc)m(hr\177) -49 b(odinger)30 b(op)s(erators",)g(In)m(v)m(en)m(t.)477 2605 y(Math.)j Fl(135)p Fx(,)g(329-367)d(\(1999\))111 2832 y([PBM])88 b(A.)38 b(P)-8 b(.)38 b(Prudnik)m(o)m(v,)k(Y)-8 b(u.)38 b(A.)g(Bryc)m(hk)m(o)m(v)i(and)e(O.)g(I.)g(Maric)m(hev.)i(\\In) m(tegrals)e(and)g(Series.)477 2977 y(V)-8 b(ol.)32 b(1.)h(Elemen)m (tary)h(functions",)g(Gordon)e(&)g(Breac)m(h)i(\(1986\))111 3204 y([S])258 b(Barry)33 b(Simon.)h(\\The)f(classical)i(momen)m(t)f (problem)f(as)h(a)e(self-adjoin)m(t)h(\014nite)h(di\013erence)477 3349 y(op)s(erator",)e(Adv.)h(Math.)g Fl(137)p Fx(,)g(82-203)e (\(1998\))111 3576 y([S1])209 b(Mic)m(hael)43 b(Reed)f(and)g(Barry)f (Simon.)h(\\Metho)s(ds)h(of)e(Mo)s(dern)h(Mathematical)g(Ph)m(ysics)477 3721 y(IV:)33 b(Analysis)h(of)e(Op)s(erators",)h(Academic)h(Press)g (1978.)1917 5587 y(26)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0306231107549--