Content-Type: multipart/mixed; boundary="-------------0409111548711" This is a multi-part message in MIME format. ---------------0409111548711 Content-Type: text/plain; name="04-284.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-284.comments" AMS classification numbers: 37D50, 37A25 ---------------0409111548711 Content-Type: text/plain; name="04-284.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-284.keywords" Decay of correlations, dispersing billiards ---------------0409111548711 Content-Type: application/postscript; name="flat.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="flat.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software %%Title: flat.dvi %%CreationDate: Sat Sep 11 15:52:31 2004 %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMR10 CMBX10 CMMI10 CMMI8 CMSY10 CMBX12 %%+ CMMI12 MSBM10 CMR8 CMR7 CMTI12 ArialMT CMEX10 CMSY8 CMMI6 CMSY7 %%+ CMSY6 CMR6 SymbolMT %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 flat %DVIPSParameters: dpi=8000 %DVIPSSource: TeX output 2004.09.11:1552 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro 0 0 %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949 8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D F02B185B692D7B27BEC7290EEFDCF92F95DDEB507068DE0B0B0351E3ECB8E443 E611BE0A41A1F8C89C3BC16B352C3443AB6F665EAC5E0CC4229DECFC58E15765 424C919C273E7FA240BE7B2E951AB789D127625BBCB7033E005050EB2E12B1C8 E5F3AD1F44A71957AD2CC53D917BFD09235601155886EE36D0C3DD6E7AA2EF9C C402C77FF1549E609A711FC3C211E64E8F263D60A57E9F2B47E3480B978AAF63 868AEA25DA3D5413467B76D2F02F8097D2841EDA6677731A6ACFEC0BABF1016A 089B2D24F47B9D66B677886B90AA787AD865B5F78EE434AA47B7B0F1244A4215 251FDCC670FD01A92226E2C667C2344298D001575BDF782D969D836ECA11E229 C7A17E28F70F9B17273FF243452DA885068A8BCB5165534F3996CBD8D97307DB 593D606C197AFC259E691C242F6E1E651575B6852AAD54567905E6F542DCA109 7F6DA24DC9112FBF7CE48B387953787B2BCB841873AED2DFA83339D39E14F4DD 3A51584527AC3A93630D121E2AE0C89D9C3F2FFA767743B1276BE1E648041010 0FD510F1A8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 14 /openbullet put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 28 /lessmuch put dup 29 /greatermuch put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 56 /universal put dup 66 /B put dup 67 /C put dup 70 /F put dup 75 /K put dup 77 /M put dup 79 /O put dup 84 /T put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 14 /delta put dup 105 /i put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 77 /M put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 73 /I put dup 76 /L put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put readonly def 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cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 13 /fl put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 127 /dieresis put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec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16:39:21 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 61 /equal put dup 97 /a put dup 105 /i put dup 109 /m put dup 110 /n put dup 120 /x put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 22 /mu put dup 25 /pi put dup 28 /tau put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 76 /L put dup 77 /M put dup 78 /N put dup 81 /Q put dup 82 /R put dup 83 /S put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 113 /q put dup 114 /r put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 3 /Lambda put dup 5 /Pi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 20 /caron put dup 34 /quotedblright put dup 35 /numbersign put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 14 /delta put dup 22 /mu put dup 25 /pi put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 76 /L put dup 78 /N put dup 87 /W put dup 88 /X put dup 97 /a put dup 98 /b put dup 99 /c put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 117 /u put dup 119 /w put dup 120 /x put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 64 /at put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 77 /M put dup 79 /O put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueID 5000795 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39139632 55387786 1000 8000 8000 (flat.dvi) @start /Fa 199[406 1[406 1[406 406 406 50[{}5 664.176 /CMR6 rf /Fb 207[243 47[640{}2 664.176 /CMSY6 rf /Fc 255[692{}1 774.872 /CMSY7 rf /Fd 231[861 24[{}1 1106.96 /CMSY10 rf /Fe 174[875 13[791 840 830 2[861 553 861 60[{}7 1106.96 /CMMI10 rf /Ff 150[296 90[376 1[464 12[{}3 664.176 /CMMI6 rf /Fg 149[261 28[1120 26[627 941 255 14[941 29[470 2[732{}8 885.568 /CMSY8 rf /Fh 142[1328 7[627 627 13[738 1697 1919 52[775 775 12[701 701 978 978 793 793 12[553 553 609 609{}18 1328.35 /CMEX10 rf /Fi 134[789 4[581 589 610 1[830 747 830 1245 415 2[415 830 2[682 3[726 12[1039 3[1021 3[898 5[939 2[1079 1062 1128 7[747 747 747 747 747 747 747 747 747 747 1[415 4[581 581 40[{}33 1328.35 /CMBX12 rf /Fj 128[664 5[631 598 863 598 697 432 531 548 598 664 664 731 1063 332 598 1[399 664 598 399 598 664 598 598 664 4[664 4[1298 1[966 930 731 948 1[881 996 966 1165 815 999 1[501 966 1[848 881 981 930 914 966 12[664 664 664 664 2[399 465 399 2[531 531 5[664 20[764 731 12[{}57 1328.35 /CMTI12 rf /Fk 206[441 49[{}1 774.872 /CMR7 rf /Fl 135[497 9[523 784 3[261 7[470 35[732 2[261 2[470 470 470 470 470 470 470 470 4[732 1[366 366 40[{}18 885.568 /CMR8 rf /Fm 173[959 82[{}1 1328.35 /MSBM10 rf /Fn 143[1107 1[664 3[369 2[664 664 9[886 886 6[723 4[1058 1[1595 1[1012 4[955 2[699 872 9[738 5[886 1328 15[1328 3[1328 1328 1[1033 1033 1033 2[1033 1033 5[664 7[1033 3[1033 369 1033{}30 1328.35 /CMSY10 rf /Fo 133[604 634 739 930 4[586 577 2[776 1138 1[676 1[444 749 623 641 603 676 560 553 683 6[890 755 1079 1228 758 2[800 990 1030 2[1042 1258 885 4[1026 842 963 1080 931 1[975 687 1[1012 650 1012 361 361 18[852 4[609 5[565 2[738 2[783 7[576 675 734 12[{}49 1328.35 /CMMI12 rf /Fp 138[1196 837 849 879 1[1196 1076 1196 1793 598 2[598 1196 1[658 982 1196 956 1[1046 12[1497 1196 1605 1[1470 6[801 20[1076 1076 1076 1076 49[{}25 1912.83 /CMBX12 rf /Fq 128[650 3[650 578 686 686 939 686 723 506 513 506 686 723 650 723 1084 361 686 397 361 723 650 397 578 723 578 723 650 3[361 650 361 795 975 1[1336 1[975 939 723 957 1[885 1012 975 1192 813 1011 668 469 975 1021 849 885 993 939 921 975 3[1012 1[361 361 650 650 650 650 650 650 650 650 650 650 650 361 434 361 1012 1[506 506 361 1012 2[1084 650 13[650 650 4[1084 723 723 759 5[975 1[903 1012 2[{}87 1328.35 /CMR12 rf /Fr 205[808 1212 49[{}2 1212.12 /CMSY10 rf /Fs 135[530 675 1[545 1[435 425 1[474 1[571 832 1[490 1[320 5[408 403 500 8[774 888 8[747 1[640 13[732 470 732 261 33[540 2[566 7[416 1[528 12[{}25 885.568 /CMMI8 rf /Ft 134[594 693 9[728 12[641 35[606 1[337 46[686 12[{}7 1212.12 /CMMI10 rf /Fu 134[640 640 875 640 673 471 478 475 1[673 606 673 1010 337 2[337 673 606 370 539 673 539 673 606 9[1246 37[606 2[337 404 337 2[471 471 40[{}29 1212.12 /CMR10 rf /Fv 139[542 550 574 14[620 774 678 31[1054 65[{}7 1212.12 /CMBX10 rf /Fw 133[492 584 1[799 584 615 430 437 434 1[615 553 615 922 307 2[307 615 553 338 492 615 492 615 553 11[830 5[861 1[1015 7[753 846 1[784 830 861 2[861 1[307 307 8[553 2[307 1[307 2[430 430 40[{}38 1106.96 /CMR10 rf /Fx 137[824 1[607 1[607 1[867 780 867 1301 4[867 780 1[694 867 1[867 780 6[954 6[867 4[1170 2[1213 2[1170 4[1127 14[780 1[780 780 780 1[434 520 434 44[{}26 1594.02 /CMR12 rf /Fy 134[1113 1113 1533 1113 1[813 825 813 2[1053 1173 1773 573 2[573 1173 1053 633 933 1173 933 1173 1053 31[1592 65[{}21 2295.84 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 8000dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 6798 12675 a Fy(A)693 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b(strips)f(b)-36 b(y)4317 8911 y(decreasing)483 b(curv)-36 b(es)483 b Fo(S)15568 8429 y Fg(0)15491 9239 y Fs(n)16117 8911 y Fq(,)495 b Fo(n)453 b Fn(\025)g Fq(1,)496 b(whic)-36 b(h)482 b(corresp)36 b(ond)482 b(to)h(the)f(discon)-36 b(tin)g(uities)483 b(of)g(the)4317 10516 y(function)614 b Fo(N)139 b Fq(\()p Fo(X)104 b Fq(\):)939 b(the)613 b(curv)-36 b(e)614 b Fo(S)21214 10034 y Fg(0)21137 10845 y Fs(n)22377 10516 y Fq(separates)g(the)g(region)g Fo(C)35737 10034 y Fg(0)35642 10845 y Fs(n)36415 10516 y Fq(:)1181 b(=)675 b Fn(f)p Fo(N)139 b Fq(\()p Fo(X)104 b Fq(\))676 b(=)g Fo(n)p Fn(g)4317 12121 y Fq(from)472 b(the)f(similar)i(region)e Fo(C)18978 11639 y Fg(0)18883 12450 y Fs(n)p Fl(+1)20712 12121 y Fq(.)692 b(The)471 b(curv)-36 b(es)472 b Fo(S)29373 11639 y Fg(0)29296 12450 y Fs(n)30393 12121 y Fq(are)g(almost)g (parallel)h(to)e Fo(S)43925 12320 y Fg(1)45393 12121 y Fq(and)4317 13726 y(accum)-36 b(ulate)434 b(to)-36 b(w)g(ard)433 b Fo(S)16281 13925 y Fg(1)17711 13726 y Fq(from)h(ab)36 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b(obtain)16822 22759 y Fo(w)17752 22958 y Fs(m)18935 22759 y Fn(\000)295 b Fo(w)21193 22958 y Fs(m)p Fl(+1)23652 22759 y Fq(=)368 b(2)p Fo(\014)74 b(x)27229 22129 y Fs(\014)46 b Fg(\000)p Fl(1)27229 23085 y Fs(m)p Fl(+1)29615 22759 y Fn(\000)295 b Fo(R)31933 22958 y Fs(w)24 b(;m)p Fl(+1)17203 24696 y Fo(x)17942 24895 y Fs(m)19125 24696 y Fn(\000)296 b Fo(x)21193 24895 y Fs(m)p Fl(+1)23652 24696 y Fq(=)368 b(2)p Fo(w)26612 24895 y Fs(m)27795 24696 y Fq(+)295 b Fo(R)30092 24895 y Fs(x;m)p Fl(+1)32973 24696 y Fo(;)4317 23678 y Fq(\(4.8\))4317 27539 y(where)4317 30473 y(\(4.9\))8156 b Fo(R)16136 30672 y Fs(w)24 b(;m)p Fl(+1)19556 30473 y Fq(=)21069 29950 y Fl(2)p 21069 30167 471 54 v 21069 30931 a(3)21673 30473 y Fo(\014)22481 29924 y Fl(3)23006 30473 y Fo(x)23745 29794 y Fl(3\()p Fs(\014)46 b Fg(\000)p Fl(1\))23745 30798 y Fs(m)p Fl(+1)27074 30473 y Fq(+)295 b Fn(O)37 b Fq(\()p Fo(x)30721 29794 y Fl(5\()p Fs(\014)46 b Fg(\000)p Fl(1\))30721 30798 y Fs(m)p Fl(+1)33754 30473 y Fq(\))368 b Fo(>)h Fq(0)4317 33406 y(and)4317 36340 y(\(4.10\))4108 b Fo(R)12738 36539 y Fs(x;m)p Fl(+1)15988 36340 y Fq(=)17501 35817 y Fl(2)p 17501 36034 V 17501 36798 a(3)18104 36340 y Fo(w)19070 35791 y Fl(3)19034 36668 y Fs(m)20217 36340 y Fq(+)295 b(\()p Fo(x)22769 35791 y Fs(\014)22769 36668 y(m)23951 36340 y Fq(+)g Fo(x)25997 35710 y Fs(\014)25997 36665 y(m)p Fl(+1)28087 36340 y Fq(\))p Fo(w)29523 36539 y Fs(m)30706 36340 y Fq(+)g Fn(O)37 b Fq(\()p Fo(x)34353 35791 y Fs(\014)34353 36668 y(m)35240 36340 y Fo(w)36206 35791 y Fl(3)36170 36668 y Fs(m)37352 36340 y Fq(+)295 b Fo(w)39625 35791 y Fl(5)39589 36668 y Fs(m)40476 36340 y Fq(\))369 b Fo(>)g Fq(0)4317 39273 y(\(the)527 b(p)36 b(ositivit)-36 b(y)528 b(of)g Fo(R)15603 39472 y Fs(w)24 b(;m)p Fl(+1)19181 39273 y Fq(and)526 b Fo(R)22793 39472 y Fs(x;m)p Fl(+1)26201 39273 y Fq(is)i(guaran)-36 b(teed)526 b(b)-36 b(y)528 b(the)e(smallness)i(of)g Fo(")p Fq(\).)4317 40878 y(Note)359 b(that)e(b)36 b(oth)358 b Fn(f)p Fo(x)14526 41077 y Fs(m)15414 40878 y Fn(g)g Fq(and)g Fn(f)p Fo(w)20484 41077 y Fs(m)21372 40878 y Fn(g)g Fq(are)g(decreasing)h(sequences)f(of)h(p)36 b(ositiv)-36 b(e)359 b(n)-36 b(um)g(b)36 b(ers)4317 42483 y(for)434 b Fo(m)369 b Fq(=)g(1)p Fo(;)221 b(:)g(:)g(:)j(;)d(n)13531 42001 y Fg(0)13842 42483 y Fq(.)4317 45195 y Fi(Lemma)499 b(3.)554 b Fj(L)-66 b(et)464 b Fo(n)14347 44713 y Fg(00)15282 45195 y Fn(2)368 b Fq([1)p Fo(;)221 b(n)18905 44713 y Fg(0)19218 45195 y Fq(])465 b Fj(b)-66 b(e)464 b(uniquely)g(de\014ne) -66 b(d)464 b(by)h(the)f(c)-66 b(ondition)4317 48129 y Fq(\(4.11\))12382 b Fo(w)20952 48328 y Fs(n)21523 48076 y Fb(00)22064 48328 y Fg(\000)p Fl(1)23690 48129 y Fo(>)369 b Fq(2)p Fo(w)26651 48328 y Fs(n)27222 48076 y Fb(0)27945 48129 y Fo(>)f(w)30255 48328 y Fs(n)30826 48076 y Fb(00)31422 48129 y Fo(:)4317 51062 y Fj(Then)464 b(for)h(al)66 b(l)466 b Fo(n)12416 50580 y Fg(00)13351 51062 y Fn(\024)369 b Fo(m)g Fn(\024)g Fo(n)18438 50580 y Fg(0)4317 53996 y Fq(\(4.12\))13207 b Fo(x)21586 54195 y Fs(m)22843 53996 y Fn(\030)369 b Fq(\()p Fo(n)25527 53447 y Fg(0)26133 53996 y Fn(\000)295 b Fo(m)p Fq(\))p Fo(w)30035 54195 y Fs(n)30606 53943 y Fb(0)4317 56929 y Fj(and)4317 60174 y Fq(\(4.13\))13936 b Fo(n)22352 59625 y Fg(0)22958 60174 y Fn(\000)296 b Fo(n)25063 59625 y Fg(00)25998 60174 y Fn(\030)369 b Fo(w)28499 58829 y Fa(2)p Fb(\000)p Ff(\014)p 28498 59038 1544 40 v 29021 59587 a(\014)28330 60550 y Fs(n)28901 60298 y Fb(0)4317 63107 y Fj(\(r)-66 b(e)g(c)g(al)66 b(l)464 b(our)i(c)-66 b(onvention)462 b(on)j(the)f(usage)i(of)e(\\)p Fn(\030)p Fj(")i(in)e(the)g(pr)-66 b(evious)465 b(se)-66 b(ction\).)25253 70015 y Fq(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 4317 7306 a Fj(Pr)-66 b(o)g(of.)649 b Fq(Due)434 b(to)f(\(4.8\))i(and)e(\(4.11\),)h(for)h(an)-36 b(y)433 b Fo(m)369 b Fn(2)g Fq([)p Fo(n)30804 6824 y Fg(00)31370 7306 y Fo(;)221 b(n)32728 6824 y Fg(0)33040 7306 y Fq(\))433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)14349 10193 y(2)p Fo(w)15929 10392 y Fs(n)16500 10140 y Fb(0)17222 10193 y Fn(\024)369 b Fq(2)p Fo(w)20204 10392 y Fs(m)21461 10193 y Fn(\024)g Fo(x)23602 10392 y Fs(m)24785 10193 y Fn(\000)295 b Fo(x)26852 10392 y Fs(m)p Fl(+1)29312 10193 y Fn(\024)369 b Fq(3)p Fo(w)32294 10392 y Fs(m)33550 10193 y Fn(\024)h Fq(6)p Fo(w)36533 10392 y Fs(n)37104 10140 y Fb(0)4317 13080 y Fq(hence)4317 15966 y(\(4.14\))7188 b(2\()p Fo(n)16760 15418 y Fg(0)17366 15966 y Fn(\000)295 b Fo(m)p Fq(\))p Fo(w)21268 16165 y Fs(n)21839 15913 y Fb(0)22561 15966 y Fn(\024)369 b Fo(x)24702 16165 y Fs(m)25959 15966 y Fn(\024)g Fq(6\()p Fo(n)29293 15418 y Fg(0)29899 15966 y Fn(\000)296 b Fo(m)f Fq(+)g(1\))p Fo(w)36054 16165 y Fs(n)36625 15913 y Fb(0)4317 18853 y Fq(\(note)433 b(that)g(0)370 b Fn(\024)f Fo(x)13692 19052 y Fs(n)14263 18800 y Fb(0)14986 18853 y Fn(\024)g Fq(3)p Fo(w)17968 19052 y Fs(n)18539 18800 y Fb(0)18892 18853 y Fq(\).)578 b(Next,)435 b(due)e(to)g(\(4.14\))i(and)e(\(4.8\)) 6956 21740 y(2)7606 21191 y Fs(\014)46 b Fg(\000)p Fl(1)9437 21740 y Fq(\()p Fo(n)10719 21191 y Fg(0)11325 21740 y Fn(\000)295 b Fo(m)g Fn(\000)h Fq(1\))16571 21191 y Fs(\014)46 b Fg(\000)p Fl(1)18402 21740 y Fo(w)19368 21110 y Fs(\014)g Fg(\000)p Fl(1)19332 22116 y Fs(n)19903 21864 y Fb(0)21568 21740 y Fn(\024)369 b Fo(w)23900 21939 y Fs(m)25082 21740 y Fn(\000)296 b Fo(w)27341 21939 y Fs(m)p Fl(+1)29799 21740 y Fn(\024)369 b Fq(2)p Fo(\014)296 b Fq(6)33531 21191 y Fs(\014)46 b Fg(\000)p Fl(1)35363 21740 y Fq(\()p Fo(n)36645 21191 y Fg(0)37250 21740 y Fn(\000)296 b Fo(m)p Fq(\))40223 21191 y Fs(\014)46 b Fg(\000)p Fl(1)42053 21740 y Fo(w)43019 21110 y Fs(\014)g Fg(\000)p Fl(1)42983 22116 y Fs(n)43554 21864 y Fb(0)4317 24627 y Fq(therefore)18238 26232 y Fo(w)19168 26431 y Fs(m)20424 26232 y Fn(\030)369 b Fo(w)22756 26431 y Fs(n)23327 26179 y Fb(0)23976 26232 y Fq(+)295 b(\()p Fo(n)26565 25683 y Fg(0)27171 26232 y Fn(\000)g Fo(m)p Fq(\))30143 25683 y Fs(\014)30771 26232 y Fo(w)31737 25602 y Fs(\014)46 b Fg(\000)p Fl(1)31701 26608 y Fs(n)32272 26356 y Fb(0)4317 28536 y Fq(Substituting)480 b Fo(m)451 b Fq(=)g Fo(n)15643 28054 y Fg(00)16209 28536 y Fq(,)494 b(then)481 b Fo(m)450 b Fq(=)h Fo(n)23902 28054 y Fg(00)24796 28536 y Fn(\000)329 b Fq(1)482 b(and)f(using)g (\(4.11\))i(implies)f(\(4.13\))h(and)4317 30141 y(completes)434 b(the)f(pro)36 b(of)434 b(of)g(the)f(lemma.)p 46548 30141 45 878 v 46593 29308 781 45 v 46593 30141 V 47373 30141 45 878 v 6268 32397 a(W)-108 b(e)434 b(note)f(that)g(\(4.12\))i(and)e (\(4.13\))h(imply)4317 35846 y(\(4.15\))15316 b Fo(x)23695 36045 y Fs(n)24266 35793 y Fb(00)25232 35846 y Fn(\030)369 b Fo(w)27779 34550 y Fa(2)p 27732 34711 498 40 v 27732 35259 a Ff(\014)27564 36222 y Fs(n)28135 35970 y Fb(0)28488 35846 y Fo(;)4317 38733 y Fq(hence)433 b Fo(x)8669 38932 y Fs(n)9240 38680 y Fb(00)10206 38733 y Fn(\034)369 b Fo(")433 b Fq(and)g(th)-36 b(us)433 b Fo(n)19112 38251 y Fg(00)20047 38733 y Fn(\035)369 b Fq(1.)579 b(Next)434 b(w)-36 b(e)434 b(consider)f(the)g(case)h(1)370 b Fo(<)e(m)h Fn(\024)g Fo(n)44571 38251 y Fg(00)45137 38733 y Fq(.)4317 41002 y Fi(Lemma)499 b(4.)554 b Fj(F)-100 b(or)465 b(al)66 b(l)466 b Fq(1)369 b Fo(<)g(m)g Fn(\024)g Fo(n)21698 40520 y Fg(00)22729 41002 y Fj(we)465 b(have)4317 44066 y Fq(\(4.16\))13088 b Fo(x)21467 43517 y Fs(\014)21467 44394 y(m)22724 44066 y Fn(\030)369 b Fo(w)25092 43517 y Fl(2)25056 44394 y Fs(m)26312 44066 y Fn(\030)g Fo(m)29305 43036 y Fa(2)p Ff(\014)p 28985 43245 1544 40 v 28985 43794 a Fa(2)p Fb(\000)p Ff(\014)30716 44066 y Fo(:)4317 46953 y Fj(Mor)-66 b(e)g(over,)4317 50409 y Fq(\(4.17\))15111 b Fo(n)23527 49860 y Fg(00)24462 50409 y Fn(\030)369 b Fo(w)26963 49065 y Fa(2)p Fb(\000)p Ff(\014)p 26962 49273 V 27485 49822 a(\014)26794 50785 y Fs(n)27365 50533 y Fb(0)28694 50409 y Fo(:)4317 53296 y Fj(Pr)-66 b(o)g(of.)649 b Fq(Due)434 b(to)f(\(4.8\))i(and)e(the)g(mean)g(v)-72 b(alue)434 b(theorem,)g(for)g(some)g Fo(x)38816 53495 y Fg(\003)39711 53296 y Fn(2)368 b Fq(\()p Fo(x)42210 53495 y Fs(m)p Fl(+1)44300 53296 y Fo(;)221 b(x)45621 53495 y Fs(m)46509 53296 y Fq(\))11504 56182 y Fo(x)12243 55634 y Fs(\014)12243 56511 y(m)13426 56182 y Fn(\000)295 b Fo(x)15493 55553 y Fs(\014)15493 56508 y(m)p Fl(+1)17952 56182 y Fq(=)369 b Fo(\014)74 b(x)20880 55634 y Fs(\014)46 b Fg(\000)p Fl(1)20880 56511 y Fg(\003)22711 56182 y Fq(\()p Fo(x)23956 56381 y Fs(m)25139 56182 y Fn(\000)295 b Fo(x)27206 56381 y Fs(m)p Fl(+1)29296 56182 y Fq(\))17952 58270 y(=)369 b(2)p Fo(\014)74 b(x)21530 57722 y Fs(\014)46 b Fg(\000)p Fl(1)21530 58599 y Fg(\003)23361 58270 y Fo(w)24291 58469 y Fs(m)25474 58270 y Fq(+)295 b Fn(O)27876 57194 y Fh(\000)28484 58270 y Fo(x)29223 57722 y Fs(\014)46 b Fg(\000)p Fl(1)29223 58599 y Fs(m)31054 58270 y Fo(w)32020 57722 y Fl(3)31984 58599 y Fs(m)33167 58270 y Fq(+)295 b Fo(x)35213 57722 y Fl(2)p Fs(\014)46 b Fg(\000)p Fl(1)35213 58599 y Fs(m)37514 58270 y Fo(w)38444 58469 y Fs(m)39332 57194 y Fh(\001)39941 58270 y Fo(:)4317 61157 y Fq(Similarly)-108 b(,)435 b(for)f(some)g Fo(w)16274 61356 y Fg(\003)17168 61157 y Fn(2)369 b Fq(\()p Fo(w)19859 61356 y Fs(m)p Fl(+1)21948 61157 y Fo(;)221 b(w)23460 61356 y Fs(m)24349 61157 y Fq(\))13994 64044 y Fo(w)14960 63495 y Fl(2)14924 64372 y Fs(m)16106 64044 y Fn(\000)296 b Fo(w)18401 63495 y Fl(2)18365 64372 y Fs(m)p Fl(+1)20823 64044 y Fq(=)369 b(2)p Fo(w)23784 64243 y Fg(\003)24310 64044 y Fq(\()p Fo(w)25746 64243 y Fs(m)26928 64044 y Fn(\000)296 b Fo(w)29187 64243 y Fs(m)p Fl(+1)31276 64044 y Fq(\))20823 66213 y(=)369 b(4)p Fo(\014)74 b(x)24401 65583 y Fs(\014)46 b Fg(\000)p Fl(1)24401 66538 y Fs(m)p Fl(+1)26491 66213 y Fo(w)27421 66412 y Fg(\003)28242 66213 y Fq(+)295 b Fn(O)30644 65137 y Fh(\000)31252 66213 y Fo(x)31991 65664 y Fl(3\()p Fs(\014)46 b Fg(\000)p Fl(1\))31991 66541 y Fs(m)35025 66213 y Fo(w)35955 66412 y Fs(m)36842 65137 y Fh(\001)37451 66213 y Fo(:)25253 70015 y Fq(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 4317 7306 a Fq(This)434 b(easily)h(implies) 19938 9666 y(1)369 b Fn(\024)22492 8728 y Fo(w)23458 8246 y Fl(2)23422 9056 y Fs(m)24604 8728 y Fn(\000)296 b Fo(w)26899 8246 y Fl(2)26863 9056 y Fs(m)p Fl(+1)p 22492 9361 6461 54 v 22683 10765 a Fo(x)23422 10135 y Fs(\014)23422 10901 y(m)24604 10765 y Fn(\000)g Fo(x)26672 10135 y Fs(\014)26672 11091 y(m)p Fl(+1)29454 9666 y Fn(\024)369 b Fq(5)p Fo(:)4317 13258 y Fq(Also,)435 b(\(4.15\))f(and)f (\(4.11\))i(imply)f(that)f Fo(w)25089 12776 y Fl(2)25053 13609 y Fs(n)25624 13357 y Fb(00)26589 13258 y Fn(\030)369 b Fo(x)28730 12628 y Fs(\014)28730 13634 y(n)29301 13382 y Fb(00)29898 13258 y Fq(,)434 b(i.e.)21486 17024 y Fo(C)22512 16476 y Fg(0)23192 17024 y Fn(\024)24727 16126 y Fo(w)25693 15644 y Fl(2)25657 16477 y Fs(n)26228 16225 y Fb(00)p 24727 16719 2098 54 v 24822 18123 a Fo(x)25561 17493 y Fs(\014)25561 18499 y(n)26132 18247 y Fb(00)27326 17024 y Fn(\024)369 b Fo(C)29754 16476 y Fg(00)4317 20934 y Fq(where)506 b(w)-36 b(e)505 b(can)h(assume)f Fo(C)18182 20452 y Fg(0)18984 20934 y Fo(<)491 b Fq(1)506 b(and)f Fo(C)25270 20452 y Fg(00)26327 20934 y Fo(>)491 b Fq(5.)795 b(No)-36 b(w)506 b(the)f(\014rst)f(relation)i(in)g(\(4.16\))4317 22539 y(follo)-36 b(ws)436 b(easily)-108 b(.)6268 24585 y(Next,)434 b(denote)f Fo(z)14603 24784 y Fs(m)15860 24585 y Fq(=)369 b Fo(x)18113 23370 y Ff(\014)34 b Fb(\000)p Fa(2)p 18113 23579 1544 40 v 18681 24127 a(2)17980 24721 y Fs(m)19844 24585 y Fq(.)579 b(Then)433 b(\(4.8\))h(and)f(the)g(mean)g (v)-72 b(alue)434 b(theorem)g(imply)17265 28032 y Fo(z)17869 28231 y Fs(m)19052 28032 y Fn(\000)296 b Fo(z)20985 28231 y Fs(m)p Fl(+1)23444 28032 y Fn(\030)369 b Fo(x)25718 26817 y Ff(\014)34 b Fb(\000)p Fa(4)p 25718 27026 V 26287 27575 a(2)25585 28169 y Fs(m)27449 28032 y Fq(\()p Fo(x)28694 28231 y Fs(m)29877 28032 y Fn(\000)295 b Fo(x)31944 28231 y Fs(m)p Fl(+1)34035 28032 y Fq(\))23444 30633 y Fn(\030)369 b Fo(x)25718 29418 y Ff(\014)34 b Fb(\000)p Fa(4)p 25718 29627 V 26287 30176 a(2)25585 30769 y Fs(m)27449 30633 y Fo(w)28379 30832 y Fs(m)23444 32721 y Fn(\030)369 b Fo(x)25585 32172 y Fs(\014)46 b Fg(\000)p Fl(2)25585 33049 y Fs(m)27785 32721 y Fq(=)369 b Fo(z)29829 32172 y Fl(2)29770 33049 y Fs(m)4317 35654 y Fq(\(w)-36 b(e)434 b(used)f(the)g(\014rst)g(relation)h(in)f(\(4.16\)\).)579 b(No)-36 b(w)434 b(let)g Fo(Z)31415 35853 y Fs(m)32672 35654 y Fq(=)369 b(1)p Fo(=z)35957 35853 y Fs(m)36845 35654 y Fq(,)434 b(then)17948 38588 y Fo(Z)18838 38787 y Fs(m)p Fl(+1)21223 38588 y Fn(\000)295 b Fo(Z)23441 38787 y Fs(m)24698 38588 y Fn(\030)369 b Fo(Z)26990 38787 y Fs(m)p Fl(+1)29081 38588 y Fo(=)-72 b(Z)30549 38787 y Fs(m)31806 38588 y Fn(\030)369 b Fq(1)4317 42035 y(\(w)-36 b(e)434 b(note)f(that)g Fo(x)13185 42234 y Fs(m)14368 42035 y Fn(\000)296 b Fo(x)16436 42234 y Fs(m)p Fl(+1)18895 42035 y Fn(\030)369 b Fo(x)21169 40820 y Ff(\014)p 21169 41029 498 40 v 21215 41578 a Fa(2)21036 42171 y Fs(m)22292 42035 y Fn(\034)h Fo(x)24729 42234 y Fs(m)25617 42035 y Fq(,)433 b(hence)g Fo(x)30763 42234 y Fs(m)31651 42035 y Fo(=x)33040 42234 y Fs(m)p Fl(+1)35499 42035 y Fn(\031)370 b Fq(1\).)578 b(Since)433 b Fo(x)43132 42234 y Fl(0)44027 42035 y Fn(\025)369 b Fo(")p Fq(,)4317 45226 y(\(4.18\))12416 b Fo(Z)20946 45425 y Fl(0)21841 45226 y Fn(\024)369 b Fo(")23852 44678 y Fg(\000)24717 44268 y Ff(\014)34 b Fb(\000)p Fa(2)p 24716 44476 1544 40 v 25285 45025 a(2)26817 45226 y Fq(=)590 b(const)p Fo(;)4317 48160 y Fq(and)433 b(w)-36 b(e)434 b(obtain)4317 51093 y(\(4.19\))9239 b Fo(Z)17769 51292 y Fs(m)19026 51093 y Fn(\030)369 b Fo(m)2601 b Fq(and)g Fo(z)29468 51292 y Fs(m)30725 51093 y Fn(\030)369 b Fq(1)p Fo(=m;)4317 54027 y Fq(whic)-36 b(h)465 b(pro)-36 b(v)g(es)466 b(the)f(second)g(relation)g(in)h(\(4.16\).)674 b(No)-36 b(w)466 b(\(4.17\))h(is)e(immediate)h(due)e(to)4317 55632 y(\(4.15\).)p 46548 55632 45 878 v 46593 54798 781 45 v 46593 55632 V 47373 55632 45 878 v 6268 57880 a(Equations)385 b(\(4.13\))h(and)f(\(4.17\))g(imply)h Fo(n)26625 57398 y Fg(00)27560 57880 y Fn(\030)369 b Fo(n)29738 57398 y Fg(0)30245 57880 y Fn(\000)196 b Fo(n)32250 57398 y Fg(00)33200 57880 y Fq(and)385 b Fo(n)36457 57398 y Fg(0)37137 57880 y Fn(\030)369 b Fo(w)39638 56536 y Fa(2)p Fb(\000)p Ff(\014)p 39637 56745 1544 40 v 40160 57293 a(\014)39469 58256 y Fs(n)40040 58004 y Fb(0)41369 57880 y Fq(.)562 b(A)385 b(similar)4317 59485 y(analysis)490 b(can)f(b)36 b(e)489 b(done)f(for)h(the)g(remaining)f(part)h(of)g(the)g (tra)72 b(jectory)-108 b(,)503 b Fo(n)41027 59003 y Fg(0)41801 59485 y Fo(<)463 b(m)f(<)h(n)p Fq(,)4317 61660 y(whic)-36 b(h)438 b(sho)-36 b(ws)438 b(that)g Fo(n)298 b Fn(\000)g Fo(n)17787 61178 y Fg(0)18475 61660 y Fn(\030)376 b Fo(w)20983 60315 y Fa(2)p Fb(\000)p Ff(\014)p 20983 60524 V 21505 61073 a(\014)20814 62036 y Fs(n)21385 61784 y Fb(0)21683 62036 y Fl(+1)22941 61660 y Fq(.)592 b(Since)437 b Fo(w)28224 61859 y Fs(n)28795 61607 y Fb(0)29525 61660 y Fn(\031)376 b Fo(w)31864 61859 y Fs(n)32435 61607 y Fb(0)32733 61859 y Fl(+1)33991 61660 y Fq(,)439 b(w)-36 b(e)439 b(obtain)e Fo(n)299 b Fn(\000)f Fo(n)43943 61178 y Fg(0)44631 61660 y Fn(\030)376 b Fo(n)46816 61178 y Fg(0)47127 61660 y Fq(,)4317 63265 y(and)433 b(so)4317 66198 y(\(4.20\))9355 b Fo(n)17771 65650 y Fg(00)18706 66198 y Fn(\030)370 b Fo(n)2601 b Fq(and)g Fo(w)29113 66397 y Fs(n)29684 66145 y Fb(0)30406 66198 y Fn(\030)370 b Fo(n)33241 65169 y Ff(\014)p 32718 65377 V 32718 65926 a Fa(2)p Fb(\000)p Ff(\014)34449 66198 y Fo(:)25253 70015 y Fq(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 4317 7306 a Fi(Lemma)499 b(5.)554 b Fj(F)-100 b(or)465 b(al)66 b(l)466 b Fo(m)369 b(<)f(n)19276 6824 y Fg(0)20052 7306 y Fj(we)465 b(have)18108 10239 y Fo(w)19074 9691 y Fl(2)19038 10568 y Fs(m)20220 10239 y Fn(\000)296 b Fq(2)p Fo(x)22938 9691 y Fs(\014)22938 10568 y(m)24195 10239 y Fo(<)368 b(w)26541 9691 y Fl(2)26505 10568 y Fs(m)p Fl(+1)28890 10239 y Fn(\000)296 b Fq(2)p Fo(x)31608 9610 y Fs(\014)31608 10565 y(m)p Fl(+1)4317 13173 y Fj(i.e.)463 b Fn(f)p Fo(w)8205 12691 y Fl(2)8169 13501 y Fs(m)9352 13173 y Fn(\000)295 b Fq(2)p Fo(x)12069 12691 y Fs(\014)12069 13501 y(m)12958 13173 y Fn(g)465 b Fj(is)f(an)h(incr)-66 b(e)g(asing)462 b(se)-66 b(quenc)g(e)463 b(for)i Fo(m)369 b Fq(=)f(1)p Fo(;)221 b(:)g(:)g(:)j(;)d(n)38050 12691 y Fg(0)38362 13173 y Fj(.)4317 15774 y(Pr)-66 b(o)g(of.)649 b Fq(By)434 b(the)f(con)-36 b(v)g(exit)g(y)435 b(of)f(the)f(function)h Fo(x)28006 15292 y Fs(\014)28635 15774 y Fq(,)15458 18708 y(2)p Fo(x)16847 18159 y Fs(\014)16847 19036 y(m)18030 18708 y Fn(\000)295 b Fq(2)p Fo(x)20747 18078 y Fs(\014)20747 19033 y(m)p Fl(+1)23207 18708 y Fn(\025)369 b Fq(2)p Fo(\014)74 b(x)26806 18078 y Fs(\014)46 b Fg(\000)p Fl(1)26806 19033 y Fs(m)p Fl(+1)28896 18708 y Fq(\()p Fo(x)30141 18907 y Fs(m)31324 18708 y Fn(\000)295 b Fo(x)33391 18907 y Fs(m)p Fl(+1)35481 18708 y Fq(\))p Fo(:)4317 21641 y Fq(No)-36 b(w)434 b(due)f(to)h(\(4.8\){\(4.10\))8927 24575 y(2)p Fo(\014)74 b(x)11124 23945 y Fs(\014)46 b Fg(\000)p Fl(1)11124 24900 y Fs(m)p Fl(+1)13214 24575 y Fq(\()p Fo(x)14459 24774 y Fs(m)15642 24575 y Fn(\000)295 b Fo(x)17709 24774 y Fs(m)p Fl(+1)19800 24575 y Fq(\))368 b Fo(>)h Fq(2)p Fo(w)23635 24774 y Fs(m)24523 24575 y Fq(\()p Fo(w)25959 24774 y Fs(m)27141 24575 y Fn(\000)295 b Fo(w)29399 24774 y Fs(m)p Fl(+1)31489 24575 y Fq(\))369 b Fo(>)f(w)34710 24026 y Fl(2)34674 24903 y Fs(m)35857 24575 y Fn(\000)295 b Fo(w)38151 24026 y Fl(2)38115 24903 y Fs(m)p Fl(+1)40205 24575 y Fo(:)p 45840 24575 45 878 v 45885 23741 781 45 v 45885 24575 V 46665 24575 45 878 v 6268 27508 a Fq(Lemma)434 b(5)f(implies)h Fo(w)17375 27026 y Fl(2)17339 27836 y Fs(m)18522 27508 y Fn(\000)295 b Fq(2)p Fo(x)21239 27026 y Fs(\014)21239 27836 y(m)22496 27508 y Fo(<)369 b(w)24843 27026 y Fl(2)24807 27860 y Fs(n)25378 27607 y Fb(0)25731 27508 y Fq(,)434 b(hence)4317 31106 y(\(4.21\))6739 b Fo(w)15309 31305 y Fs(m)16565 31106 y Fo(<)17946 29405 y Fh(q)p 19274 29405 5734 54 v 1701 x Fq(2)p Fo(x)20663 30476 y Fs(\014)20663 31242 y(m)21846 31106 y Fq(+)295 b Fo(w)24119 30649 y Fl(2)24083 31482 y Fs(n)24654 31230 y Fb(0)25376 31106 y Fo(<)26757 29943 y Fn(p)p 27864 29943 651 54 v 1163 x Fq(2)222 b Fo(x)29608 29891 y Ff(\014)p 29608 30100 498 40 v 29654 30649 a Fa(2)29475 31242 y Fs(m)30658 31106 y Fq(+)32097 30583 y Fl(1)p 32097 30801 471 54 v 32097 31564 a(2)32922 31106 y Fo(x)33661 30301 y Fg(\000)34526 29891 y Ff(\014)p 34526 30100 498 40 v 34572 30649 a Fa(2)33661 31242 y Fs(m)35212 31106 y Fo(w)36178 30558 y Fl(2)36142 31435 y Fs(n)36713 31182 y Fb(0)37066 31106 y Fo(:)6268 34040 y Fq(Next)434 b(w)-36 b(e)434 b(deriv)-36 b(e)434 b(a)g(more)f(precise) h(estimate)g(on)f(the)g Fo(x)h Fq(co)36 b(ordinate:)4317 36309 y Fi(Lemma)499 b(6.)554 b Fj(F)-100 b(or)465 b(al)66 b(l)466 b Fq(1)369 b Fn(\024)h Fo(m)e Fn(\024)h Fo(n)21719 35827 y Fg(00)22750 36309 y Fj(we)466 b(have)4317 39977 y Fq(\(4.22\))6149 b Fo(x)14661 38762 y Fa(2)p Fb(\000)p Ff(\014)p 14661 38971 1544 40 v 15229 39520 a Fa(2)14528 40113 y Fs(m)16761 39977 y Fn(\024)369 b Fo(Lm)295 b Fq(+)g Fo(C)22719 40176 y Fl(1)23465 39977 y Fq(ln)222 b Fo(m)295 b Fq(+)f Fo(C)28441 40176 y Fl(2)28967 39977 y Fo(m)30105 38503 y Fh(\020)31031 39079 y Fo(m)p 31031 39672 1138 54 v 31212 40888 a(n)32301 38503 y Fh(\021)33547 38390 y Fa(2)p Ff(\014)p 33227 38599 1544 40 v 33227 39147 a(\014)34 b Fb(\000)p Fa(2)35254 39977 y Fq(+)295 b Fo(C)37492 40176 y Fl(3)4317 43533 y Fj(wher)-66 b(e)465 b Fo(L)369 b Fq(=)f(\()p Fo(\014)h Fn(\000)296 b Fq(2\))14715 42434 y Fn(p)p 15822 42434 651 54 v 1099 x Fq(2)465 b Fj(and)g Fo(C)20392 43732 y Fl(1)20917 43533 y Fo(;)221 b(C)22430 43732 y Fl(2)22956 43533 y Fo(;)g(C)24469 43732 y Fl(3)25364 43533 y Fo(>)369 b Fq(0)465 b Fj(ar)-66 b(e)465 b(some)f(c)-66 b(onstants.)4317 46134 y(Pr)g(o)g(of.)649 b Fq(Due)434 b(to)f(\(4.8\))i(and)e(\(4.21\))14692 49581 y Fo(x)15431 49780 y Fs(m)16614 49581 y Fn(\000)295 b Fo(x)18681 49780 y Fs(m)p Fl(+1)21140 49581 y Fo(<)369 b Fq(2)23171 48418 y Fn(p)p 24278 48418 V 1163 x Fq(2)222 b Fo(x)26022 48366 y Ff(\014)p 26022 48575 498 40 v 26068 49124 a Fa(2)25889 49718 y Fs(m)27072 49581 y Fq(+)294 b Fo(x)29117 48777 y Fg(\000)29982 48366 y Ff(\014)p 29982 48575 V 30028 49124 a Fa(2)29117 49718 y Fs(m)30668 49581 y Fo(w)31634 49033 y Fl(2)31598 49910 y Fs(n)32169 49658 y Fb(0)32818 49581 y Fq(+)g Fo(C)95 b(x)36022 48366 y Fa(3)p Ff(\014)p 36022 48575 904 40 v 36271 49124 a Fa(2)35889 49718 y Fs(m)4317 52515 y Fq(for)623 b(some)g(large)g Fo(C)786 b(>)690 b Fq(0)623 b(\(w)-36 b(e)623 b(used)f(the)g(fact)h Fo(w)29919 52033 y Fl(2)29883 52843 y Fs(m)31461 52515 y Fn(\030)691 b Fo(x)33924 52033 y Fs(\014)33924 52843 y(m)34812 52515 y Fq(\).)1145 b(As)622 b(b)36 b(efore,)671 b(w)-36 b(e)623 b(put)4317 54634 y Fo(z)4921 54833 y Fs(m)6434 54634 y Fq(=)i Fo(x)8943 53419 y Ff(\014)34 b Fb(\000)p Fa(2)p 8943 53628 1544 40 v 9512 54176 a(2)8810 54770 y Fs(m)10674 54634 y Fq(.)1030 b(W)-108 b(e)585 b(consider)f(t)-36 b(w)g(o)584 b(cases.)1030 b(If)585 b Fo(\014)699 b Fn(\025)626 b Fq(4,)c(then)583 b(the)h(function)g Fo(x)44299 53733 y Ff(\014)34 b Fb(\000)p Fa(2)p 44299 53942 V 44867 54491 a(2)46614 54634 y Fq(is)4317 56239 y(con)-36 b(v)g(ex)434 b(do)-36 b(wn,)434 b(and)13363 59613 y Fo(z)13967 59812 y Fs(m)15150 59613 y Fn(\000)295 b Fo(z)17082 59812 y Fs(m)p Fl(+1)19541 59613 y Fn(\024)21076 59029 y Fs(\014)46 b Fg(\000)p Fl(2)p 21076 59307 1776 54 v 21729 60071 a(2)23206 59613 y Fo(x)24078 58398 y Ff(\014)34 b Fb(\000)p Fa(4)p 24078 58607 1544 40 v 24647 59155 a(2)23945 59749 y Fs(m)25810 59613 y Fq(\()p Fo(x)27055 59812 y Fs(m)28237 59613 y Fn(\000)296 b Fo(x)30305 59812 y Fs(m)p Fl(+1)32395 59613 y Fq(\))19541 61736 y Fn(\024)369 b Fo(L)221 b(x)22788 61188 y Fs(\014)46 b Fg(\000)p Fl(2)22788 62064 y Fs(m)24915 61736 y Fq(+)26355 61152 y Fs(\014)g Fg(\000)p Fl(2)p 26355 61431 1776 54 v 27008 62194 a(2)28485 61736 y Fo(x)29224 61188 y Fg(\000)p Fl(2)29224 62064 y Fs(m)30482 61736 y Fo(w)31448 61188 y Fl(2)31412 62064 y Fs(n)31983 61812 y Fb(0)32631 61736 y Fq(+)295 b Fo(C)95 b(x)35703 61188 y Fl(2)p Fs(\014)46 b Fg(\000)p Fl(2)35703 62064 y Fs(m)19541 64466 y Fn(\024)369 b Fo(L)221 b(z)22712 63917 y Fl(2)22653 64794 y Fs(m)23837 64466 y Fq(+)25276 63882 y Fs(\014)46 b Fg(\000)p Fl(2)p 25276 64160 V 25929 64924 a(2)27406 64466 y Fo(z)28069 63532 y Fg(\000)29503 63170 y Fa(4)p 28935 63331 1544 40 v 28935 63879 a Ff(\014)34 b Fb(\000)p Fa(2)28010 64602 y Fs(m)30666 64466 y Fo(w)31632 63917 y Fl(2)31596 64794 y Fs(n)32167 64542 y Fb(0)32815 64466 y Fq(+)295 b Fo(C)95 b(z)35944 63122 y Fa(4)p Ff(\014)34 b Fb(\000)p Fa(4)p 35944 63330 1950 40 v 36147 63879 a Ff(\014)g Fb(\000)p Fa(2)35752 64602 y Fs(m)38082 64466 y Fo(:)25253 70015 y Fq(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 4317 7339 a Fq(If)434 b Fo(\014)443 b(<)369 b Fq(4,)434 b(then)f(the)g(function)g Fo(x)20789 6439 y Ff(\014)34 b Fb(\000)p Fa(2)p 20789 6648 1544 40 v 21358 7196 a(2)22954 7339 y Fq(is)434 b(con)-36 b(v)g(ex)434 b(up,)f(and)11568 10098 y Fo(z)12172 10297 y Fs(m)13355 10098 y Fn(\000)295 b Fo(z)15287 10297 y Fs(m)p Fl(+1)17746 10098 y Fn(\024)19281 9514 y Fs(\014)46 b Fg(\000)p Fl(2)p 19281 9792 1776 54 v 19934 10556 a(2)21411 10098 y Fo(x)22283 8883 y Ff(\014)34 b Fb(\000)p Fa(4)p 22283 9092 1544 40 v 22852 9640 a(2)22150 10424 y Fs(m)p Fl(+1)24240 10098 y Fq(\()p Fo(x)25485 10297 y Fs(m)26668 10098 y Fn(\000)295 b Fo(x)28735 10297 y Fs(m)p Fl(+1)30825 10098 y Fq(\))17746 12699 y Fn(\024)369 b Fo(L)221 b(x)21126 11483 y Ff(\014)p 21127 11692 498 40 v 21173 12241 a Fa(2)20993 12835 y Fs(m)22103 12699 y Fo(x)22975 11483 y Ff(\014)34 b Fb(\000)p Fa(4)p 22975 11692 1544 40 v 23544 12241 a(2)22842 13024 y Fs(m)p Fl(+1)25227 12699 y Fq(+)26667 12115 y Fs(\014)46 b Fg(\000)p Fl(2)p 26667 12393 1776 54 v 27319 13157 a(2)28797 12699 y Fo(x)29536 12150 y Fg(\000)p Fl(2)29536 13027 y Fs(m)p Fl(+1)31626 12699 y Fo(w)32592 12150 y Fl(2)32556 13027 y Fs(n)33127 12775 y Fb(0)33775 12699 y Fq(+)295 b Fo(C)95 b(x)36847 12150 y Fl(2)p Fs(\014)46 b Fg(\000)p Fl(2)36847 13027 y Fs(m)17746 15428 y Fn(\024)369 b Fo(L)221 b(z)21574 14084 y Ff(\014)p 21051 14293 1544 40 v 21051 14842 a(\014)34 b Fb(\000)p Fa(2)20858 15565 y Fs(m)23004 15428 y Fo(z)23800 14084 y Ff(\014)g Fb(\000)p Fa(4)p 23800 14293 V 23800 14842 a Ff(\014)g Fb(\000)p Fa(2)23608 15754 y Fs(m)p Fl(+1)25993 15428 y Fq(+)27433 14844 y Fs(\014)46 b Fg(\000)p Fl(2)p 27433 15123 1776 54 v 28085 15887 a(2)29563 15428 y Fo(z)30226 14494 y Fg(\000)31659 14132 y Fa(4)p 31091 14293 1544 40 v 31091 14842 a Ff(\014)34 b Fb(\000)p Fa(2)30167 15754 y Fs(m)p Fl(+1)32822 15428 y Fo(w)33788 14880 y Fl(2)33752 15757 y Fs(n)34323 15505 y Fb(0)34972 15428 y Fq(+)294 b Fo(C)95 b(z)38100 14084 y Fa(4)p Ff(\014)34 b Fb(\000)p Fa(4)p 38101 14293 1950 40 v 38304 14842 a Ff(\014)g Fb(\000)p Fa(2)37908 15565 y Fs(m)4317 17747 y Fq(As)434 b(b)36 b(efore,)434 b(let)g Fo(Z)13271 17946 y Fs(m)14528 17747 y Fq(=)368 b(1)p Fo(=z)17812 17946 y Fs(m)18701 17747 y Fq(.)578 b(Then)433 b(in)h(the)f(case)h Fo(\014)443 b(>)368 b Fq(4)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)8599 20724 y Fo(Z)9489 20923 y Fs(m)p Fl(+1)11874 20724 y Fn(\000)295 b Fo(Z)14092 20923 y Fs(m)15349 20724 y Fn(\024)369 b Fo(L)17991 19825 y(Z)18881 20024 y Fs(m)p Fl(+1)p 17991 20418 2981 54 v 18592 21635 a Fo(Z)19482 21834 y Fs(m)21399 20724 y Fq(+)22838 19825 y Fo(\014)h Fn(\000)295 b Fq(2)p 22838 20418 3082 54 v 24054 21635 a(2)26275 20724 y Fo(Z)27713 19379 y Fa(2)p Ff(\014)p 27393 19588 1544 40 v 27393 20137 a(\014)34 b Fb(\000)p Fa(2)27165 21049 y Fs(m)p Fl(+1)29255 20724 y Fo(w)30221 20175 y Fl(2)30185 21052 y Fs(n)30756 20800 y Fb(0)31404 20724 y Fq(+)295 b Fo(C)95 b(Z)34722 19790 y Fg(\000)35907 19379 y Fa(2)p Ff(\014)p 35587 19588 V 35587 20137 a(\014)34 b Fb(\000)p Fa(2)34627 21049 y Fs(m)p Fl(+1)15349 24098 y Fn(\024)369 b Fo(L)296 b Fq(+)e Fo(L)20478 23199 y(Z)21368 23398 y Fs(m)p Fl(+1)23753 23199 y Fn(\000)h Fo(Z)25971 23398 y Fs(m)p 20478 23792 6382 54 v 22779 25009 a Fo(Z)23669 25208 y Fs(m)27287 24098 y Fq(+)28727 23199 y Fo(\014)369 b Fn(\000)295 b Fq(2)p 28727 23792 3082 54 v 29943 25009 a(2)32163 24098 y Fo(Z)33601 22754 y Fa(2)p Ff(\014)p 33281 22963 1544 40 v 33281 23511 a(\014)34 b Fb(\000)p Fa(2)33053 24424 y Fs(m)p Fl(+1)35143 24098 y Fo(w)36109 23550 y Fl(2)36073 24426 y Fs(n)36644 24174 y Fb(0)37293 24098 y Fq(+)294 b Fo(C)95 b(Z)40610 23164 y Fg(\000)41795 22754 y Fa(2)p Ff(\014)p 41476 22963 V 41476 23511 a(\014)34 b Fb(\000)p Fa(2)40515 24424 y Fs(m)p Fl(+1)4317 27029 y Fq(Solving)435 b(the)e(last)h(inequalit)-36 b(y)434 b(for)g Fo(Z)22520 27228 y Fs(m)p Fl(+1)24906 27029 y Fn(\000)295 b Fo(Z)27124 27228 y Fs(m)28446 27029 y Fq(and)433 b(using)g(\(4.19\))i(and)e(\(4.20\))h(giv)-36 b(es)15758 30233 y Fo(Z)16648 30432 y Fs(m)p Fl(+1)19033 30233 y Fn(\000)296 b Fo(Z)21252 30432 y Fs(m)22509 30233 y Fn(\024)369 b Fo(L)295 b Fq(+)26530 29334 y Fo(C)27556 28852 y Fg(0)p 26530 29928 1337 54 v 26629 31144 a Fo(m)28295 30233 y Fq(+)g Fo(C)30628 29685 y Fg(00)31194 28759 y Fh(\020)32120 29334 y Fo(m)p 32120 29928 1138 54 v 32301 31144 a(n)33390 28759 y Fh(\021)34636 28646 y Fa(2)p Ff(\014)p 34316 28855 1544 40 v 34316 29403 a(\014)34 b Fb(\000)p Fa(2)4317 32964 y Fq(for)540 b(some)f(large)h Fo(C)14085 32482 y Fg(0)14396 32964 y Fo(;)221 b(C)16004 32482 y Fg(00)17119 32964 y Fo(>)548 b Fq(0.)895 b(Summing)539 b(up)f(o)-36 b(v)g(er)539 b Fo(m)g Fq(implies)h(\(4.22\))g(with)f Fo(C)45403 33163 y Fl(3)46477 32964 y Fq(=)4317 34769 y Fo(")4926 34279 y Fg(\000)5791 33868 y Ff(\014)34 b Fb(\000)p Fa(2)p 5790 34077 V 6359 34626 a(2)7522 34769 y Fq(,)434 b(see)f(\(4.18\).)6268 36374 y(In)h(the)f(other)g(case,)h Fo(\014)443 b(<)369 b Fq(4,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)8528 39822 y Fo(Z)9418 40021 y Fs(m)p Fl(+1)11804 39822 y Fn(\000)295 b Fo(Z)14022 40021 y Fs(m)15279 39822 y Fn(\024)369 b Fo(L)17566 37949 y Fh(\022)18677 38923 y Fo(Z)19567 39122 y Fs(m)p Fl(+1)p 18677 39516 2981 54 v 19278 40733 a Fo(Z)20168 40932 y Fs(m)21790 37949 y Fh(\023)23469 37884 y Fa(2)p 22900 38045 1544 40 v 22900 38594 a Ff(\014)34 b Fb(\000)p Fa(2)24927 39822 y Fq(+)26366 38923 y Fo(\014)370 b Fn(\000)295 b Fq(2)p 26366 39516 3082 54 v 27582 40733 a(2)29802 39822 y Fo(Z)31241 38478 y Fa(2)p Ff(\014)p 30921 38686 1544 40 v 30921 39235 a(\014)34 b Fb(\000)p Fa(2)30692 40148 y Fs(m)p Fl(+1)32783 39822 y Fo(w)33749 39273 y Fl(2)33713 40150 y Fs(n)34284 39898 y Fb(0)34932 39822 y Fq(+)295 b Fo(C)95 b(Z)38250 38888 y Fg(\000)39435 38478 y Fa(2)p Ff(\014)p 39115 38686 V 39115 39235 a(\014)34 b Fb(\000)p Fa(2)38155 40148 y Fs(m)p Fl(+1)15279 43348 y Fn(\024)369 b Fo(L)295 b Fq(+)g Fo(G)20548 42449 y(Z)21438 42648 y Fs(m)p Fl(+1)23823 42449 y Fn(\000)h Fo(Z)26042 42648 y Fs(m)p 20548 43042 6382 54 v 22850 44259 a Fo(Z)23740 44458 y Fs(m)27358 43348 y Fq(+)28797 42449 y Fo(\014)369 b Fn(\000)296 b Fq(2)p 28797 43042 3082 54 v 30013 44259 a(2)32233 43348 y Fo(Z)33672 42003 y Fa(2)p Ff(\014)p 33352 42212 1544 40 v 33352 42761 a(\014)34 b Fb(\000)p Fa(2)33123 43673 y Fs(m)p Fl(+1)35214 43348 y Fo(w)36180 42799 y Fl(2)36144 43676 y Fs(n)36715 43424 y Fb(0)37363 43348 y Fq(+)295 b Fo(C)95 b(Z)40681 42414 y Fg(\000)41866 42003 y Fa(2)p Ff(\014)p 41546 42212 V 41546 42761 a(\014)34 b Fb(\000)p Fa(2)40586 43673 y Fs(m)p Fl(+1)4317 46410 y Fq(with)313 b Fo(G)369 b Fq(=)10400 45887 y Fl(3)p Fs(L)p 10068 46105 1776 54 v 10068 46868 a(\014)46 b Fg(\000)p Fl(2)11976 46410 y Fq(,)338 b(and)312 b(the)h(subsequen)-36 b(t)312 b(analysis)i(is)f(similar)h(to)g(the)e(previous)h(case.)p 46548 46410 45 878 v 46593 45576 781 45 v 46593 46410 V 47373 46410 45 878 v 4317 49168 a Fi(Corollary)521 b(7.)554 b Fj(F)-100 b(or)465 b(al)66 b(l)466 b Fq(1)369 b Fn(\024)g Fo(m)g Fn(\024)g Fo(n)23083 48686 y Fg(00)24114 49168 y Fj(we)465 b(have)4317 52396 y Fq(\(4.23\))3921 b(2)p Fn(K)19 b Fq(\()p Fo(r)14334 52595 y Fs(m)15223 52396 y Fq(\))368 b Fn(\025)h Fo(D)18615 50523 y Fh(\024)19317 52396 y Fo(m)294 b Fq(+)h Fo(C)23082 51848 y Fg(0)22987 52725 y Fl(1)23734 52396 y Fq(ln)221 b Fo(m)295 b Fq(+)g Fo(C)28805 51848 y Fg(0)28710 52725 y Fl(2)29235 52396 y Fo(m)30373 50922 y Fh(\020)31299 51498 y Fo(m)p 31299 52091 1138 54 v 31480 53307 a(n)32570 50922 y Fh(\021)33816 50809 y Fa(2)p Ff(\014)p 33496 51018 1544 40 v 33496 51566 a(\014)34 b Fb(\000)p Fa(2)35523 52396 y Fq(+)294 b Fo(C)37855 51848 y Fg(0)37760 52725 y Fl(3)38286 50523 y Fh(\025)38987 50821 y Fg(\000)p Fl(2)4317 55779 y Fj(wher)-66 b(e)465 b Fo(D)405 b Fq(=)10985 55146 y Fs(\014)46 b Fl(\()p Fs(\014)g Fg(\000)p Fl(1\))p 10985 55474 3082 54 v 11041 56237 a(\()p Fs(\014)g Fg(\000)p Fl(2\))13549 55985 y Fa(2)14664 55779 y Fj(and)465 b Fo(C)18214 55297 y Fg(0)18119 56108 y Fl(1)18644 55779 y Fo(;)221 b(C)20252 55297 y Fg(0)20157 56108 y Fl(2)20683 55779 y Fo(;)g(C)22291 55297 y Fg(0)22196 56108 y Fl(3)23091 55779 y Fo(>)369 b Fq(0)465 b Fj(ar)-66 b(e)465 b(some)g(c)-66 b(onstants.)4317 58386 y(Pr)g(o)g(of.)649 b Fq(Equation)434 b(\(4.6\))g(implies)15876 60704 y Fn(K)19 b Fq(\()p Fo(r)17999 60903 y Fs(m)18887 60704 y Fq(\))369 b(=)f Fo(\014)74 b Fq(\()p Fo(\014)369 b Fn(\000)295 b Fq(1\))p Fo(x)26782 60156 y Fs(\014)46 b Fg(\000)p Fl(2)26782 61033 y Fs(m)28909 60704 y Fq(+)295 b Fn(O)31311 59628 y Fh(\000)31919 60704 y Fo(x)32658 60156 y Fl(3)p Fs(\014)46 b Fg(\000)p Fl(4)32658 61033 y Fs(m)34960 59628 y Fh(\001)35569 60704 y Fo(:)4317 63484 y Fq(T)-108 b(o)289 b(estimate)h(the)e(main)i(term)e(w)-36 b(e)289 b(use)g(\(4.22\),)319 b(and)289 b(the)f(remainder)h(term)g(is)g Fn(O)42893 62408 y Fh(\000)43501 63484 y Fo(m)44639 62865 y Fg(\000)45504 62454 y Fa(6)p Ff(\014)34 b Fb(\000)p Fa(8)p 45504 62663 1950 40 v 45707 63212 a Ff(\014)g Fb(\000)p Fa(2)47641 62408 y Fh(\001)4317 65089 y Fq(b)-36 b(y)579 b(\(4.16\),)617 b(so)580 b(it)g(can)f(b)36 b(e)579 b(incorp)36 b(orated)579 b(in)-36 b(to)580 b(the)e(righ)-36 b(t)579 b(hand)g(side)g(of)h(\(4.23\))h(b)-36 b(y)4317 66694 y(c)g(ho)36 b(osing)434 b(su\016cien)-36 b(tly)434 b(large)g(constan)-36 b(ts)433 b Fo(C)26074 66212 y Fg(0)25979 67023 y Fl(1)26505 66694 y Fo(;)221 b(C)28113 66212 y Fg(0)28018 67023 y Fl(2)28544 66694 y Fo(;)g(C)30152 66212 y Fg(0)30057 67023 y Fl(3)30952 66694 y Fo(>)368 b Fq(0.)p 46548 66694 45 878 v 46593 65861 781 45 v 46593 66694 V 47373 66694 45 878 v 25253 70015 a(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 4317 7306 a Fi(Lemma)499 b(8.)554 b Fj(F)-100 b(or)465 b(al)66 b(l)466 b Fq(1)369 b Fn(\024)h Fo(m)e(<)h(n)21698 6824 y Fg(00)22729 7306 y Fj(we)465 b(have)4317 11072 y Fq(\(4.24\))3528 b Fn(B)40 b Fq(\()p Fo(X)13665 11271 y Fs(m)p Fg(\000)p Fl(1)15756 11072 y Fq(\))369 b Fn(\025)g Fo(A)19008 9199 y Fh(\024)19710 11072 y Fo(m)294 b Fq(+)h Fo(C)23380 11271 y Fl(4)24127 11072 y Fq(ln)221 b Fo(m)295 b Fq(+)g Fo(C)29103 11271 y Fl(5)29628 11072 y Fo(m)30766 9597 y Fh(\020)31692 10173 y Fo(m)p 31692 10766 1138 54 v 31873 11983 a(n)32963 9597 y Fh(\021)34209 9484 y Fa(2)p Ff(\014)p 33889 9693 1544 40 v 33889 10242 a(\014)34 b Fb(\000)p Fa(2)35915 11072 y Fq(+)295 b Fo(C)38153 11271 y Fl(6)38679 9199 y Fh(\025)39380 9496 y Fg(\000)p Fl(1)4317 14972 y Fj(wher)-66 b(e)482 b Fo(A)400 b(>)g Fq(0)482 b Fj(satis\014es)g Fq(2)p Fo(A)18447 14490 y Fl(2)19280 14972 y Fn(\000)308 b Fo(A)401 b Fq(=)e Fo(D)36 b Fj(,)486 b(henc)-66 b(e)480 b Fo(A)401 b Fq(=)31932 14388 y Fs(\014)46 b Fg(\000)p Fl(1)p 31932 14666 1776 54 v 31932 15430 a Fs(\014)g Fg(\000)p Fl(2)33841 14972 y Fj(,)486 b(and)481 b Fo(C)38197 15171 y Fl(4)38723 14972 y Fo(;)221 b(C)40236 15171 y Fl(5)40762 14972 y Fo(;)g(C)42275 15171 y Fl(6)43201 14972 y Fo(>)400 b Fq(0)482 b Fj(ar)-66 b(e)4317 16577 y(lar)g(ge)464 b(c)-66 b(onstants.)4317 19178 y(Pr)g(o)g(of.)649 b Fq(W)-108 b(e)341 b(use)g(induction)f(on)g Fo(m)p Fq(.)548 b(F)-108 b(or)340 b Fo(m)369 b Fq(=)f(1)341 b(the)g(v)-72 b(alidit)-36 b(y)342 b(of)f(\(4.24\))h(is)f(guaran)-36 b(teed)4317 20783 y(b)g(y)309 b(c)-36 b(ho)36 b(osing)309 b Fo(C)12087 20982 y Fl(6)12920 20783 y Fq(large)h(enough.)536 b(Assume)308 b(that)g(\(4.24\))h(is)g(v)-72 b(alid)309 b(for)g(some)g Fo(m)369 b(<)f(n)44798 20301 y Fg(00)45404 20783 y Fn(\000)40 b Fq(1.)4317 22388 y(Due)434 b(to)f(\(4.5\))i(and)e (\(4.23\))h(it)g(is)g(enough)f(to)g(v)-36 b(erify)13929 25067 y Fo(D)p 4450 25660 20076 54 v 4450 26093 a Fh(h)5077 27568 y Fo(m)295 b Fq(+)g Fo(C)8843 27110 y Fg(0)8748 27893 y Fl(1)9495 27568 y Fq(ln)221 b Fo(m)295 b Fq(+)g Fo(C)14566 27110 y Fg(0)14471 27893 y Fl(2)14996 27568 y Fo(m)16134 26492 y Fh(\000)16875 27045 y Fs(m)p 16875 27262 833 54 v 17006 28026 a(n)17841 26492 y Fh(\001)18902 26379 y Fa(2)p Ff(\014)p 18582 26588 1544 40 v 18582 27136 a(\014)34 b Fb(\000)p Fa(2)20609 27568 y Fq(+)295 b Fo(C)22942 27110 y Fg(0)22847 27893 y Fl(3)23372 26093 y Fh(i)23999 26391 y Fl(2)24953 25966 y Fq(+)37067 25067 y Fo(A)p 26393 25660 22325 54 v 26393 27470 a(A\034)27933 27669 y Fs(m)29115 27470 y Fq(+)g Fo(m)g Fq(+)g Fo(C)34093 27669 y Fl(4)34840 27470 y Fq(ln)221 b Fo(m)295 b Fq(+)g Fo(C)39816 27669 y Fl(5)40341 27470 y Fo(m)41479 26394 y Fh(\000)42220 26947 y Fs(m)p 42220 27164 833 54 v 42351 27928 a(n)43186 26394 y Fh(\001)44247 26281 y Fa(2)p Ff(\014)p 43927 26490 1544 40 v 43927 27039 a(\014)34 b Fb(\000)p Fa(2)45954 27470 y Fq(+)295 b Fo(C)48192 27669 y Fl(6)19148 30680 y Fo(>)34201 29782 y(A)p 20661 30375 28056 54 v 20661 32184 a(m)g Fq(+)g(1)g(+)g Fo(C)26584 32383 y Fl(4)27331 32184 y Fq(ln\()p Fo(m)f Fq(+)h(1\))h(+)e Fo(C)35349 32383 y Fl(5)35875 32184 y Fq(\()p Fo(m)g Fq(+)h(1\))40276 31108 y Fh(\000)41018 31661 y Fs(m)p Fl(+1)p 41018 31879 2035 54 v 41750 32642 a Fs(n)43186 31108 y Fh(\001)44247 30995 y Fa(2)p Ff(\014)p 43927 31204 1544 40 v 43927 31753 a(\014)34 b Fb(\000)p Fa(2)45954 32184 y Fq(+)295 b Fo(C)48192 32383 y Fl(6)4317 35011 y Fq(pro)-36 b(vided)433 b Fo(C)10595 35210 y Fl(4)11121 35011 y Fo(;)221 b(C)12634 35210 y Fl(5)13160 35011 y Fo(;)g(C)14673 35210 y Fl(6)15568 35011 y Fo(>)368 b Fq(0)434 b(are)g(large)g(enough.)578 b(Here)13655 38199 y Fo(\034)14220 38398 y Fs(m)15476 38199 y Fq(=)369 b Fo(\034)148 b Fq(\()p Fo(X)19155 38398 y Fs(m)20043 38199 y Fq(\))369 b(=)f(2)296 b(+)f Fn(O)37 b Fq(\()p Fo(w)27082 38398 y Fs(m)27968 38199 y Fq(\))369 b(=)g(2)295 b(+)g Fn(O)33571 37123 y Fh(\000)34179 38199 y Fo(m)35972 37169 y Ff(\014)p 35450 37378 V 35450 37927 a(\014)34 b Fb(\000)p Fa(2)37181 37123 y Fh(\001)37790 38199 y Fo(:)4317 41132 y Fq(It)434 b(is)g(easy)g(to)g(see)g(that)26449 43737 y Fo(A)p 12908 44331 28056 54 v 12908 46140 a(m)295 b Fq(+)g(1)g(+)g Fo(C)18831 46339 y Fl(4)19578 46140 y Fq(ln\()p Fo(m)g Fq(+)f(1\))i(+)f Fo(C)27597 46339 y Fl(5)28122 46140 y Fq(\()p Fo(m)g Fq(+)f(1\))32523 45064 y Fh(\000)33265 45617 y Fs(m)p Fl(+1)p 33265 45835 2035 54 v 33997 46598 a Fs(n)35433 45064 y Fh(\001)36494 44951 y Fa(2)p Ff(\014)p 36174 45160 1544 40 v 36174 45709 a(\014)34 b Fb(\000)p Fa(2)38201 46140 y Fq(+)295 b Fo(C)40439 46339 y Fl(6)10709 48999 y Fn(\000)22549 48101 y Fo(A)p 11875 48694 22325 54 v 11875 50503 a(A\034)13415 50702 y Fs(m)14598 50503 y Fq(+)f Fo(m)h Fq(+)g Fo(C)19575 50702 y Fl(4)20322 50503 y Fq(ln)221 b Fo(m)295 b Fq(+)g Fo(C)25298 50702 y Fl(5)25823 50503 y Fo(m)26961 49427 y Fh(\000)27703 49980 y Fs(m)p 27703 50198 833 54 v 27834 50961 a(n)28668 49427 y Fh(\001)29729 49314 y Fa(2)p Ff(\014)p 29409 49523 1544 40 v 29409 50072 a(\014)34 b Fb(\000)p Fa(2)31436 50503 y Fq(+)295 b Fo(C)33674 50702 y Fl(6)34701 48999 y Fo(<)36214 48101 y Fq(2)p Fo(A)37839 47618 y Fl(2)38661 48101 y Fn(\000)g Fo(A)p 36214 48694 4750 54 v 38083 49910 a Fq(\002)4317 53330 y(where)419 b(\002)g(denotes)g(the)g(pro)36 b(duct)418 b(of)i(the)e(t)-36 b(w)g(o)420 b(denominators.)573 b(Th)-36 b(us)419 b(it)g(is)h(enough)e(to)4317 54935 y(v)-36 b(erify)21782 56356 y Fo(D)p 12302 56950 20076 54 v 12302 57382 a Fh(h)12929 58857 y Fo(m)295 b Fq(+)g Fo(C)16695 58399 y Fg(0)16600 59182 y Fl(1)17347 58857 y Fq(ln)221 b Fo(m)295 b Fq(+)g Fo(C)22418 58399 y Fg(0)22323 59182 y Fl(2)22848 58857 y Fo(m)23986 57781 y Fh(\000)24728 58334 y Fs(m)p 24728 58551 833 54 v 24859 59315 a(n)25693 57781 y Fh(\001)26754 57668 y Fa(2)p Ff(\014)p 26434 57877 1544 40 v 26434 58425 a(\014)34 b Fb(\000)p Fa(2)28461 58857 y Fq(+)295 b Fo(C)30794 58399 y Fg(0)30699 59182 y Fl(3)31224 57382 y Fh(i)31851 57680 y Fl(2)32879 57255 y Fo(>)34393 56356 y Fq(2)p Fo(A)36018 55874 y Fl(2)36839 56356 y Fn(\000)g Fo(A)p 34393 56950 4750 54 v 36262 58166 a Fq(\002)39275 57255 y Fo(:)4317 61603 y Fq(W)-108 b(e)434 b(recall)g(that)f(2)p Fo(A)14468 61121 y Fl(2)15290 61603 y Fn(\000)295 b Fo(A)369 b Fq(=)g Fo(D)36 b Fq(.)578 b(Th)-36 b(us)433 b(it)h(is)g(enough)f(to)g (v)-36 b(erify)4317 65447 y(\(4.25\))6253 b(\002)369 b Fo(>)16654 63574 y Fh(\024)17355 65447 y Fo(m)295 b Fq(+)g Fo(C)21121 64898 y Fg(0)21026 65775 y Fl(1)21773 65447 y Fq(ln)221 b Fo(m)295 b Fq(+)g Fo(C)26844 64898 y Fg(0)26749 65775 y Fl(2)27274 65447 y Fo(m)28412 63972 y Fh(\020)29338 64548 y Fo(m)p 29338 65141 1138 54 v 29519 66358 a(n)30609 63972 y Fh(\021)31855 63859 y Fa(2)p Ff(\014)p 31535 64068 1544 40 v 31535 64617 a(\014)34 b Fb(\000)p Fa(2)33561 65447 y Fq(+)295 b Fo(C)35894 64898 y Fg(0)35799 65775 y Fl(3)36324 63574 y Fh(\025)37026 63871 y Fl(2)37551 65447 y Fo(:)25253 70015 y Fq(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 4317 7306 a Fq(The)488 b(leading)h(term)e Fo(m)15879 6824 y Fl(2)16893 7306 y Fq(app)36 b(ears)487 b(on)h(b)36 b(oth)488 b(sides)g(and)f(cancels)i(out.)741 b(Keeping)488 b(only)4317 8911 y(the)433 b(largest)h(non-cancelling)g (terms)f(on)h(b)36 b(oth)433 b(sides)g(of)i(\(4.25\))f(w)-36 b(e)434 b(obtain)8269 12247 y(2)221 b Fo(C)10071 12446 y Fl(4)10598 12247 y Fo(m)g Fq(ln)g Fo(m)295 b Fq(+)g(2)221 b Fo(C)17804 12446 y Fl(5)18330 12247 y Fo(m)19468 11698 y Fl(2)19993 10772 y Fh(\020)20919 11348 y Fo(m)p 20919 11941 1138 54 v 21100 13158 a(n)22190 10772 y Fh(\021)23436 10659 y Fa(2)p Ff(\014)p 23116 10868 1544 40 v 23116 11417 a(\014)34 b Fb(\000)p Fa(2)25216 12247 y Fo(>)369 b Fq(2)221 b Fo(C)28494 11698 y Fg(0)28399 12575 y Fl(1)28925 12247 y Fo(m)g Fq(ln)g Fo(m)295 b Fq(+)g(2)221 b Fo(C)36226 11698 y Fg(0)36131 12575 y Fl(2)36657 12247 y Fo(m)37795 11698 y Fl(2)38321 10772 y Fh(\020)39247 11348 y Fo(m)p 39247 11941 1138 54 v 39428 13158 a(n)40517 10772 y Fh(\021)41763 10659 y Fa(2)p Ff(\014)p 41444 10868 1544 40 v 41444 11417 a(\014)34 b Fb(\000)p Fa(2)43175 12247 y Fo(;)4317 15035 y Fq(whic)-36 b(h)577 b(can)f(b)36 b(e)577 b(ensured)e(b)-36 b(y)577 b(c)-36 b(ho)36 b(osing)577 b Fo(C)25848 15234 y Fl(4)26951 15035 y Fq(and)f Fo(C)30554 15234 y Fl(5)31656 15035 y Fq(large)i(enough.)1007 b(This)577 b(implies)4317 16640 y(\(4.25\))435 b(and)e(then)f(Lemma)i(8.)p 46548 16640 45 878 v 46593 15807 781 45 v 46593 16640 V 47373 16640 45 878 v 4317 18910 a Fi(Corollary)521 b(9.)4317 21995 y Fq(\(4.26\))6957 b Fn(B)40 b Fq(\()p Fo(X)17094 22194 y Fs(m)p Fg(\000)p Fl(1)19185 21995 y Fq(\))369 b Fn(\025)21677 21097 y Fo(A)p 21595 21690 1138 54 v 21595 22907 a(m)23161 21995 y Fq(+)24601 21097 y Fo(C)25627 20615 y Fg(0)25532 21425 y Fl(4)26278 21097 y Fq(ln)222 b Fo(m)p 24601 21690 4121 54 v 25829 22907 a(m)26967 22523 y Fl(2)29149 21995 y Fq(+)30589 21097 y Fo(C)31615 20615 y Fg(0)31520 21425 y 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Fl(\(2\))13815 11394 y Fs(n)15072 11065 y Fq(\()p Fo(X)104 b Fq(\))148 b(:)813 b(=)20299 9405 y Fs(n)p Fg(\000)p Fl(1)20337 9804 y Fh(Y)19969 12639 y Fs(m)p Fl(=)p Fs(n)22104 12386 y Fb(0)22402 9990 y Fh(\000)23011 11065 y Fq(1)296 b(+)f Fo(\034)148 b Fq(\()p Fo(X)27562 11264 y Fs(m)28450 11065 y Fq(\))p Fn(B)40 b Fq(\()p Fo(X)31453 11264 y Fs(m)32341 11065 y Fq(\))32847 9990 y Fh(\001)33825 11065 y Fn(\025)369 b Fo(C)95 b(n)37685 10036 y Ff(\014)p 37162 10245 1544 40 v 37162 10793 a(\014)34 b Fb(\000)p Fa(2)4317 15089 y Fj(wher)-66 b(e)465 b Fo(C)f(>)368 b Fq(0)466 b Fj(is)e(a)h(c)-66 b(onstant.)4317 17678 y(Pr)g(o)g(of.)649 b Fq(This)505 b(can)f(b)36 b(e)503 b(obtained)h(b)-36 b(y)504 b(a)g(detailed)g (analysis)i(of)e(the)g(dynamics)g(on)g(the)4317 19284 y(in)-36 b(terv)-72 b(al)556 b(\()p Fo(n)10418 18801 y Fg(0)10729 19284 y Fo(;)221 b(n)p Fq(\))556 b(similar)h(to)f(the)f (one)h(done)f(for)h(the)g(in)-36 b(terv)-72 b(al)556 b(\(0)p Fo(;)221 b(n)39157 18801 y Fg(0)39468 19284 y 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/_ArialMT 12 tfn (O) 343.356 583.937 tpt T %%PageTrailer _PDX_savepage restore %%Trailer end % showpage %%EOF %%EndDocument @endspecial 5685 38771 a /End PSfrag 5685 38771 a 5685 26089 a /Hide PSfrag 5685 26089 a -5901 27011 a Fq(PSfrag)h(replacemen)-36 b(ts)p -5901 27482 11587 54 v 5685 27535 a /Unhide PSfrag 5685 27535 a 4710 29140 a { 4710 29140 a Fq(\005)4710 29140 y } 0/Place PSfrag 4710 29140 a 3691 30064 a { 3691 30064 a 1696 30546 a Fo(D)2776 30745 y Fs(X)3674 30546 y Fn(F)4761 30064 y Fs(n)5332 29752 y Fb(0)3691 30064 y } 1/Place PSfrag 3691 30064 a 4038 32351 a { 4038 32351 a Fo(V)5085 31869 y Fs(u)4038 32351 y } 2/Place PSfrag 4038 32351 a 4148 33956 a { 4148 33956 a Fo(V)5195 33474 y Fs(s)4148 33956 y } 3/Place PSfrag 4148 33956 a 4502 35561 a { 4502 35561 a Fo(X)4502 35561 y } 4/Place PSfrag 4502 35561 a 3786 36967 a { 3786 36967 a Fq(\005)4761 37166 y Fs(n)5332 36914 y Fb(0)3786 36967 y } 5/Place PSfrag 3786 36967 a 2743 38313 a { 2743 38313 a Fn(O)37 b Fq(\()4527 37790 y Fl(1)p 4477 38007 571 54 v 4477 38771 a Fs(n)5180 38313 y Fq(\))2743 38313 y } 6/Place PSfrag 2743 38313 a 23780 41483 a Fq(Fig.)435 b(7:)6268 44539 y(Consider)649 b(the)f(parallelogram)i(\005)23720 44738 y Fs(n)24291 44486 y Fb(0)25379 44539 y Fq(=)735 b Fo(D)28206 44738 y Fs(X)29104 44539 y Fn(F)30191 44057 y Fs(n)30762 43745 y Fb(0)31115 44539 y Fq(\(\005\))648 b(spanned)g(b)-36 b(y)648 b(the)g(v)-36 b(ectors)4317 46144 y Fo(V)5364 45662 y Fs(u)5075 46496 y(n)5646 46244 y Fb(0)6456 46144 y Fq(=)456 b Fo(D)9004 46343 y Fs(X)9902 46144 y Fn(F)10989 45662 y Fs(n)11560 45350 y Fb(0)11913 46144 y Fq(\()p Fo(V)13466 45662 y Fs(u)14067 46144 y Fq(\))484 b(and)h Fo(V)18685 45662 y Fs(s)18396 46496 y(n)18967 46244 y Fb(0)19776 46144 y Fq(=)456 b Fo(D)22324 46343 y Fs(X)23223 46144 y Fn(F)24310 45662 y Fs(n)24881 45350 y Fb(0)25234 46144 y Fq(\()p Fo(V)26787 45662 y Fs(s)27277 46144 y Fq(\).)732 b(Since)485 b(the)f(map)g Fn(F)38643 45662 y Fs(n)39214 45350 y Fb(0)40052 46144 y Fq(preserv)-36 b(es)485 b(the)4317 47749 y(measure)434 b Fo(d\026)368 b Fq(=)h(cos)221 b Fo(')g(dr)258 b(d')p Fq(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)15969 50639 y(cos)222 b Fo(')18784 50838 y Fs(n)19355 50587 y Fb(0)20151 50639 y Fq(Area)q(\(\005)24342 50838 y Fs(n)24913 50587 y Fb(0)25266 50639 y Fq(\))369 b(=)g(cos)221 b Fo(')443 b Fq(Area\(\005\))p Fo(:)4317 53529 y Fq(Note)434 b(that)f(cos)222 b Fo(')369 b Fn(\031)g Fq(1)434 b(and)f(cos)222 b Fo(')21292 53728 y Fs(n)21863 53477 y Fb(0)22585 53529 y Fn(\031)370 b Fq(1,)434 b(hence)18468 56419 y(Area\(\005)22658 56618 y Fs(n)23229 56366 y Fb(0)23582 56419 y Fq(\))369 b Fn(\030)g Fq(Area)q(\(\005\))f Fn(\030)h Fq(1)p Fo(:)4317 59309 y Fq(On)433 b(the)g(other)g(hand,)17756 62199 y(Area)q(\(\005)21947 62398 y Fs(n)22518 62146 y Fb(0)22871 62199 y Fq(\))369 b(=)f Fn(j)p Fo(V)26543 61651 y Fs(u)26253 62528 y(n)26824 62276 y Fb(0)27178 62199 y Fn(j)221 b(j)p Fo(V)29185 61651 y Fs(s)28895 62528 y(n)29466 62276 y Fb(0)29820 62199 y Fn(j)443 b Fq(sin)221 b Fo(\015)33125 62398 y Fs(n)33696 62146 y Fb(0)4317 65089 y Fq(where)575 b Fn(j)p Fo(V)9632 64607 y Fs(u)9343 65441 y(n)9914 65189 y Fb(0)10267 65089 y Fn(j)g Fq(and)f Fn(j)p Fo(V)15297 64607 y Fs(s)15008 65441 y(n)15579 65189 y Fb(0)15933 65089 y Fn(j)g Fq(denote)g(the)g (lengths)g(of)h(these)f(v)-36 b(ectors)575 b(in)g(the)f(Euclidean)4317 66694 y(norm)433 b(\(4.4\))h(and)g Fo(\015)14025 66893 y Fs(n)14596 66641 y Fb(0)15382 66694 y Fq(denotes)f(the)g(angle)i(b)36 b(et)-36 b(w)g(een)433 b(them.)25253 70015 y(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 6268 7306 a Fq(Next)434 b(w)-36 b(e)434 b(estimate)g Fo(\015)17247 7505 y Fs(n)17818 7253 y Fb(0)18171 7306 y Fq(.)578 b(It)434 b(easily)h(follo)-36 b(ws)436 b(from)e(\(4.5\))g(that)12292 11489 y Fn(B)40 b Fq(\()p Fo(X)14789 11688 y Fs(n)15360 11436 y Fb(0)15714 11489 y Fq(\))369 b Fn(\025)17991 9218 y Fh(")18766 9829 y Fs(n)19337 9516 y Fb(0)19635 9829 y Fg(\000)p Fl(1)18842 10227 y Fh(X)18785 13017 y Fs(m)p Fl(=0)21059 11489 y Fo(\034)148 b Fq(\()p Fo(X)23357 11688 y Fs(m)24245 11489 y Fq(\))295 b(+)g(1)p Fo(=)p Fn(B)40 b Fq(\()p Fo(X)30150 11688 y Fl(0)30678 11489 y Fq(\))31184 9218 y Fh(#)31958 9515 y Fg(\000)p Fl(1)33585 11489 y Fn(\030)35338 10590 y Fq(1)p 35120 11184 1087 54 v 35120 12400 a Fo(n)35896 12016 y Fg(0)36709 11489 y Fn(\030)38307 10590 y Fq(1)p 38244 11184 777 54 v 38244 12400 a Fo(n)39153 11489 y(:)4317 15452 y Fq(No)-36 b(w)434 b(\(4.2\))g(implies)g(that)g(the)f(slop)36 b(e)434 b(of)g(the)f(v)-36 b(ector)434 b Fo(V)31825 14970 y Fs(u)31536 15804 y(n)32107 15551 y Fb(0)32894 15452 y Fq(is)17440 18032 y Fo(d')p 17440 18625 1528 54 v 17555 19842 a(dr)19469 18931 y Fq(=)369 b(cos)222 b Fo(')23665 19130 y Fs(n)24236 18878 y Fb(0)25032 18931 y Fn(B)40 b Fq(\()p Fo(X)27529 19130 y Fs(n)28100 18878 y Fb(0)28455 18931 y Fq(\))295 b Fn(\000)g(K)19 b Fq(\()p Fo(r)32707 19130 y Fs(n)33278 18878 y Fb(0)33632 18931 y Fq(\))p Fo(:)4317 22245 y Fq(W)-108 b(e)585 b(note)f(that)g(cos)222 b Fo(')15533 22444 y Fs(n)16104 22192 y Fb(0)17084 22245 y Fn(\031)626 b Fq(1)585 b(and)f Fn(K)19 b Fq(\()p Fo(r)24781 22444 y Fs(n)25352 22192 y Fb(0)25706 22245 y Fq(\))625 b Fn(\030)i Fo(n)29273 21763 y Fg(\000)p Fs(\014)31218 22245 y Fq(due)584 b(to)h(\(4.6\),)623 b(b)36 b(ecause)584 b Fo(x)44926 22444 y Fs(n)45497 22192 y Fb(0)46477 22245 y Fo(<)4317 24179 y Fq(3)p Fo(w)5897 24378 y Fs(n)6468 24126 y Fb(0)7191 24179 y Fn(\030)369 b Fo(n)10025 23149 y Ff(\014)p 9502 23358 1544 40 v 9502 23907 a Fa(2)p Fb(\000)p Ff(\014)11234 24179 y Fq(,)433 b(cf.)i(\(4.13\).)579 b(Therefore,)23618 26740 y Fo(d')p 23618 27333 1528 54 v 23733 28550 a(dr)25648 27638 y(>)27161 26740 y(C)p 27161 27333 1026 54 v 27286 28550 a(n)4317 30852 y Fq(for)450 b(some)f(constan)-36 b(t)449 b Fo(C)490 b(>)395 b Fq(0.)626 b(Hence)449 b(the)f(v)-36 b(ector)450 b Fo(V)30422 30370 y Fs(u)30133 31204 y(n)30704 30952 y Fb(0)31507 30852 y Fq(mak)-36 b(es)449 b(an)g(angle)h Fn(\025)396 b Fo(C)59 b(=n)450 b Fq(with)4317 32457 y(the)352 b(horizon)-36 b(tal)353 b Fo(r)36 b Fq(-axis.)552 b(By)353 b(the)f(time)h(rev)-36 b(ersibilit)g(y)-108 b(,)369 b(the)352 b(v)-36 b(ector)353 b Fo(V)37984 31975 y Fs(s)37695 32809 y(n)38266 32557 y Fb(0)38972 32457 y Fq(mak)-36 b(es)354 b(an)e(angle)4317 34062 y Fn(\024)400 b(\000)p Fo(C)59 b(=n)452 b Fq(with)g(the)f (horizon)-36 b(tal)452 b Fo(r)36 b Fq(-axis,)457 b(see)452 b(Fig.)h(7,)j(hence)451 b(sin)222 b Fo(\015)37485 34261 y Fs(n)38056 34009 y Fb(0)38809 34062 y Fo(>)399 b(c=n)452 b Fq(for)g(some)4317 35667 y(constan)-36 b(t)433 b Fo(c)369 b(>)g Fq(0,)434 b(and)f(w)-36 b(e)434 b(obtain)21829 38468 y Fn(j)p Fo(V)23245 37920 y Fs(u)22956 38796 y(n)23527 38544 y Fb(0)23880 38468 y Fn(j)221 b(j)p Fo(V)25887 37920 y Fs(s)25597 38796 y(n)26168 38544 y Fb(0)26522 38468 y Fn(j)369 b Fo(<)g(cn)4317 41269 y Fq(for)434 b(some)g(constan)-36 b(t)433 b Fo(c)369 b(>)g Fq(0.)6268 42874 y(Next,)423 b(the)c(Euclidean)h(norm)f Fn(j)p Fo(V)289 b Fn(j)420 b Fq(de\014ned)e(b)-36 b(y)419 b(\(4.4\))h(is)g(uniformly)g (equiv)-72 b(alen)-36 b(t)420 b(to)4317 44479 y(the)439 b(p-norm)g(\(4.3\))h(for)h(b)36 b(oth)439 b(stable)h(and)f(unstable)h (v)-36 b(ectors)440 b(in)g(our)f(considerations.)4317 46084 y(Indeed,)546 b(cos)222 b Fo(')522 b Fn(\031)h Fq(1)h(and)g Fn(j)p Fo(d')p Fn(j)e(\024)h Fo(C)316 b Fn(j)p Fo(dr)36 b Fn(j)524 b Fq(for)h(some)f(constan)-36 b(t)523 b Fo(C)618 b(>)522 b Fq(0,)547 b(as)524 b(it)g(easily)4317 47689 y(follo)-36 b(ws)436 b(from)e(\(4.2\).)579 b(Therefore,)434 b(w)-36 b(e)434 b(obtain)21300 50490 y Fn(j)p Fo(V)22716 49941 y Fs(u)22427 50818 y(n)22998 50566 y Fb(0)23351 50490 y Fn(j)23720 50689 y Fs(p)24471 50490 y Fn(j)p Fo(V)25887 49941 y Fs(s)25598 50818 y(n)26169 50566 y Fb(0)26522 50490 y Fn(j)26891 50689 y Fs(p)27789 50490 y Fo(<)369 b(cn)4317 53290 y Fq(for)434 b(some)g(constan)-36 b(t)433 b Fo(c)369 b(>)g Fq(0.)578 b(Ob)-36 b(viously)-108 b(,)16563 56091 y Fn(j)p Fo(V)17979 55542 y Fs(u)17690 56419 y(n)18261 56167 y Fb(0)18615 56091 y 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Fq(\),)435 b(hence)14353 66694 y Fn(j)p Fo(V)15770 66146 y Fs(s)15480 67023 y(n)16051 66771 y Fb(0)16405 66694 y Fn(j)16774 66893 y Fs(p)17672 66694 y Fn(\030)19074 65618 y Fh(\002)19628 66694 y Fq(\003)20531 66015 y Fl(\(2\))20531 67070 y Fs(n)21102 66818 y Fb(0)21788 66694 y Fq(\()p Fo(X)104 b Fq(\))23983 65618 y Fh(\003)24537 65916 y Fg(\000)p Fl(1)25795 66694 y Fn(j)p Fo(V)27211 66146 y Fs(s)27701 66694 y Fn(j)28070 66893 y Fs(p)28968 66694 y Fn(\030)30371 65618 y Fh(\002)30924 66694 y Fq(\003)31827 66015 y Fl(\(2\))31827 67070 y Fs(n)32398 66818 y Fb(0)33084 66694 y Fq(\()p Fo(X)g Fq(\))35279 65618 y Fh(\003)35833 65916 y Fg(\000)p Fl(1)37091 66694 y Fo(:)25253 70015 y Fq(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 4317 7306 a Fq(Therefore,)4317 10239 y(\(4.30\))12039 b(\003)20582 9560 y Fl(\(2\))20582 10615 y Fs(n)21153 10363 y Fb(0)21839 10239 y Fq(\()p Fo(X)104 b Fq(\))370 b Fo(>)e(c)p Fq(\003)27247 9560 y Fl(\(1\))27247 10615 y Fs(n)27818 10363 y Fb(0)28505 10239 y 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b(wn)433 b(b)-36 b(y)434 b(the)f(solid)h(line)g(on)f(Fig.)i(3\).)6268 35229 y(In)357 b(that)f(case)h Fo(n)14010 34747 y Fg(0)14678 35229 y Fq(can)f(b)36 b(e)357 b(de\014ned)e(as)i(the)f(turning)g(p)36 b(oin)-36 b(t,)372 b(i.e.)358 b(b)-36 b(y)356 b Fo(x)39496 35428 y Fs(n)40067 35176 y Fb(0)40790 35229 y Fo(<)368 b(x)42909 35428 y Fs(n)43480 35176 y Fb(0)43779 35428 y Fg(\000)p Fl(1)45393 35229 y Fq(and)4317 36834 y Fo(x)5056 37033 y Fs(n)5627 36781 y Fb(0)6502 36834 y Fo(<)520 b(x)8773 37033 y Fs(n)9344 36781 y Fb(0)9643 37033 y Fl(+1)10900 36834 y Fq(.)846 b(Observ)-36 b(e)523 b(that)f(if)i Fo(X)22562 36352 y Fg(0)23394 36834 y Fq(=)d(\()p Fo(r)26055 36352 y Fg(0)26365 36834 y Fo(;)221 b(')27799 36352 y Fg(0)28111 36834 y Fq(\))521 b Fn(2)f Fo(C)31570 36352 y Fg(0)31475 37163 y Fs(n)32101 36834 y Fq(,)545 b(then)522 b(there)g(exists)i(another)4317 38439 y(p)36 b(oin)-36 b(t)620 b Fo(X)790 b Fq(=)686 b(\()p Fo(r)-36 b(;)221 b(')p Fq(\))687 b Fn(2)e Fo(C)17747 37957 y Fg(00)17652 38768 y Fs(n)18933 38439 y Fq(with)620 b Fo(r)722 b Fq(=)685 b Fo(r)25709 37957 y Fg(0)26640 38439 y Fq(and)619 b Fo(')687 b(<)e(')33443 37957 y Fg(0)33754 38439 y Fq(,)666 b(whose)621 b(tra)72 b(jectory)621 b(go)36 b(es)4317 40044 y(through)452 b(the)g(windo)-36 b(w,)459 b(as)453 b(it)g(is)g(clear)h(from)f(Fig.)g(4.)637 b(Since)452 b Fo(')35697 39562 y Fg(0)36410 40044 y Fo(<)401 b(')p Fq(,)458 b(it)453 b(follo)-36 b(ws)455 b(that)4317 41649 y(the)e Fo(x)p Fq(-co)36 b(ordinate)454 b Fo(x)14904 41848 y Fs(m)16245 41649 y Fq(of)g(the)f(p)36 b(oin)-36 b(t)453 b Fn(F)24509 41167 y Fs(m)25396 41649 y Fq(\()p Fo(X)104 b Fq(\))454 b(will)h(b)36 b(e)453 b(alw)-36 b(a)g(ys)455 b(smaller)f(than)f(the)g Fo(x)p Fq(-)4317 43255 y(co)36 b(ordinate)386 b Fo(x)11403 42773 y Fg(0)11403 43583 y Fs(m)12676 43255 y Fq(of)g(the)e(p)36 b(oin)-36 b(t)385 b Fn(F)20735 42773 y Fs(m)21623 43255 y Fq(\()p Fo(X)23312 42773 y Fg(0)23623 43255 y Fq(\),)395 b(for)386 b(all)g(1)369 b Fn(\024)g Fo(m)g Fn(\024)g Fo(n)p Fq(.)563 b(This)385 b(observ)-72 b(ation)386 b(and)4317 44860 y(the)498 b(b)36 b(ound)498 b(\(4.22\))i(that)e(w)-36 b(e)499 b(ha)-36 b(v)g(e)499 b(pro)-36 b(v)g(ed)499 b(for)g Fo(x)29532 45059 y Fs(m)30919 44860 y Fq(implies)g(that)f(the)h(same)g (b)36 b(ound)4317 46465 y(holds)448 b(for)h Fo(x)10476 45983 y Fg(0)10476 46793 y Fs(m)11811 46465 y Fq(and)f(for)g(all)h(1) 394 b Fn(\024)g Fo(m)f Fn(\024)h Fo(n)24382 45983 y Fg(00)24948 46465 y Fq(.)621 b(The)449 b(rest)e(of)i(the)e(pro)36 b(of)449 b(of)g(Prop)36 b(osition)448 b(2)4317 48070 y(for)434 b Fo(X)7487 47588 y Fg(0)8168 48070 y Fn(2)368 b Fo(C)10448 47588 y Fg(0)10353 48398 y Fs(n)11413 48070 y Fq(is)434 b(iden)-36 b(tical)433 b(to)h(that)f(of)h(the)f(case)i Fo(X)473 b Fn(2)369 b Fo(C)32675 47588 y Fg(00)32580 48398 y Fs(n)33241 48070 y Fq(.)6268 49675 y(Prop)36 b(osition)434 b(2)g(is)g(no)-36 b(w)434 b(pro)-36 b(v)g(en.)p 46548 49675 45 878 v 46593 48841 781 45 v 46593 49675 V 47373 49675 45 878 v 4317 54112 a Fp(References)4967 57033 y Fq([1])652 b(Bunimo)-36 b(vic)g(h,)380 b(L.)366 b(A.;)390 b(Sinai,)380 b(Y)-108 b(a.)367 b(G.)g(&)f(Cherno)-36 b(v,)380 b(N.)367 b(I.)g Fj(Markov)403 b(p)-66 b(artitions)402 b(for)6991 58638 y(two-dimensional)517 b(hyp)-66 b(erb)g(olic)518 b(bil)66 b(liar)-66 b(ds,)506 b Fq(Russian)492 b(Math.)g(Surv)-36 b(eys)492 b Fi(45)h Fq(\(1990\))6991 60243 y(105{152.)4967 62955 y([2])652 b(Bunimo)-36 b(vic)g(h)549 b(L.)h(A.;)609 b(Sinai,)579 b(Y)-108 b(a.)550 b(G.)f(&)h(Cherno)-36 b(v,)579 b(N.)550 b(I.,)580 b Fj(Statistic)-66 b(al)569 b(pr)-66 b(op)g(er-)6991 64560 y(ties)523 b(of)h(two-dimensional)f(hyp) -66 b(erb)g(olic)523 b(bil)66 b(liar)-66 b(ds,)513 b Fq(Russian)498 b(Math.)f(Surv)-36 b(eys)498 b Fi(46)6991 66165 y Fq(\(1991\))434 b(47{106.)25253 70015 y(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 4967 7306 a Fq([3])652 b(Cherno)-36 b(v,)493 b(N.,)h Fj(Entr)-66 b(opy,)519 b(Lyapunov)509 b(exp)-66 b(onents)508 b(and)g(me)-66 b(an-fr)g(e)g(e)506 b(p)-66 b(ath)509 b(for)f(bil-)6991 8911 y(liar)-66 b(ds)p Fq(,)433 b(J.)h(Statist.)g(Ph)-36 b(ys.)434 b Fi(88)g Fq(\(1997\),)h(1{29.)4967 11522 y([4])652 b(Cherno)-36 b(v,)691 b(N.,)h Fj(De)-66 b(c)g(ay)653 b(of)h(c)-66 b(orr)g(elations)653 b(in)g(disp)-66 b(ersing)653 b(bil)66 b(liar)-66 b(ds)p Fq(,)691 b(J.)641 b(Statist.)6991 13127 y(Ph)-36 b(ys.)433 b Fi(94)h Fq(\(1999\),)h(513{556.)4967 15738 y([5])652 b(Cherno)-36 b(v,)632 b(N.)593 b(and)e(Hask)-36 b(ell,)634 b(C.,)f Fj(Nonuniformly)609 b(hyp)-66 b(erb)g(olic)609 b(K-systems)i(ar)-66 b(e)6991 17343 y(Bernoul)66 b(li)p Fq(,)434 b(Ergo)36 b(dic)434 b(Theory)g(and)f(Dynamical)j(Systems)d Fi(16)h Fq(\(1996\),)h(19{44.)4967 19955 y([6])652 b(Cherno)-36 b(v,)504 b(N.)490 b(and)g(Mark)-72 b(arian,)505 b(R.,)g Fj(Intr)-66 b(o)g(duction)514 b(to)j(the)f(Er)-66 b(go)g(dic)516 b(The)-66 b(ory)516 b(of)6991 21560 y(Chaotic)464 b(Bil)66 b(liar)-66 b(ds)p Fq(,)434 b(2nd)f(Ed.,)h(IMP)-108 b(A,)433 b(Rio,)i(Brasil,)g(2003.)4967 24171 y([7])652 b(Cherno)-36 b(v,)364 b(N.)347 b(and)f(T)-108 b(roub)36 b(etzk)-36 b(o)g(y)-108 b(,)365 b(S.,)f Fj(Er)-66 b(go)g(dicity)383 b(of)i(bil)66 b(liar)-66 b(ds)385 b(in)f(p)-66 b(olygons)385 b(with)6991 25776 y(p)-66 b(o)g(ckets)p Fq(,)432 b(Nonlinearit)-36 b(y)435 b Fi(11)f Fq(\(1998\),)h(1095{1102.)4967 28387 y([8])652 b(Cherno)-36 b(v,)491 b(N.)479 b(and)g(Zhang,)491 b(H.-K.,)g Fj(Bil)66 b(liar)-66 b(ds)507 b(with)g(p)-66 b(olynomial)506 b(mixing)f(r)-66 b(ates)p Fq(,)6991 29992 y(man)-36 b(uscript,)432 b(arc)-36 b(hiv)g(ed)434 b(in)f(h)-36 b(ttp://www.ma.utexas.edu/mp)p 38731 29992 391 45 v 470 w(arc/)434 b(#04-261.)4967 32603 y([9])652 b(Galla)-36 b(v)g(otti,)927 b(G.)827 b(and)g(Ornstein,)925 b(D.,)h Fj(Bil)66 b(liar)-66 b(ds)827 b(and)g(Bernoul)66 b(li)827 b(schemes)p Fq(,)6991 34208 y(Comm.)434 b(Math.)f(Ph)-36 b(ys.)434 b Fi(38)g Fq(\(1974\),)h(83{101.)4317 36819 y([10])652 b(Mac)-36 b(h)g(ta,)648 b(J.,)h Fj(Power)623 b(law)g(de)-66 b(c)g(ay)622 b(of)g(c)-66 b(orr)g(elations)621 b(in)h(a)h(bil)66 b(liar)-66 b(d)622 b(pr)-66 b(oblem)p Fq(,)647 b(J.)6991 38425 y(Statist.)433 b(Ph)-36 b(ys.)434 b Fi(32)g Fq(\(1983\),)h(555{564.)4317 41036 y([11])652 b(Mark)-72 b(arian,)505 b(R.,)g Fj(Bil)66 b(liar)-66 b(ds)518 b(with)f(p)-66 b(olynomial)516 b(de)-66 b(c)g(ay)516 b(of)h(c)-66 b(orr)g(elations)p Fq(,)503 b(Er.)490 b(Th.)6991 42641 y(Dynam.)434 b(Syst.)g Fi(24)g Fq(\(2004\),)h(177{197.)4317 45252 y([12])652 b(Ornstein,)606 b(D.)574 b(and)e(W)-108 b(eiss,)608 b(B.,)g Fj(On)593 b(the)f(Bernoul)66 b(li)593 b(natur)-66 b(e)592 b(of)h(systems)g(with)6991 46857 y(some)617 b(hyp)-66 b(erb)g(olic)616 b(structur)-66 b(e)p Fq(,)640 b(Ergo)36 b(d.)600 b(Th.)f(Dynam.)h(Syst.)g Fi(18)f Fq(\(1998\),)642 b(441{)6991 48462 y(456.)4317 51073 y([13])652 b(Reh\023)-650 b(a)-36 b(\024)-614 b(cek,)390 b(J.,)g(On)378 b(the)f(ergo)36 b(dicit)-36 b(y)380 b(of)f(disp)36 b(ersing)378 b(billiards.)h(Random)f(Comput.)6991 52678 y(Dynam.)434 b Fi(3)g Fq(\(1995\),)h(35{55.)4317 55289 y([14])652 b(Sinai,)447 b(Y)-108 b(a.)445 b(G.,)i Fj(Dynamic)-66 b(al)474 b(systems)h(with)g(elastic)f(r)-66 b(e\015e)g(ctions.)472 b(Er)-66 b(go)g(dic)474 b(pr)-66 b(op-)6991 56894 y(erties)414 b(of)g(diep)-66 b(ersing)414 b(bil)66 b(liar)-66 b(ds,)389 b Fq(Russian)379 b(Math.)g(Surv)-36 b(eys)379 b Fi(25)h Fq(\(1970\))g(137{189.)4317 59506 y([15])652 b(Y)-108 b(oung)626 b(L.-S.,)675 b Fj(Statistic)-66 b(al)640 b(pr)-66 b(op)g(erties)641 b(of)h(systems)h(with)f(some)h(hyp)-66 b(erb)g(olicity)6991 61111 y(including)463 b(c)-66 b(ertain)463 b(bil)66 b(liar)-66 b(ds)p Fq(,)434 b(Ann.)f(Math.,)g Fi(147)h Fq(\(1998\),)h(585{650.)4317 63722 y([16])652 b(Y)-108 b(oung,)399 b(L.-S.,)g Fj(R)-66 b(e)g(curr)g(enc)g(e)422 b(times)j(and)h(r)-66 b(ates)425 b(of)g(mixing)p Fq(,)398 b(Israel)391 b(J.)g(Math.)g Fi(110)6991 65327 y Fq(\(1999\),)435 b(153{188.)25253 70015 y(23)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0409111548711--