Content-Type: multipart/mixed; boundary="-------------0507280458471" This is a multi-part message in MIME format. ---------------0507280458471 Content-Type: text/plain; name="05-256.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-256.keywords" KAM theory, Divergent Series, Lindstedt Series, Lower-dimensional elliptic tory, Degeneracy, Bryuno condition, Resummations, Renormalization Group, Quantum Field Theory ---------------0507280458471 Content-Type: application/postscript; name="bryuno.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bryuno.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: bryuno10.dvi %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMBX12 CMR10 CMCSC10 CMTI10 CMMI10 CMSY10 %%+ CMMIB10 MSBM10 CMMI7 CMEX10 CMSY7 MSBM7 CMMI5 CMR7 CMR5 CMBX7 CMR8 %%+ CMMI8 EUFM10 CMSY5 EUFM7 EUFM5 CMR9 CMSY8 %%DocumentPaperSizes: a4 %%EndComments 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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 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 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 /.notdef put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM5 %!PS-AdobeFont-1.1: EUFM5 2.1 %%CreationDate: 1992 Nov 20 17:36:22 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM5) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM5 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 118 /v put dup 119 /w put readonly def /FontBBox{0 -251 1451 753}readonly def /UniqueID 5031998 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM7 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 118 /v put readonly def /FontBBox{0 -250 1193 750}readonly def /UniqueID 5031992 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % 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 (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 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 /.notdef put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 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 66 /B put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec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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 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 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 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % 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 (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 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 /.notdef put readonly def /FontBBox{-55 -250 1289 751}readonly def /UniqueID 5000765 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % 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 (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 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 /.notdef put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % 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 (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 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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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 (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 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 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 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 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % 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 (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 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 /.notdef put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueID 5087392 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D004B836D34E88C25F6CE738846C8E2E59A2BCF 4ACF80A26D78872E9343A0537BC3BD7715F32ACD958D5AAED865BFE129278935 063A31C2634DE2F9077E0AAAAEB224466B779096D8E3FF0A12AD5157F6603DED 1A82F3511359143311179080C510740B401C930C96270FD1AB3ECBCFEF5DE53F E846BAAE95828D5790922640EF8AB9D7CEBE7669FEA02B72F86872D3D8754A18 A1629C40A7C00C956F140BC63362478279C36EE353638CD3E249897207A94504 4400668C8E702058EBF7284C9BDF830A3FC79C7EE900CC4C3664F9767A237275 CEE3671644A75F1E696DA906B4C66870DBE87F5B4A176920C078ADBE24F55C09 3D18CDE21B5FBC1C6A8AB18E05EDBEF9D1C1C18B3E6377BA2A688579D4F708F9 A5CF4F56C5E39E2726106E9713E638775E606464CD674E5DC25CE9A696A65806 C8E9D206B421E246F18013ACC6C7B2985BA93B1B7D7745CCB25B09957F50128C B523A55ACA6A7A2A0193A536E590291ED9D577B527CAD0372E05BFCA1829FED1 662D06144A5FFA628C587A4FA05B179F1A7E3B23B47765FDC054271A0DBF9C2B B4F4771F80D1F7AAD9024868C30DAD5CF728DB2A71D86D53B0E674996E8C01F7 EF97B225A28872F8AD4752A466E078C2B020EB832F237CB9B5631EB2D2EDDB00 709D3864CA3A6C3EF18085EAEABC011E9F35C9BE4B5D0B608361F329B5784DAC 5557A602E9E3C204909D84DB988F0BAB914E87CD685C7DA55C5E0B9F0176184F FC39B570873BBF346A0D1DE3942DA05434949A65CE64D8BAB0A091C40F7FF47A 4FC57CB4420221C7B3EB8B891044B5FB0227009F0F6028D3F28545E6396CA41F 228D8839A35712ED56A98AF56EB1FC55B662BB1DBEC166EEC9D5B0B61391E394 831AAC9D49C2DA84E03F2313E1F8FF53F287268589291FF1B79D9D1AAD045464 9B510168771CF93A5D438AF5DB83C512B57566396F1615B6D1AD034F5215070C F5193B20838C5DED83D1EB2473CAC0D1F8339428A5B98E5C88281461E0419C46 FDBAB512FE612C437371822150AE79EC2EB2D1F4501D9F9DE15D6AFB0A020401 91FB4474DF088AD245265B73F5D01108DA4F89694557807EBBBA6894D2A75F29 40B052859CDF3E69052CFA2E1A60DB51145C7C91FEB64F82DEFBDE21431B128C 2C47AABF3DE42AF31A0A977FCE870411B852FEC674AC91836E6C7BC947E9F2F7 B892D6DADD5DC5AF9B0D89FC3393320FD920137E8771E2A9DF6698A124EAE663 AF4DCAFE6C8BBD41B20981AC9D5370312B74D7D920AC44FD3CBB034EBCFC5FB0 0F23871CFB51D10A2DB8B775F3145C9BA49285E53CC93213B5E698DD7F10411E 8E596530248F97E754FDF8304EBE3D58F1AF15AEB50B7B369DA5CE886F342E45 207D846E35C4D55EA2AA2319CB7D09221FBAAE80F576FD9E69C54059BEC406D8 0244E1CFD7156D5B6D114B6388B68575551589B02616808AD262898ACC8B3A93 2FEC34EC6FADD13EB1AFBE6772DA17EE6CABA32AF3572817B998983636139B5D C0110CCF3087093A702C2EA6CB9D7BE0FB016D9F1A2EFAD44DCCA933420FE9B8 EFA0C90008513F3206C9ACE118D036D0DD35E7B5DB673666866C65AE3ED03328 1F24A5FDA73D63E1631A6D3160387F666BA54CEF1F42231A6BEF36C7BA9DFA09 3166E4C002FBC9C94AAB45C795F8535DF417ACDBA497D43E649F510327671D0A BCA6EDDA0B440B4B7B5FB5921D6527DF0AD11E05526CD5952C250F7757094B16 75BC2EA8BC166FB0DA505534474FC52DAFFE66F893A9F8CE82B78E816ED09B7E 9EBF2344CF8C1B234BEF12B08FB73458BD0AC43FB72B4E187AE8F9CBBC22A143 29B1B8D47BFD04CEB88A115AC4B1E03903863E2AFDE294CF0C364C5D6CAE12BD 4D1A7C1C8676B5BD10837F4933EE4964B2355C19D8943F14072D2115C7F89269 C89B5A78118DCC52E229700B5BAB069AFB355D7697B1060FB021BFE69B2483F5 15492E4C5833D6248D8F9FBFB60B21A0041228D5F32B2F45E96B0B21FA227291 B09B479FDC96D54056608F6F95FE714E5FB1F923665FF80B542B7712C6027FE6 A29B25D6E57FC50267F0861BFA442CC93C0889E0AA2144D9A402A9C2E743975D B27E647E2DB3DDD787ED7E05E275FF5DBF648BFBADB4BD166ABFE369057641A2 FAA937D6852D2113A56ABBE374D225555474893E89511A8A0E600698639BBBE7 665B325625E4ACFF9EA516555D24D2B990710B0760678D5246685575DDBC7D47 746D4C5FE2633267ED07C008A106CBEE574FABFD56DF569E5C52B00F81D52461 AF5FC0A417371272B2847FB099D6E67188CCA4BE585CB6E3EEC391782B286327 F7406F9715AC109C863A10DE205CD37C1EB455E526ECCF76D929CC09F3799395 B2A1F3CA2534817DE38A3DD2F4A924FB2CE2C5F44D0EE76589D73A98C763912D E84F7CCE155AD232149860FB076EEE11EA32C9F30B1B0966167BD629535628A6 119462D043F891E63B91C2E6E1278D69C1113A4982AC812A4B46939D09CA9A6D 7C0AF3B93CC78CFB491A75E87004EA06F051FFD58F1DF49B029A476C94899760 C27707A884474FF127151D7DE7DD76B45C22A6794100CE580AE4E8E25E263CDB F59B2ADFE49E8300C49D59DC4709E50DC87174BFEF274DED64D21F6CE301C4FB 582CE3D97A235720C925ACC1C70B5F559D00385D3AAB27BB82CBB137A53FB671 EA1FD8E8CA42F2435B53284F90FD7F93B32DD3668AA6DB873279AF7A4C8534A2 4CBBE8A8F6A16A1922626040B77526AF7D5BFF6FCDAB7379F30A2A6087A1438E 010F1964B21F0F079AD2547C005D1740DE05C3CB65E5 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 81 /producttext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 110 /braceleftBig put dup 112 /radicalbig 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 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % 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 (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 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 /.notdef put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F C0962EAC6E3EBFB8AD256A3C654EAAF9A50C51BC6FA90B61B60401C235AFAB7B 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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 65 /A put dup 77 /M put dup 78 /N put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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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 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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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. 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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. 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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. 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y(iden)n(tit)n(y\),)42 b(so)37 b(that)i(the)f(condition)g Ft(B)j Fy(=)g Ft(0)d Fy(de\014nes)g(an)g Fv(r)r Fy(-dimensional)g (manifold)g Fr(M)3009 1319 y Fq(r)3084 1307 y Fy(\()p Fw(r)l(esonant)118 1414 y(submanifold)p Fy(\),)c(whic)n(h)d(is)g (determined)g(b)n(y)g(the)g(space)f(of)h(the)g(non-resonan)n(t)e (action)i(v)-5 b(ariable)30 b Ft(A)p Fy(;)j(w)n(e)118 1520 y(call)27 b(the)h(latter)g(the)g Fw(non-r)l(esonant)g(action)j (variable)h(sp)l(ac)l(e)p Fy(.)189 1634 y(W)-7 b(e)28 b(are)f(in)n(terested)g(in)h(t)n(w)n(o)e(di\013eren)n(t)i(problems.)118 1740 y(\(A\))34 b(One)e(can)h(\014x)f(the)h(p)r(erturbation)g (parameter)e(\(small)i(enough\))f(and)h(study)f(for)h(whic)n(h)f (rotation)118 1847 y(v)n(ectors)g(some)h(in)n(v)-5 b(arian)n(t)32 b(tori)h(are)g(conserv)n(ed,)g(in)h(the)g(spirit)f(of)h(the)f(KAM)h (theorem)f(for)g(maximal)118 1953 y(tori,)27 b(and)f(as)h(done)f(in)h (most)g(of)g(the)g(pap)r(ers)g(on)f(suc)n(h)h(a)f(sub)5 b(ject,)28 b(as)e(Refs.)37 b([35],)26 b([36],)g([31],)h([32],)f([14],) 118 2059 y([38],)h([9],)g([41],)g([5],)h([28],)e(and)i(man)n(y)f (others.)118 2165 y(\(B\))33 b(Either)e(one)h(can)f(lo)r(ok)h(at)g(a)f (lo)n(w)n(er-dimensional)f(in)n(v)-5 b(arian)n(t)31 b(torus)g(with)i (\014xed)f(rotation)f(v)n(ector,)118 2272 y(and)26 b(study)h(the)g(dep) r(endence)f(of)h(suc)n(h)f(a)g(torus)f(on)i(the)f(p)r(erturbation)g (parameter.)35 b(F)-7 b(or)26 b(instance)g(this)118 2378 y(has)h(b)r(een)h(done,)g(with)g(the)g(tec)n(hniques)f(used)h(here,)f (in)h(Refs.)f([17])g(and)h([23].)189 2492 y(The)h(same)g(t)n(w)n(ofold) f(program)g(has)g(b)r(een)i(follo)n(w)n(ed,)f(under)g(the)g(usual)g (Diophan)n(tine)h(condition,)f(in)118 2598 y(Refs.)35 b([21])e(and)i([22])e(in)i(the)g(study)f(of)g(the)h(quasi-p)r(erio)r (dic)e(solutions)h(and)g(of)h(the)f(sp)r(ectrum)h(for)f(a)118 2705 y(class)27 b(of)g(t)n(w)n(o-lev)n(el)f(systems)h(in)h(a)f(strong)g (quasi-p)r(erio)r(dic)f(external)h(\014eld.)189 2819 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b([45],)e([26])h(and)h([27]\).)118 852 y Ft(Theorem)34 b(2.)42 b Fw(L)l(et)31 b Fs(!)j Fw(b)l(e)d(a)h(ve)l(ctor)g(in)g Fr(R)1474 811 y Fq(r)1543 852 y Fw(satisfying)h(the)e(Diophantine)i(c)l (ondition)g(\(1.4\).)45 b(Supp)l(ose)118 959 y Fs(\014)173 971 y Fk(0)234 959 y Fw(to)24 b(b)l(e)g(such)g(that)g Fv(@)815 976 y Fo(\014)865 959 y Fv(f)906 971 y Fi(0)947 959 y Fy(\()p Fs(\014)1034 971 y Fk(0)1072 959 y Fy(\))f(=)g(0)p Fw(,)i(and)f(assume)g(that)g(the)g(eigenvalues)h(of)g(the)f(matrix)g Fv(@)3006 929 y Fk(2)3001 988 y Fo(\014)3051 959 y Fv(f)3092 971 y Fi(0)3133 959 y Fy(\()p Fs(\014)3220 971 y Fk(0)3258 959 y Fy(\))g Fw(ar)l(e)118 1065 y(al)t(l)33 b(di\013er)l(ent)f(fr)l (om)g(zer)l(o)g(and)h(p)l(airwise)g(distinct)f(\(that)g(is)g Fv(a)2075 1077 y Fq(i)2130 1065 y Fu(6)p Fy(=)26 b(0)31 b Fw(for)i(al)t(l)g Fv(i)26 b Fy(=)h(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(s)31 b Fw(and)i Fv(a)3201 1077 y Fq(i)3255 1065 y Fu(6)p Fy(=)26 b Fv(a)3390 1077 y Fq(j)118 1171 y Fw(for)31 b(al)t(l)g Fy(1)23 b Fu(\024)g Fv(i)g(<)g(j)28 b Fu(\024)c Fv(s)p Fw(\).)39 b(Then)30 b(ther)l(e)g(exists)g Fv(")1641 1183 y Fk(0)1708 1171 y Fw(and)g(a)h(set)e Fu(E)i(\032)23 b Fy(\()p Fu(\000)p Fv(")2370 1183 y Fk(0)2407 1171 y Fv(;)14 b(")2483 1183 y Fk(0)2520 1171 y Fy(\))p Fw(,)30 b(with)h(a)f(density)h(p)l(oint)f(at)118 1278 y(the)c(origin,)i(such)e (that)f(for)i(al)t(l)f Fv(")d Fu(2)g(E)33 b Fw(ther)l(e)26 b(is)g(a)g(lower-dimensional)i(torus)d(for)h(the)g(system)g(describ)l (e)l(d)118 1384 y(by)31 b(the)g(Hamiltonian)g(\(1.1\))h(with)f(r)l (otation)g(ve)l(ctor)g Fs(!)s Fw(,)f(which)i(c)l(an)f(b)l(e)f(p)l(ar)l (ameterise)l(d)i(as)f(\(1.3\),)i(with)118 1490 y Fs( )26 b Fu(2)e Fr(T)345 1449 y Fq(d)414 1490 y Fw(and)30 b(the)g(functions)g Ft(a)g Fw(and)g Ft(b)g Fw(vanishing)h(at)f Fv(")22 b Fy(=)h(0)p Fw(,)30 b(analytic)h(and)f(p)l(erio)l(dic)i(in)e Fs( )s Fw(.)189 1681 y Fy(A)i(densit)n(y)f(p)r(oin)n(t)h(for)f Fu(E)39 b Fy(at)31 b(the)h(origin)e(means)h(that)h(the)g(relativ)n(e)e 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y(corresp)r(onding)i(torus)g(is)h(h)n(yp)r(erb)r(olic.)51 b(In)33 b(the)g(\014rst)f(case)f(the)i(allo)n(w)n(ed)e(v)-5 b(alues)32 b(of)g Fv(")h Fy(form)f(a)g(Can)n(tor)118 2332 y(set)d(in)g([0)p Fv(;)14 b(")488 2344 y Fk(0)525 2332 y Fy(\))29 b(with)h(large)d(relativ)n(e)h(measure,)h(in)g(the)g (second)g(one)f(all)h(v)-5 b(alues)29 b(in)g([0)p Fv(;)14 b(")2897 2344 y Fk(0)2933 2332 y Fy(\))30 b(are)e(allo)n(w)n(ed.)118 2438 y(The)g(ob)n(vious)e(analogue)g(holds)h(for)g Fv(")c(<)g Fy(0.)189 2558 y(F)-7 b(or)25 b(b)r(oth)h(theorems)e(w)n(e)h(shall)h (giv)n(e)e(the)i(pro)r(of)f(in)g(the)h(case)f(in)h(whic)n(h)f(all)g (eigen)n(v)-5 b(alues)24 b(of)i Fv(@)3141 2528 y Fk(2)3136 2587 y Fo(\014)3186 2558 y Fv(f)3227 2570 y Fi(0)3269 2558 y Fy(\()p Fs(\014)3356 2570 y Fk(0)3393 2558 y Fy(\))118 2664 y(are)21 b(strictly)h(p)r(ositiv)n(e)f(and)h Fv(")h(>)f Fy(0,)h(whic)n(h)f(is)g(the)g(di\016cult)h(case.)34 b(All)22 b(the)h(other)e(cases)g(can)g(b)r(e)i(obtained)118 2771 y(with)e(trivial)g(adaptations)e(of)i(the)g(pro)r(of.)35 b(Note)20 b(also)g(that)h(the)h(case)d(of)i(maximal)g(tori)f(can)g(b)r (e)i(obtained)118 2877 y(as)27 b(a)g(b)n(ypro)r(duct)g(b)n(y)h(setting) f Fv(r)f Fy(=)d Fv(d)28 b Fy(in)g(the)g(follo)n(wing.)189 2997 y(Finally)33 b(w)n(e)g(men)n(tion)g(that)g(if)h(do)e(not)h (require)f(that)i(degeneracy)d(of)i(the)g(normal)f(frequencies)h(b)r(e) 118 3103 y(remo)n(v)n(ed)c(to)h(\014rst)h(order)e(then)i(the)g(problem) f(can)g(b)r(ecome)g(m)n(uc)n(h)g(harder.)45 b(F)-7 b(or)30 b(instance)g(if)h(no)f(con-)118 3209 y(dition)e(at)g(all)f(is)g(imp)r (osed)h(on)f(the)h(p)r(erturbation)g(only)f(partial)g(results)g(exist,) g(and)h(only)f(for)g(the)h(case)118 3316 y Fv(s)23 b Fy(=)g(1)k(and)g Fs(!)k Fy(a)c(Diophan)n(tine)g(rotation)g(v)n(ector;)f (cf.)38 b(Refs.)27 b([8])h(and)f([9])g(\(see)h(also)e(Ref.)j([18]\).) 189 3435 y(The)g(pap)r(er)f(is)g(organised)f(as)h(follo)n(ws.)39 b(In)28 b(Section)h(2)f(w)n(e)g(in)n(tro)r(duce)g(the)h(Bryuno)f(v)n (ectors,)f(and)i(w)n(e)118 3542 y(brie\015y)20 b(review)g(some)g(prop)r (erties)g(of)g(theirs,)i(whic)n(h)f(will)g(b)r(e)g(used)f(in)h(the)g (forthcoming)f(analysis.)33 b(Then)118 3648 y(Sections)24 b(2)f(and)h(3)f(are)g(dev)n(oted)h(to)f(the)i(pro)r(of)e(of)h(Theorem)f (1)g(and)h(of)g(Theorem)f(2,)h(resp)r(ectiv)n(ely)-7 b(.)35 b(The)118 3754 y(pro)r(ofs)23 b(hea)n(v)n(enly)f(rely)-7 b(,)24 b(b)r(oth)f(for)g(notations)g(and)g(results,)h(on)f(Ref.)h ([23],)f(and)h(w)n(e)f(con\014ne)g(ourselv)n(es)e(to)118 3861 y(giv)n(e)i(full)h(details)f(only)g(for)g(the)g(parts)g(whic)n(h)g (are)g(really)f(di\013eren)n(t.)36 b(In)23 b(particular)f(the)i(more)f (tec)n(hnical)118 3967 y(asp)r(ects)34 b(are)f(deferred)h(to)g(App)r (endix)h(A1.)56 b(Ho)n(w)n(ev)n(er,)34 b(b)n(y)g(assuming)f(the)h (results)g(of)g(Ref.)h([23],)g(the)118 4073 y(discussion)k(b)r(elo)n(w) f(b)r(ecomes)h(completely)g(self-con)n(tained.)71 b(Finally)39 b(in)h(App)r(endix)g(A2)f(w)n(e)g(brie\015y)118 4179 y(discuss)f(ho)n(w)f(the)h(analysis)f(can)h(b)r(e)g(adapted)g(to)g (deal)g(with)g(more)f(general)g(con)n(v)n(ex)f(Hamiltonian)118 4286 y(systems.)1160 4534 y Fz(2.)50 b(The)38 b(Bryuno)g(condition)118 4724 y Fy(Giv)n(en)25 b Fs(!)h Fu(2)d Fr(R)585 4683 y Fk(2)647 4724 y Fy(set)i Fv(!)h Fu(\021)d Fy(min)p Fu(fj)p Fv(!)1195 4736 y Fk(1)1231 4724 y Fu(j)p Fv(;)14 b Fu(j)p Fv(!)1366 4736 y Fk(2)1403 4724 y Fu(jg)p Fv(=)g Fy(max)o Fu(fj)p Fv(!)1795 4736 y Fk(1)1831 4724 y Fu(j)p Fv(;)g Fu(j)p Fv(!)1966 4736 y Fk(2)2003 4724 y Fu(jg)p Fy(.)36 b(Let)25 b Fu(f)p Fv(q)2352 4736 y Fq(n)2397 4724 y Fu(g)2439 4694 y Fn(1)2439 4745 y Fq(n)p Fk(=0)2593 4724 y Fy(b)r(e)g(the)h (denominators)d(of)1751 4924 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 118 555 a Fy(the)28 b(con)n(v)n(ergen)n(ts)d(of)j Fv(!)s Fy(.)189 662 y(The)g Fw(Bryuno)i(function)d Fu(B)s Fy(\()p Fv(!)s Fy(\))g(is)h(de\014ned)g(as)f(the)h(solution)f(of)g(the) h(functional)g(equation)f([49])952 731 y Fp(\032)1028 798 y Fu(B)s Fy(\()p Fv(!)21 b Fy(+)d(1\))k(=)h Fu(B)s Fy(\()p Fv(!)s Fy(\))p Fv(;)1028 897 y Fu(B)s Fy(\()p Fv(!)s Fy(\))f(=)h Fu(\000)14 b Fy(log)g Fv(!)21 b Fy(+)d Fv(!)e Fu(B)s Fy(\(1)p Fv(=!)s Fy(\))p Fv(;)82 b Fy(if)28 b Fv(!)d Fu(2)f Fy(\(0)p Fv(;)14 b Fy(1\).)3255 848 y(\(2)p Fv(:)p Fy(1\))118 1029 y(De\014ne)1383 1174 y Fv(D)r Fy(\()p Fs(!)s Fy(\))23 b Fu(\021)1721 1070 y Fn(1)1694 1095 y Fp(X)1691 1271 y Fq(n)p Fk(=0)1840 1118 y Fy(log)14 b Fv(q)1998 1130 y Fq(n)p Fk(+1)p 1840 1155 288 4 v 1943 1231 a Fv(q)1980 1243 y Fq(n)2138 1174 y Fv(:)1094 b Fy(\(2)p Fv(:)p Fy(2\))118 1380 y(Then)28 b(it)g(is)f(easy)g(to)h(sho)n (w)e(that)i Fu(B)s Fy(\()p Fv(!)s Fy(\))22 b Fv(<)h Fu(1)28 b Fy(if)g(and)f(only)h(if)g Fv(D)r Fy(\()p Fs(!)s Fy(\))23 b Fv(<)g Fu(1)k Fy([49].)189 1486 y(Giv)n(en)g Fs(!)f Fu(2)e Fr(R)658 1445 y Fq(r)723 1486 y Fy(and)j Fv(n)c Fu(2)g Fr(Z)1095 1498 y Fk(+)1178 1486 y Fy(set)1320 1629 y Fv(\013)1373 1641 y Fq(n)1419 1629 y Fy(\()p Fs(!)s Fy(\))g(=)119 b(inf)1657 1686 y Fk(0)p Fq(<)p Fn(j)p Fo(\027)5 b Fn(j\024)p Fk(2)1910 1670 y Fl(n)1964 1629 y Fu(j)p Fs(!)21 b Fu(\001)d Fs(\027)6 b Fu(j)14 b Fv(;)1032 b Fy(\(2)p Fv(:)p Fy(3\))118 1812 y(and)28 b(de\014ne)f(the)h Fw(gener)l(alise)l(d)k(Bryuno)e(function)d Fy(as)1295 2010 y Fv(B)t Fy(\()p Fs(!)s Fy(\))c(=)1629 1906 y Fn(1)1602 1931 y Fp(X)1600 2107 y Fq(n)p Fk(=0)1771 1953 y Fy(1)p 1749 1991 87 4 v 1749 2067 a(2)1791 2043 y Fq(n)1859 2010 y Fy(log)2082 1953 y(1)p 1990 1991 226 4 v 1990 2067 a Fv(\013)2043 2079 y Fq(n)2088 2067 y Fy(\()p Fs(!)s Fy(\))2226 2010 y Fv(:)1006 b Fy(\(2)p Fv(:)p Fy(4\))118 2297 y Ft(De\014nition)32 b(1.)39 b Fw(We)31 b(shal)t(l)h(c)l(al)t(l)g Ff(B)1262 2309 y Fq(r)1324 2297 y Fy(=)24 b Fu(f)p Fs(!)h Fu(2)f Fr(R)1683 2255 y Fq(r)1743 2297 y Fy(:)f Fv(B)t Fy(\()p Fs(!)s Fy(\))h Fv(<)e Fu(1g)31 b Fw(the)f(set)h(of)g(Bryuno)g (ve)l(ctors)g(in)g Fr(R)3363 2255 y Fq(r)3400 2297 y Fw(.)118 2403 y(F)-6 b(or)30 b(any)g(op)l(en)g(set)g Fy(\012)23 b Fu(\032)f Fr(R)992 2362 y Fq(r)1059 2403 y 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b(taking)f(p)r(ossibly)h(a)f(subsequence,)k(w)n(e)c(can)h(alw)n (a)n(ys)e(supp)r(ose)i Fv(\013)2775 4310 y Fq(n)p Fk(+1)2904 4298 y Fy(\()p Fs(!)s Fy(\))j Fv(<)f(\013)3230 4310 y Fq(n)3275 4298 y Fy(\()p Fs(!)s Fy(\),)118 4405 y(strictly)-7 b(.)118 4582 y Ft(De\014nition)31 b(2.)36 b Fw(Set)29 b Fr(Z)877 4540 y Fq(r)877 4602 y Fn(\003)938 4582 y Fy(=)23 b Fr(Z)1086 4540 y Fq(r)1141 4582 y Fu(n)18 b(f)p Ft(0)p Fu(g)p Fw(,)29 b(and)h(de\014ne)681 4724 y Fv(n)p Fy(\()p Fs(\027)6 b Fy(\))23 b(=)960 4657 y Fp(\010)1008 4724 y Fv(n)g Fu(2)g Fr(Z)1219 4736 y Fk(+)1297 4724 y Fy(:)g(2)1385 4690 y Fq(n)p Fn(\000)p Fk(1)1538 4724 y Fv(<)g Fu(j)p Fs(\027)6 b Fu(j)23 b(\024)f Fy(2)1878 4690 y Fq(n)1923 4657 y Fp(\011)1994 4724 y Fy(=)h(inf)e Fu(f)o Fv(n)i Fu(2)h Fr(N)f Fy(:)g Fu(j)p Fs(\027)6 b Fu(j)23 b(\024)f Fy(2)2776 4690 y Fq(n)2821 4724 y Fu(g)392 b Fy(\(2)p Fv(:)p Fy(6\))1751 4924 y(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 118 555 a Fw(for)31 b(any)f Fs(\027)f Fu(2)23 b Fr(Z)625 514 y Fq(r)625 576 y Fn(\003)663 555 y Fw(.)189 733 y Fy(F)-7 b(or)27 b(all)g Fs(\027)i Fu(2)23 b Fr(Z)668 691 y Fq(r)668 753 y Fn(\003)734 733 y Fy(one)k(has,)g(b)n (y)h(de\014nition,)g Fu(j)p Fs(!)21 b Fu(\001)d Fs(\027)6 b Fu(j)23 b(\025)g Fv(\013)1951 748 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))2091 733 y Fy(\()p Fs(!)s Fy(\))28 b(and)f(2)2449 703 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))p Fn(\000)p Fk(1)2697 733 y Fv(<)22 b Fu(j)p Fs(\027)6 b Fu(j)23 b(\024)g Fy(2)3037 703 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))3176 733 y Fy(.)118 910 y Ft(De\014nition)31 b(3.)36 b Fw(Given)31 b(a)f(non-incr)l(e)l(asing)g(se)l(quenc)l(e)f Fu(f)p Fv(\013)1979 880 y Fn(\003)1979 930 y Fq(n)2024 910 y Fu(g)2066 880 y Fn(1)2066 930 y Fq(n)p Fk(=0)2224 910 y Fw(c)l(onver)l(ging)i(to)e Fy(0)p Fw(,)h(de\014ne)1020 1169 y Fv(B)1087 1135 y Fn(\003)1148 1169 y Fy(=)1265 1065 y Fn(1)1238 1090 y Fp(X)1236 1266 y Fq(n)p Fk(=0)1407 1113 y Fy(1)p 1385 1150 87 4 v 1385 1226 a(2)1427 1202 y Fq(n)1495 1169 y Fy(log)1655 1113 y(1)p 1626 1150 99 4 v 1626 1226 a Fv(\013)1679 1202 y Fn(\003)1679 1247 y Fq(n)1735 1169 y Fv(;)183 b Fy(\000)1993 1135 y Fn(\003)1993 1190 y Fq(p)2055 1169 y Fu(\021)2172 1065 y Fn(1)2145 1090 y Fp(X)2142 1266 y Fq(n)p Fk(=0)2281 1169 y Fv(\013)2334 1135 y Fn(\003)2334 1190 y Fq(n)2379 1169 y Fy(2)2421 1135 y Fq(np)2500 1169 y Fv(;)732 b Fy(\(2)p Fv(:)p Fy(7\))118 1438 y Fw(for)31 b Fv(p)23 b Fu(2)g Fr(Z)454 1450 y Fk(+)509 1438 y Fw(.)38 b(One)30 b(has)g Fy(\000)951 1407 y Fn(\003)951 1458 y Fq(p)p Fk(+1)1097 1438 y Fv(>)22 b Fy(\000)1236 1407 y Fn(\003)1236 1458 y Fq(p)1304 1438 y Fw(for)31 b(al)t(l)g Fv(p)22 b Fu(2)i Fr(Z)1759 1450 y Fk(+)1843 1438 y Fw(such)30 b(that)g Fy(\000)2254 1407 y Fn(\003)2254 1458 y Fq(p)2322 1438 y Fw(is)g(\014nite.)118 1615 y Ft(Lemma)c(2.)35 b Fw(L)l(et)25 b Fy(\012)e Fu(\032)f Fr(R)935 1574 y Fq(r)997 1615 y Fw(b)l(e)k(an)f(op)l(en)h(set,)h(and)f (let)f Fu(f)p Fv(\013)1916 1585 y Fn(\003)1916 1635 y Fq(n)1961 1615 y Fu(g)2003 1585 y Fn(1)2003 1635 y Fq(n)p Fk(=0)2157 1615 y Fw(b)l(e)g(a)h(de)l(cr)l(e)l(asing)g(se)l(quenc)l(e)f (c)l(onver)l(ging)118 1721 y(to)i(zer)l(o)h(such)g(that)f(one)h(has)g Fv(B)1110 1691 y Fn(\003)1171 1721 y Fv(<)23 b Fu(1)k Fw(and)h Fy(\000)1580 1691 y Fn(\003)1580 1742 y Fq(r)1641 1721 y Fy(=)23 b Fv(C)1788 1733 y Fk(0)1853 1721 y Fw(for)28 b(some)g(\014nite)f(c)l(onstant)f Fv(C)2792 1733 y Fk(0)2830 1721 y Fw(.)38 b(Cal)t(l)29 b Fy(\012\()p Fv(C)3220 1733 y Fk(0)3257 1721 y Fy(\))f Fw(the)118 1827 y(subset)h(of)i(Bryuno)g(ve) l(ctors)f(in)g Ff(B)1222 1839 y Fq(r)1259 1827 y Fy(\(\012\))g Fw(such)h(that)f Fv(\013)1826 1839 y Fq(n)1871 1827 y Fy(\()p Fs(!)s Fy(\))24 b Fu(\025)f Fv(\013)2163 1797 y Fn(\003)2163 1848 y Fq(n)2238 1827 y Fw(for)31 b(al)t(l)g Fv(n)23 b Fu(\025)g Fy(1)p Fw(.)39 b(Then)31 b(the)f(L)l(eb)l(esgue)118 1934 y(me)l(asur)l(e)g(of)g(the)g(set)f Fy(\012)864 1904 y Fk(c)898 1934 y Fy(\()p Fv(C)989 1946 y Fk(0)1027 1934 y Fy(\))23 b(=)g(\012)18 b Fu(n)g Fy(\012\()p Fv(C)1459 1946 y Fk(0)1497 1934 y Fy(\))30 b Fw(is)g(b)l(ounde)l(d)g(pr)l(op)l (ortional)i(to)e Fv(C)2590 1946 y Fk(0)2627 1934 y Fw(.)118 2111 y(Pr)l(o)l(of.)39 b Fy(The)27 b(measure)g(of)g(the)h(set)g(\012) 1295 2081 y Fk(c)1329 2111 y Fy(\()p Fv(C)1420 2123 y Fk(0)1457 2111 y Fy(\))g(can)g(b)r(e)g(b)r(ounded)g(b)n(y)553 2370 y(meas\(\012)826 2336 y Fk(c)860 2370 y Fy(\()p Fv(C)951 2382 y Fk(0)989 2370 y Fy(\)\))23 b Fu(\024)g Fy(const)p Fv(:)1420 2266 y Fn(1)1393 2291 y Fp(X)1390 2467 y Fq(n)p Fk(=0)1673 2291 y Fp(X)1529 2474 y Fk(2)1562 2457 y Fl(n)p Fe(\000)p Fj(1)1676 2474 y Fq(<)p Fn(j)p Fo(\027)5 b Fn(j\024)p Fk(2)1896 2457 y Fl(n)1961 2314 y Fv(\013)2014 2284 y Fn(\003)2014 2335 y Fq(n)p 1960 2351 100 4 v 1960 2427 a Fu(j)p Fs(\027)h Fu(j)1076 2667 y(\024)23 b Fy(const)p Fv(:)1420 2563 y Fn(1)1393 2588 y Fp(X)1390 2764 y Fq(n)p Fk(=0)1529 2667 y Fv(\013)1582 2633 y Fn(\003)1582 2688 y Fq(n)1628 2667 y Fy(2)1670 2633 y Fq(nr)1747 2667 y Fy(2)1789 2633 y Fn(\000)p Fk(\()p Fq(n)p Fn(\000)p Fk(1\))2045 2667 y Fu(\024)g Fy(const)p Fv(:)13 b Fy(\000)2411 2633 y Fn(\003)2411 2688 y Fq(r)r Fn(\000)p Fk(1)2556 2667 y Fu(\024)23 b Fy(const)p Fv(:)13 b(C)2929 2679 y Fk(0)2967 2667 y Fv(;)3255 2523 y Fy(\(2)p Fv(:)p Fy(8\))118 2931 y(so)27 b(that)h(the)g(assertion)e(follo)n(ws.)p 3384 2923 42 42 v 189 3109 a(The)i Fw(Diophantine)j(ve)l(ctors)p Fy(,)d(that)g(is)f(the)h(v)n(ectors)e(satisfying)h(the)h(usual)f (Diophan)n(tine)h(condition)1513 3339 y Fu(j)p Fs(!)21 b Fu(\001)e Fs(\027)6 b Fu(j)22 b Fv(>)1878 3283 y(C)1937 3295 y Fk(0)p 1856 3320 142 4 v 1856 3396 a Fu(j)p Fs(\027)6 b Fu(j)1956 3372 y Fq(\034)2007 3339 y Fv(;)1225 b Fy(\(2)p Fv(:)p Fy(9\))118 3585 y(for)27 b(all)g Fs(\027)i Fu(2)23 b Fr(Z)575 3543 y Fq(r)575 3605 y Fn(\003)641 3585 y Fy(and)k(for)g(suitable)h(p)r(ositiv)n(e)f(constan)n(ts)f Fv(C)1972 3597 y Fk(0)2038 3585 y Fy(and)h Fv(\034)9 b Fy(,)28 b(are)f(a)g(particular)f(case)h(of)g(Bryuno)118 3691 y(v)n(ectors,)g(with)i Fv(\013)667 3703 y Fq(n)712 3691 y Fy(\()p Fs(!)s Fy(\))24 b 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Fv(\015)1584 4721 y Fq(n)1630 4709 y Fy(\()p Fs(!)s Fy(\))1767 4652 y Fv(;)p 1970 4586 68 4 v 180 w(B)2037 4600 y Fn(\003)2098 4652 y Fy(=)2215 4549 y Fn(1)2188 4573 y Fp(X)2186 4749 y Fq(n)p Fk(=0)2357 4596 y Fy(1)p 2335 4633 87 4 v 2335 4709 a(2)2377 4685 y Fq(n)2445 4652 y Fy(log)2600 4596 y(1)p 2576 4633 89 4 v 2576 4709 a Fv(\015)2624 4685 y Fn(\003)2619 4730 y Fq(n)2674 4652 y Fv(;)3213 4519 y Fy(\(2)p Fv(:)p Fy(10\))1751 4924 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 118 555 a Fy(one)27 b(has)p 418 489 52 4 v 27 w(\000)470 567 y Fq(r)507 555 y Fy(\()p Fs(!)s Fy(\))c Fu(\025)p 745 489 V 23 w Fy(\000)797 503 y Fn(\003)797 576 y Fq(r)863 555 y Fy(and)p 1024 489 68 4 v 27 w Fv(B)t Fy(\()p Fs(!)s Fy(\))h Fu(\024)p 1329 489 V 22 w Fv(B)1396 503 y Fn(\003)1462 555 y Fy(for)j(all)h(\012)23 b Fu(\032)f Fr(R)1940 514 y Fq(r)2004 555 y Fy(and)28 b(all)f Fs(!)f Fu(2)d Fy(\012\()p Fv(C)2596 567 y Fk(0)2634 555 y Fy(\).)189 668 y(Note)30 b(that)g(if)h Fu(j)p Fs(!)22 b Fu(\001)e Fs(\027)6 b Fu(j)27 b Fv(<)f(C)1055 680 y Fk(0)1093 668 y Fv(\015)1136 680 y Fq(n)1181 668 y Fy(\()p Fs(!)s Fy(\))k(then)h Fu(j)p Fs(\027)6 b Fu(j)26 b Fv(>)h Fy(2)1790 638 y Fq(n)1835 668 y Fy(.)43 b(This)30 b(is)g(easily)f(c)n (hec)n(k)n(ed)g(b)n(y)h(con)n(tradiction:)40 b(if)118 775 y Fu(j)p Fs(\027)6 b Fu(j)23 b(\024)g Fy(2)371 744 y Fq(n)443 775 y Fy(then)28 b Fu(j)p Fs(!)21 b Fu(\001)e Fs(\027)6 b Fu(j)22 b(\025)h Fv(C)1024 787 y Fk(0)1062 775 y Fv(\015)1105 787 y Fq(n)1150 775 y Fy(\()p Fs(!)s Fy(\).)333 1023 y Fz(3.)50 b(Fixing)39 b(the)e(p)s(erturbation)h (parameter:)51 b(pro)s(of)38 b(of)g(Theorem)g(1)118 1206 y Fy(W)-7 b(e)33 b(follo)n(w)f(v)n(ery)g(closely)g(Ref.)i([23])e(\(and) h(Ref.)g([22]\),)h(b)n(y)e(con\014ning)h(ourselv)n(es)e(to)i(sho)n(w)f (where)g(the)118 1313 y(analysis)39 b(di\013ers.)76 b(Also)40 b(notations)f(whic)n(h)i(are)e(not)i(de\014ned)g(b)r(elo)n(w)f(are)f (mean)n(t)h(the)h(same)f(as)g(in)118 1419 y(Ref.)28 b([23].)189 1532 y(The)22 b(m)n(ultiscale)f(decomp)r(osition)g(is)h(p)r(erformed)f (as)h(in)g(Ref.)g([23],)g(b)n(y)g(using)f(the)h(C)2797 1502 y Fn(1)2889 1532 y Fy(non-decreasing)118 1638 y(function)28 b(de\014ned)g(as)1259 1786 y Fv(\037)p Fy(\()p Fv(x)p Fy(\))c(=)1534 1669 y Fp(\032)1610 1739 y Fy(1)p Fv(;)83 b Fy(if)28 b Fu(j)p Fv(x)p Fu(j)23 b Fv(<)g(C)2103 1709 y Fk(2)2097 1760 y(0)2140 1739 y Fv(=)p Fy(4,)1610 1839 y(0)p Fv(;)83 b Fy(if)28 b Fu(j)p Fv(x)p Fu(j)23 b Fv(>)g(C)2103 1809 y Fk(2)2097 1860 y(0)2140 1839 y Fy(,)3255 1786 y(\(3)p Fv(:)p Fy(1\))118 2019 y(with)34 b(the)f(only)f(di\013erence)h (that)g(no)n(w)g Fv(\037)1441 2031 y Fq(n)1519 2019 y Fy(for)f Fv(n)g Fu(\025)f Fy(0)i(is)g(de\014ned)g(as)f Fv(\037)2443 2031 y Fq(n)2488 2019 y Fy(\()p Fv(x)p Fy(\))h(=)f Fv(\037)p Fy(\()p Fv(\014)2864 1989 y Fn(\000)p Fk(2)2953 2019 y Fy(\()p Fv(\015)3033 1989 y Fn(\003)3028 2040 y Fq(n)3074 2019 y Fy(\))3106 1989 y Fn(\000)p Fk(2)3195 2019 y Fy(\()p Fs(!)s Fy(\))p Fv(x)p Fy(\),)118 2125 y(with)c Fv(\014)g Fy(=)22 b(1)p Fv(=)p Fy(4.)36 b(W)-7 b(e)28 b(set)f(also)g Fv( )1147 2137 y Fq(n)1192 2125 y Fy(\()p Fv(x)p Fy(\))d(=)f(1)18 b Fu(\000)g Fv(\037)1610 2137 y Fq(n)1655 2125 y Fy(\()p Fv(x)p Fy(\))29 b(for)e Fv(n)c Fu(\025)f Fy(0.)189 2238 y(W)-7 b(e)24 b(de\014ne)h Fw(clusters)e Fy(and)h Fw(self-ener)l(gy)k(clusters)23 b Fy(as)h(in)g(Ref.)h([23],)f(and)g(w)n(e)f(in)n(tro)r(duce)h(the)g Fw(self-ener)l(gy)118 2344 y(value)h Fu(V)378 2356 y Fq(T)430 2344 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))24 b(and)g(the)g Fw(tr)l(e)l(e)i(value)f Fy(V)-7 b(al\()p Fv(\022)r Fy(\))25 b(according)d(to)i(Ref.)g([23],)g (equations)f(\(5.8\))h(and)g(\(5.11\).)118 2451 y(Of)30 b(course)f(one)h(has)f(d)p Fu(V)897 2463 y Fq(T)950 2451 y Fv(=)p Fy(d)p Fs(!)g Fy(=)e Fv(@)1263 2463 y Fq(x)1305 2451 y Fu(V)1356 2463 y Fq(T)1408 2451 y Fv(@)1452 2463 y Fo(!)1507 2451 y Fv(x)20 b Fy(+)g Fv(@)1703 2463 y Fo(!)1757 2451 y Fu(V)1808 2463 y Fq(T)1860 2451 y Fy(,)31 b(and)f(d)p Fu(V)2175 2463 y Fq(T)2228 2451 y Fv(=)p Fy(d)p Fv(")c Fy(=)h Fv(@)2517 2463 y Fq(")2553 2451 y Fu(V)2604 2463 y Fq(T)2656 2451 y Fy(.)44 b(Note)30 b(that)h(here)e(and)118 2557 y(henceforth,)24 b(with)h(resp)r(ect)e(to) h(Ref.)g([23],)g(w)n(e)f(are)f(making)h(explicit)h(the)g(dep)r(endence) h(of)e(all)h(quan)n(tities)118 2663 y(on)j Fs(!)s Fy(,)h(as)f(w)n(e)g (are)g(in)n(terested)g(also)f(in)i(c)n(hanging)e Fs(!)31 b Fy(for)c(\014xed)g Fv(")p Fy(.)189 2776 y(In)h(terms)f(of)h(the)g (self-energy)e(v)-5 b(alues)27 b(w)n(e)g(can)g(de\014ne)h(the)g Fw(self-ener)l(gy)j(matric)l(es)660 3057 y Fu(M)760 3023 y Fk([)p Fn(\024)p Fq(n)p Fk(])895 3057 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))23 b(=)1332 2953 y Fq(n)1292 2978 y Fp(X)1293 3154 y Fq(p)p Fk(=0)1426 3057 y Fu(M)1526 3023 y Fk([)p Fq(n)p Fk(])1609 3057 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))p Fv(;)712 3352 y Fu(M)812 3318 y Fk([)p Fq(n)p Fk(])895 3352 y Fy(\()p Fv(x)p Fy(;)g Fv(";)g Fs(!)s Fy(\))23 b(=)1292 3210 y Fp( )1397 3248 y Fq(n)1364 3273 y Fp(Y)1358 3449 y Fq(p)p Fk(=0)1490 3352 y Fv(\037)1542 3364 y Fq(p)1581 3352 y Fy(\(\001)1682 3318 y Fk([)p Fq(p)p Fk(])1758 3352 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\)\))2077 3210 y Fp(!)2247 3273 y(X)2157 3460 y Fq(T)9 b Fn(2S)2295 3440 y Fe(R)2291 3482 y Fl(k)q(;n)p Fe(\000)p Fj(1)2470 3352 y Fu(V)2521 3364 y Fq(T)2573 3352 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))p Fv(;)3255 3230 y Fy(\(3)p Fv(:)p Fy(2\))118 3696 y(where)37 b Fu(S)424 3666 y Fn(R)418 3719 y Fq(k)q(;n)557 3696 y Fy(denotes)h(the)f(set)h(of)f Fw(r)l(enormalise)l(d)j(self-ener) l(gy)g(clusters)d Fy(of)h(degree)e Fv(k)k Fy(and)e(scale)e([)p Fv(n)p Fy(])118 3802 y(\(renormalised)22 b(means)h(that)h(they)f(do)g (not)h(con)n(tain)e(an)n(y)h(other)g(self-energy)f(clusters\).)35 b(Suc)n(h)23 b(matrices)118 3908 y(are)31 b(formally)g(Hermitian)h (\(cf.)51 b(Lemma)32 b(2)g(in)g(Ref.)h([23]\),)g(so)e(that)h(they)h (admit)f Fv(d)g Fy(real)f(eigen)n(v)-5 b(alues,)118 4024 y(whic)n(h)28 b(w)n(e)f(denote)g(b)n(y)h Fv(\025)909 3981 y Fk([)p Fq(n)p Fk(])909 4046 y(1)992 4024 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\),)28 b Fv(:)14 b(:)g(:)p Fy(,)27 b Fv(\025)1525 3981 y Fk([)p Fq(n)p Fk(])1525 4049 y Fq(d)1609 4024 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\).)189 4137 y(The)28 b(matrices)e Fu(M)792 4107 y Fk([)p Fq(n)p Fk(])875 4137 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))28 b(in)g(\(3.2\))f(can)g(b)r(e)h(written)g(as) 866 4431 y Fu(M)966 4396 y Fk([)p Fq(n)p Fk(])1048 4431 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))24 b(=)1446 4289 y Fp( )1526 4373 y Fu(M)1626 4330 y Fk([)p Fq(n)p Fk(])1626 4383 y Fq(\013\013)1716 4373 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))83 b Fu(M)2186 4330 y Fk([)p Fq(n)p Fk(])2186 4398 y Fq(\013\014)2274 4373 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))1527 4505 y Fu(M)1627 4461 y Fk([)p Fq(n)p Fk(])1627 4530 y Fq(\014)s(\013)1715 4505 y Fy(\()p Fv(x)p Fy(;)g Fv(";)g Fs(!)s Fy(\))85 b Fu(M)2187 4461 y Fk([)p Fq(n)p Fk(])2187 4530 y Fq(\014)s(\014)2273 4505 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))2575 4289 y Fp(!)2654 4431 y Fv(;)578 b Fy(\(3)p Fv(:)p Fy(3\))118 4724 y(where)27 b(the)h(lab)r(els)g Fv(\013)g Fy(and)f Fv(\014)32 b Fy(run)c(o)n(v)n(er)d Fu(f)p Fy(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(r)r Fu(g)28 b Fy(and)f Fu(f)p Fv(r)21 b Fy(+)d(1)p Fv(;)c(:)g(:)g(:)f(;)h(d)p Fu(g)p Fy(,)27 b(resp)r(ectiv)n(ely)-7 b(.)1751 4924 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 189 555 a Fy(With)31 b(resp)r(ect)e(to)g(Ref.)i ([23],)e(w)n(e)g(sligh)n(tly)g(c)n(hange)g(the)h(de\014nition)g(of)g (the)g Fw(pr)l(op)l(agator)j(divisors)f Fy(for)118 662 y Fv(n)23 b Fu(\025)g Fy(0)k(\(cf.)37 b(De\014nition)29 b(6)e(in)h(Ref.)g([23]\);)f(see)g(also)g(Ref.)h([19].)36 b(W)-7 b(e)28 b(set)901 972 y(\001)970 938 y Fk([)p Fq(n)p Fk(])1053 972 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))23 b(=)1451 805 y Fp(0)1451 955 y(@)1534 916 y Fy(1)p 1534 953 44 4 v 1534 1029 a Fv(d)1643 869 y Fq(d)1601 893 y Fp(X)1603 1070 y Fq(j)s Fk(=1)2035 916 y Fy(1)p 1744 953 623 4 v 1744 1050 a(\()p Fv(x)1823 1026 y Fk(2)1880 1050 y Fu(\000)18 b Fv(\025)p 1963 1063 49 4 v -43 x Fk([)p Fq(n)p Fk(])2011 1073 y Fq(j)2094 1050 y Fy(\()p Fv(";)c Fs(!)s Fy(\)\))2329 1026 y Fk(2)2376 805 y Fp(1)2376 955 y(A)2449 823 y Fn(\000)p Fk(1)p Fq(=)p Fk(2)2619 972 y Fv(;)613 b Fy(\(3)p Fv(:)p Fy(4\))118 1255 y(and)28 b(de\014ne)f(the)h Fw(pr)l(op)l(agators)i Fy(as)214 1517 y Fv(g)257 1483 y Fk([)p Fq(n)p Fk(])339 1517 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))23 b(=)737 1375 y Fp( )803 1414 y Fq(n)p Fn(\000)p Fk(1)813 1439 y Fp(Y)806 1614 y Fq(p)p Fk(=0)942 1517 y Fv(\037)994 1529 y Fq(p)1033 1517 y Fy(\(\001)1134 1483 y Fk([)p Fq(p)p Fk(])1210 1517 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\)\))1529 1375 y Fp(!)1609 1517 y Fv( )1663 1529 y Fq(n)1709 1517 y Fy(\(\001)1810 1483 y Fk([)p Fq(n)p Fk(])1893 1517 y Fy(\()p Fv(x)p Fy(;)g Fv(";)g Fs(!)s Fy(\)\))2226 1425 y Fp(\020)2276 1517 y Fv(x)2323 1483 y Fk(2)2379 1517 y Fu(\000)k(M)2562 1483 y Fk([)p Fn(\024)p Fq(n)p Fk(])2696 1517 y Fy(\()p Fv(x)p Fy(;)c Fv(";)g Fs(!)s Fy(\))2983 1425 y Fp(\021)3033 1442 y Fn(\000)p Fk(1)3136 1517 y Fv(;)96 b Fy(\(3)p Fv(:)p Fy(5\))118 1838 y(where)21 b(the)g Fw(self-ener)l(gies)h Fv(\025)p 947 1851 V 996 1795 a Fk([)p Fq(n)p Fk(])996 1862 y Fq(j)1078 1838 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))21 b(are)f(de\014ned)i(recursiv)n(ely)d (as)h Fv(\025)p 2216 1851 V 2265 1795 a Fk([)p Fq(n)p Fk(])2265 1862 y Fq(j)2347 1838 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))23 b(=)g Fv(\025)2709 1795 y Fk([)p Fq(n)p Fk(])2709 1862 y Fq(j)2792 1838 y Fy(\()2824 1737 y Fp(q)p 2907 1737 420 4 v 101 x Fv(\025)p 2907 1851 49 4 v 2956 1795 a Fk([)p Fq(n)p Fn(\000)p Fk(1])2956 1862 y Fq(j)3124 1838 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\);)g Fv(";)118 1978 y Fs(!)s Fy(\))28 b(for)f Fv(n)c Fu(\025)f Fy(1,)28 b(with)g Fv(\025)p 810 1991 V -43 x Fk([0])858 2001 y Fq(j)933 1978 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))23 b(=)g Fv("a)1330 1990 y Fq(j)s Fn(\000)p Fq(r)1476 1978 y Fy(for)k Fv(j)h Fy(=)23 b Fv(r)e Fy(+)d(1)p Fv(;)c(:)g(:)g(:)f(;)h(d)28 b Fy(and)g Fv(\025)p 2353 1991 V -43 x Fk([0])2401 2001 y Fq(j)2476 1978 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))23 b(=)f(0)28 b(for)f Fv(j)h Fy(=)22 b(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(r)r Fy(.)189 2095 y(Therefore)26 b(if)i(a)g(line)f Fv(`)h Fy(is)f(on)h(scale)e([)p Fv(n)p Fy(],)i(with)g Fv(n)23 b Fu(\025)g Fy(1,)k(suc)n(h)g(that)h Fv(g)2326 2065 y Fk([)p Fq(n)p Fk(])2408 2095 y Fy(\()p Fs(!)22 b Fu(\001)c Fs(\027)2611 2107 y Fq(`)2643 2095 y Fy(;)c Fv(";)g Fs(!)s Fy(\))23 b Fu(6)p Fy(=)f(0)27 b(one)h(has)692 2336 y(min)660 2390 y Fk(1)p Fn(\024)p Fq(j)s Fn(\024)p Fq(d)876 2240 y Fp(\014)876 2290 y(\014)876 2340 y(\014)903 2336 y Fy(\()p Fs(!)22 b Fu(\001)c Fs(\027)1106 2348 y Fq(`)1138 2336 y Fy(\))1170 2302 y Fk(2)1226 2336 y Fu(\000)g Fv(\025)p 1309 2349 V 1358 2293 a Fk([)p Fq(n)p Fk(])1358 2359 y Fq(j)1440 2336 y Fy(\()p Fv(";)c Fs(!)s Fy(\))1643 2240 y Fp(\014)1643 2290 y(\014)1643 2340 y(\014)1694 2336 y Fu(\025)1817 2280 y Fv(C)1882 2249 y Fk(2)1876 2300 y(0)p 1792 2317 154 4 v 1792 2404 a Fy(4)1834 2333 y Fu(p)p 1902 2333 44 4 v 1902 2404 a Fv(d)1955 2336 y(\014)2006 2302 y Fk(2)2044 2336 y Fy(\()p Fv(\015)2124 2302 y Fn(\003)2119 2356 y Fq(n)2164 2336 y Fy(\))2196 2302 y Fk(2)2234 2336 y Fv(;)698 2542 y Fy(min)666 2596 y Fk(1)p Fn(\024)p Fq(j)s Fn(\024)p Fq(d)883 2446 y Fp(\014)883 2496 y(\014)883 2546 y(\014)910 2542 y Fy(\()p Fs(!)22 b Fu(\001)c Fs(\027)1113 2554 y Fq(`)1145 2542 y Fy(\))1177 2507 y Fk(2)1233 2542 y Fu(\000)g Fv(\025)p 1316 2555 49 4 v -43 x Fk([)p Fq(p)p Fk(])1364 2565 y Fq(j)1440 2542 y Fy(\()p Fv(";)c Fs(!)s Fy(\))1643 2446 y Fp(\014)1643 2496 y(\014)1643 2546 y(\014)1694 2542 y Fu(\024)23 b Fv(C)1847 2507 y Fk(2)1841 2562 y(0)1884 2542 y Fv(\014)1935 2507 y Fk(2)1973 2542 y Fy(\()p Fv(\015)2053 2507 y Fn(\003)2048 2562 y Fq(p)2091 2542 y Fy(\))2123 2507 y Fk(2)2160 2542 y Fv(;)180 b Fy(0)23 b Fu(\024)f Fv(p)h Fu(\024)g Fv(n)18 b Fu(\000)g Fy(1)p 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Fy(\))p Fu(j)23 b(\025)f Fv(C)2872 3068 y Fk(2)2866 3119 y(0)2910 3098 y Fv(\014)2961 3068 y Fk(2)2999 3098 y Fy(\()p Fv(\015)3079 3068 y Fn(\003)3074 3119 y Fk(0)3117 3098 y Fy(\))3149 3068 y Fk(2)3186 3098 y Fv(=)p Fy(4)3270 3027 y Fu(p)p 3338 3027 44 4 v 3338 3098 a Fv(d)q Fy(.)189 3204 y(W)-7 b(e)30 b(de\014ne)f(the)h(renormalised)e(expansion)g(for)h Ft(h)p Fy(\()p Fs( )s Fv(;)14 b Fs(\014)1971 3216 y Fk(0)2008 3204 y Fv(;)g Fs(!)s Fv(;)g(")p Fy(\))25 b(=)h(\()p Ft(a)p Fy(\()p Fs( )s Fv(;)14 b Fs(\014)2600 3216 y Fk(0)2638 3204 y Fv(;)g Fs(!)s Fv(;)g(")p Fy(\))p Fv(;)g Ft(b)p Fy(\()p Fs( )s Fv(;)g Fs(\014)3126 3216 y Fk(0)3162 3204 y Fv(;)g Fs(!)s Fv(;)g(")p Fy(\)\),)118 3311 y(b)n(y)27 b(setting)638 3510 y Ft(h)p Fy(\()p Fs( )s Fv(;)14 b Fs(\014)881 3522 y Fk(0)918 3510 y Fv(;)g Fs(!)s Fv(;)g(")p Fy(\))23 b(=)1257 3432 y Fp(X)1236 3611 y Fo(\027)5 b Fn(2)p Fm(Z)1364 3591 y Fl(r)1411 3510 y Fy(e)1448 3476 y Fq(i)p Fo(\027)g Fn(\001)p Fo( )1605 3510 y Ft(h)1658 3522 y Fo(\027)1705 3510 y Fy(\()p Fs(\014)1792 3522 y Fk(0)1829 3510 y Fv(;)14 b Fs(!)s Fv(;)g(")p Fy(\))p Fv(;)180 b Ft(h)2293 3522 y Fo(\027)2340 3510 y Fy(\()p Fs(\014)2427 3522 y Fk(0)2464 3510 y Fv(;)14 b Fs(!)s Fv(;)g(")p Fy(\))23 b Fu(\021)f Ft(h)2835 3522 y Fo(\027)2882 3510 y Fv(;)638 3794 y Ft(h)691 3806 y Fo(\027)761 3794 y Fy(=)h(\()p Fv(h)929 3806 y Fo(\027)5 b Fq(;)p Fk(1)1029 3794 y Fv(;)14 b(:)g(:)g(:)f(;)h(h)1261 3806 y Fo(\027)5 b Fq(;d)1362 3794 y Fy(\))p Fv(;)181 b(h)1646 3806 y Fo(\027)5 b Fq(;\015)1774 3794 y Fy(=)1889 3690 y Fn(1)1862 3715 y Fp(X)1861 3894 y Fq(k)q Fk(=1)2071 3715 y Fp(X)1996 3902 y Fq(\022)r Fn(2)p Fk(\002)2126 3882 y Fe(R)2126 3924 y Fl(k)q(;\027)s(;\015)2280 3794 y Fy(V)-7 b(al)o(\()p Fv(\022)r Fy(\))p Fv(;)3255 3703 y Fy(\(3)p Fv(:)p Fy(7\))118 4107 y(where)27 b(the)h(set)g(of)f(trees)g(\002)989 4077 y Fn(R)989 4130 y Fq(k)q(;)p Fo(\027)5 b Fq(;\015)1178 4107 y Fy(is)28 b(de\014ned)g(as)f(in)g(Ref.)i([23],)d(De\014nition)j (5.)189 4213 y(W)-7 b(e)28 b(shall)f(imp)r(ose)h(the)g(follo)n(wing)e (Diophan)n(tine)i(conditions:)949 4326 y Fp(\014)949 4376 y(\014)949 4426 y(\014)949 4475 y(\014)977 4446 y Fs(!)21 b Fu(\001)d Fs(\027)24 b Fu(\006)1254 4335 y Fp(q)p 1338 4335 335 4 v 1338 4446 a Fv(\025)p 1338 4459 49 4 v -43 x Fk([)p Fq(n)p Fk(])1386 4469 y Fq(i)1469 4446 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))1672 4326 y Fp(\014)1672 4376 y(\014)1672 4426 y(\014)1672 4475 y(\014)1722 4446 y Fu(\025)23 b Fv(C)1869 4458 y Fk(0)1907 4446 y Fv(\015)1955 4412 y Fn(\003)1950 4469 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))2089 4446 y Fv(;)949 4558 y Fp(\014)949 4608 y(\014)949 4658 y(\014)949 4708 y(\014)977 4679 y Fs(!)21 b Fu(\001)d Fs(\027)24 b Fu(\006)1254 4568 y Fp(q)p 1338 4568 335 4 v 1338 4679 a Fv(\025)p 1338 4692 49 4 v -43 x Fk([)p Fq(n)p Fk(])1386 4702 y Fq(i)1469 4679 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))k Fu(\006)1773 4573 y Fp(q)p 1856 4573 335 4 v 106 x Fv(\025)p 1856 4692 49 4 v 1905 4636 a Fk([)p Fq(n)p Fk(])1905 4702 y Fq(j)1987 4679 y Fy(\()p Fv(";)c Fs(!)s Fy(\))2190 4558 y Fp(\014)2190 4608 y(\014)2190 4658 y(\014)2190 4708 y(\014)2241 4679 y Fu(\025)23 b Fv(C)2388 4691 y Fk(0)2425 4679 y Fv(\015)2473 4644 y Fn(\003)2468 4701 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))3255 4563 y Fy(\(3)p Fv(:)p Fy(8\))1751 4924 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 118 555 a Fy(for)23 b(all)h Fv(i;)14 b(j)27 b Fy(=)c(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(d)p Fy(,)25 b(for)e(all)h Fs(\027)k Fu(2)c Fr(Z)1335 514 y Fq(r)1335 576 y Fn(\003)1397 555 y Fy(and)f(for)h(all)f Fv(n)g Fu(\025)g Fy(0.)35 b(W)-7 b(e)24 b(shall)g(refer)f(to)g(conditions)h (\(3.8\))f(as)g(to)118 662 y(the)28 b Fw(\014rst)h(Mel)567 632 y Fn(0)576 662 y Fw(nikov)j(c)l(onditions)d Fy(\(\014rst)e(line\))h (and)g(the)g Fw(se)l(c)l(ond)i(Mel)2286 632 y Fn(0)2295 662 y Fw(nikov)i(c)l(onditions)d Fy(\(second)e(line\).)118 839 y Ft(De\014nition)33 b(4)c Fw(Given)k Fv(C)947 851 y Fk(0)1011 839 y Fu(2)26 b Fr(R)1157 851 y Fk(+)1243 839 y Fw(and)32 b(an)g(op)l(en)g(set)f 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b(of)f(all)g(note)h(that)f(one)g(can)h (ha)n(v)n(e)e Fv(N)1734 2193 y Fq(n)1779 2181 y Fy(\()p Fv(\022)r Fy(\))i(only)f(if)h Fv(M)9 b Fy(\()p Fv(\022)r Fy(\))27 b Fu(\025)f Fy(2)2531 2151 y Fq(n)p Fn(\000)p Fk(1)2660 2181 y Fy(.)43 b(Indeed)30 b(if)f(a)h(line)f Fv(`)g Fy(is)118 2287 y(on)e(scale)g([)p Fv(n)p Fy(])h(then)g(there)f (exists)h Fv(i)22 b Fu(2)i(f)p Fy(1)p Fv(;)14 b(:)g(:)g(:)e(;)i(d)p Fu(g)28 b Fy(suc)n(h)f(that)693 2528 y Fv(C)752 2540 y Fk(0)790 2528 y Fv(\015)838 2494 y Fn(\003)833 2549 y Fq(n)p Fn(\000)p Fk(1)986 2528 y Fv(>)22 b(C)1132 2540 y Fk(0)1170 2528 y Fv(\014)t(\015)1269 2494 y Fn(\003)1264 2549 y Fq(n)p Fn(\000)p Fk(1)1418 2528 y Fu(\025)1505 2408 y Fp(\014)1505 2458 y(\014)1505 2508 y(\014)1505 2558 y(\014)1533 2528 y Fu(j)p Fs(!)f Fu(\001)e Fs(\027)1727 2540 y Fq(`)1758 2528 y Fu(j)g(\000)1883 2417 y Fp(q)p 1966 2417 420 4 v 111 x Fv(\025)p 1966 2541 49 4 v -43 x Fk([)p Fq(n)p Fn(\000)p Fk(1])2014 2552 y Fq(i)2182 2528 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))2385 2408 y Fp(\014)2385 2458 y(\014)2385 2508 y(\014)2385 2558 y(\014)2436 2528 y Fv(>)22 b(C)2582 2540 y Fk(0)2620 2528 y Fv(\015)2668 2494 y Fn(\003)2663 2551 y Fq(n)p Fk(\()p Fo(\027)2768 2560 y Fl(`)2797 2551 y Fk(\))2827 2528 y Fv(;)363 b Fy(\(3)p Fv(:)p Fy(10\))118 2781 y(so)28 b(that)g Fv(n)p Fy(\()p Fs(\027)531 2793 y Fq(`)563 2781 y Fy(\))c Fu(\025)g Fv(n)p Fy(.)38 b(Then)29 b(one)e(m)n(ust)i(ha)n(v)n(e)e Fu(j)p Fs(\027)1659 2793 y Fq(`)1691 2781 y Fu(j)d Fv(>)f Fy(2)1868 2751 y Fq(n)p Fk(\()p Fo(\027)1973 2760 y Fl(`)2002 2751 y Fk(\))p Fn(\000)p Fk(1)2141 2781 y Fu(\025)h Fy(2)2272 2751 y Fq(n)p Fn(\000)p Fk(1)2401 2781 y Fy(,)29 b(hence)f Fv(M)9 b Fy(\()p Fv(\022)r Fy(\))24 b Fu(\025)g(j)p Fs(\027)3063 2793 y Fq(`)3095 2781 y Fu(j)g Fv(>)f Fy(2)3272 2751 y Fq(n)p Fn(\000)p Fk(1)3402 2781 y Fy(,)118 2887 y(thence)28 b Fv(K)6 b Fy(2)500 2857 y Fn(\000)p Fq(n)596 2887 y Fv(M)j Fy(\()p Fv(\022)r Fy(\))24 b Fu(\025)e Fy(1)27 b(if)h Fv(K)h Fu(\025)23 b Fy(2.)189 2994 y(Then)30 b(one)f(pro)n(v)n (es)e(the)j(b)r(ound)g Fv(N)1290 3006 y Fq(n)1335 2994 y Fy(\()p Fv(\022)r Fy(\))d Fu(\024)f Fy(max)o Fu(f)p Fy(2)1796 2964 y Fn(\000)p Fq(n)1892 2994 y Fv(K)6 b(M)j Fy(\()p Fv(\022)r Fy(\))20 b Fu(\000)f Fy(1)p Fv(;)14 b Fy(0)p Fu(g)28 b Fy(for)h(all)g Fv(n)d Fu(\025)g Fy(0,)k(b)n(y)f (induction)118 3100 y(on)j(the)h(n)n(um)n(b)r(er)f(of)g(v)n(ertices)f (of)h(the)h(tree.)50 b(The)33 b(only)e(case)h(whic)n(h)g(requires)f(a)h (di\013eren)n(t)g(discussion)118 3206 y(with)h(resp)r(ect)f(to)h(Ref.)g ([23],)g(App)r(endix)h(A3,)f(is)g(the)g(one)f(in)h(whic)n(h)f(the)h(ro) r(ot)f(line)h Fv(`)f Fy(is)g(on)g(scale)g([)p Fv(n)p Fy(])118 3313 y(and)26 b(exits)h(a)f(cluster)g(on)g(scale)g([)p Fv(n)1194 3325 y Fq(T)1246 3313 y Fy(],)h(whic)n(h)f(has)g(only)g(one)g (en)n(tering)g(line,)h(sa)n(y)e Fv(`)2710 3282 y Fn(0)2733 3313 y Fy(,)i(on)f(scale)g([)p Fv(n)3168 3282 y Fn(0)3191 3313 y Fy(],)h(with)118 3419 y Fv(n)168 3389 y Fn(0)214 3419 y Fu(\025)c Fv(n)p Fy(.)37 b(In)28 b(suc)n(h)f(a)g(case)g(one)g (has)g Fv(n)1298 3431 y Fq(T)1373 3419 y Fv(<)c(n)k Fy(of)h(course,)e (and,)i(for)f(suitable)g Fv(i)h Fy(and)f Fv(j)5 b Fy(,)1099 3535 y Fp(\014)1099 3585 y(\014)1099 3635 y(\014)1099 3685 y(\014)1127 3656 y Fu(j)p Fs(!)21 b Fu(\001)d Fs(\027)6 b Fu(j)18 b(\000)1450 3545 y Fp(q)p 1533 3545 420 4 v 111 x Fv(\025)p 1533 3669 49 4 v 1582 3612 a Fk([)p Fq(n)p Fn(\000)p Fk(1])1582 3679 y Fq(i)1750 3656 y Fy(\()p Fv(";)c Fs(!)s Fy(\))1952 3535 y Fp(\014)1952 3585 y(\014)1952 3635 y(\014)1952 3685 y(\014)2003 3656 y Fu(\024)23 b Fv(C)2150 3668 y Fk(0)2187 3656 y Fv(\014)t(\015)2286 3621 y Fn(\003)2281 3676 y Fq(n)p Fn(\000)p Fk(1)2412 3656 y Fv(;)1099 3768 y Fp(\014)1099 3817 y(\014)1099 3867 y(\014)1099 3917 y(\014)1127 3888 y Fu(j)p Fs(!)e Fu(\001)d Fs(\027)1326 3854 y Fn(0)1349 3888 y Fu(j)h(\000)1474 3783 y Fp(q)p 1557 3783 420 4 v 105 x Fv(\025)p 1557 3901 49 4 v -43 x Fk([)p Fq(n)p Fn(\000)p Fk(1])1605 3911 y Fq(j)1773 3888 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))1976 3768 y Fp(\014)1976 3817 y(\014)1976 3867 y(\014)1976 3917 y(\014)2026 3888 y Fu(\024)23 b Fv(C)2173 3900 y Fk(0)2211 3888 y Fv(\014)t(\015)2310 3854 y Fn(\003)2305 3908 y Fq(n)p Fn(\000)p Fk(1)2435 3888 y Fv(;)3213 3772 y Fy(\(3)p Fv(:)p Fy(11\))118 4129 y(where)k Fs(\027)i Fy(=)22 b Fs(\027)570 4141 y Fq(`)630 4129 y Fy(and)27 b Fs(\027)845 4099 y Fn(0)891 4129 y Fy(=)c Fs(\027)1027 4141 y Fq(`)1055 4125 y Fe(0)1081 4129 y Fy(,)28 b(so)f(that,)h(for)f (suitable)g Fv(\021)s(;)14 b(\021)1998 4099 y Fn(0)2045 4129 y Fu(2)24 b(f\006)p Fy(1)p Fu(g)p Fy(,)673 4250 y Fp(\014)673 4300 y(\014)673 4350 y(\014)673 4400 y(\014)701 4371 y Fs(!)d Fu(\001)e Fy(\()p Fs(\027)24 b Fu(\000)18 b Fs(\027)1065 4336 y Fn(0)1088 4371 y Fy(\))h(+)f Fv(\021)1266 4260 y Fp(q)p 1349 4260 420 4 v 111 x Fv(\025)p 1349 4384 49 4 v -44 x Fk([)p Fq(n)p Fn(\000)p Fk(1])1397 4394 y Fq(i)1565 4371 y Fy(\()p Fv(";)c Fs(!)s Fy(\))k(+)g Fv(\021)1913 4336 y Fn(0)1937 4265 y Fp(q)p 2020 4265 420 4 v 106 x Fv(\025)p 2020 4384 49 4 v -44 x Fk([)p Fq(n)p Fn(\000)p Fk(1])2068 4394 y Fq(j)2236 4371 y Fy(\()p Fv(";)c Fs(!)s Fy(\))2439 4250 y Fp(\014)2439 4300 y(\014)2439 4350 y(\014)2439 4400 y(\014)2490 4371 y Fv(<)22 b(C)2636 4383 y Fk(0)2674 4371 y Fv(\015)2722 4336 y Fn(\003)2717 4391 y Fq(n)p Fn(\000)p Fk(1)2847 4371 y Fv(;)343 b Fy(\(3)p Fv(:)p Fy(12\))118 4612 y(whic)n(h)29 b(b)n(y)f(the)i(Diophan)n(tine)e (conditions)h(\(3.8\))f(implies)h Fv(n)p Fy(\()p Fs(\027)c Fu(\000)19 b Fs(\027)2264 4582 y Fn(0)2287 4612 y Fy(\))25 b Fu(\025)g Fv(n)p Fy(,)k(hence)g(one)f(\014nds)h Fv(M)9 b Fy(\()p Fv(T)j Fy(\))25 b Fu(\025)118 4718 y(j)p Fs(\027)k Fu(\000)22 b Fs(\027)359 4688 y Fn(0)382 4718 y Fu(j)34 b Fv(>)g Fy(2)580 4688 y Fq(n)p Fn(\000)p Fk(1)709 4718 y Fy(,)i(if)f Fv(M)9 b Fy(\()p Fv(T)j Fy(\))33 b(=)1198 4656 y Fp(P)1285 4743 y Fd(v)p Fn(2)p Fq(V)14 b Fk(\()p Fq(T)9 b Fk(\))1544 4718 y Fu(j)p Fs(\027)1615 4730 y Fd(v)1662 4718 y Fu(j)p Fy(.)57 b(Call)34 b Fv(\022)1988 4688 y Fn(0)2046 4718 y Fy(the)g(tree)g(ha)n(ving)f Fv(`)2677 4688 y Fn(0)2734 4718 y Fy(as)h(ro)r(ot)f(line.)57 b(Then)1730 4924 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 118 555 a Fy(b)n(y)36 b(the)h(inductiv)n(e)f(h) n(yp)r(othesis)g Fv(N)1247 567 y Fq(n)1292 555 y Fy(\()p Fv(\022)r Fy(\))j(=)e(1)23 b(+)h Fv(N)1759 567 y Fq(n)1804 555 y Fy(\()p Fv(\022)1877 525 y Fn(0)1901 555 y Fy(\))38 b Fu(\024)f Fy(1)24 b(+)g(max)o Fu(f)p Fy(2)2466 525 y Fn(\000)p Fq(n)2562 555 y Fv(K)6 b(M)j Fy(\()p Fv(\022)2802 525 y Fn(0)2825 555 y Fy(\))25 b Fu(\000)f Fy(1)p Fv(;)14 b Fy(0)p Fu(g)p Fy(.)61 b(If)37 b(the)118 662 y(maxim)n(um)24 b(is)g(0)g(the)h(b)r(ound)f(is)h(trivially)e(satis\014ed,)i(b)r(ecause) f(in)g(suc)n(h)g(a)g(case)f Fv(N)2657 674 y Fq(n)2702 662 y Fy(\()p Fv(\022)r Fy(\))h(=)f(1)g(and)i(w)n(e)e(ha)n(v)n(e)118 768 y(seen)32 b(that)g(in)g(order)f(to)h(ha)n(v)n(e)f(a)g(line)i(on)e (scale)h([)p Fv(n)p Fy(])g(one)f(needs)h Fv(M)9 b Fy(\()p Fv(\022)r Fy(\))31 b Fv(>)f Fy(2)2548 738 y Fq(n)p Fn(\000)p Fk(1)2678 768 y Fy(.)50 b(Otherwise)31 b(one)h(has)118 874 y Fv(N)185 886 y Fq(n)230 874 y Fy(\()p Fv(\022)r Fy(\))24 b Fu(\024)f Fy(1)16 b(+)h(2)629 844 y Fn(\000)p Fq(n)725 874 y Fv(K)6 b(M)j Fy(\()p Fv(\022)965 844 y Fn(0)988 874 y Fy(\))17 b Fu(\000)g Fy(1)23 b Fu(\024)f Fy(2)1313 844 y Fn(\000)p Fq(n)1410 874 y Fv(K)6 b(M)j Fy(\()p Fv(\022)r Fy(\))17 b Fu(\000)f Fy(1)h(+)f(\(1)h Fu(\000)g Fy(2)2135 844 y Fn(\000)p Fq(n)2231 874 y Fv(K)6 b(M)j Fy(\()p Fv(T)j Fy(\)\),)26 b(where)h(2)2886 844 y Fn(\000)p Fq(n)2982 874 y Fv(K)6 b(M)j Fy(\()p Fv(T)j Fy(\))22 b Fu(\025)h Fy(1)118 981 y(b)n(y)k(the)h(inequalit)n(y)g Fu(j)p Fs(\027)c Fu(\000)18 b Fs(\027)994 951 y Fn(0)1017 981 y Fu(j)23 b Fv(>)f Fy(2)1192 951 y Fq(n)p Fn(\000)p Fk(1)1322 981 y Fy(,)28 b(pro)n(vided)e(that)i(one)f(tak)n(es)g Fv(K)i Fu(\025)22 b Fy(2.)p 3384 973 42 42 v 118 1158 a Ft(Lemma)36 b(4.)47 b Fw(Cal)t(l)35 b Fv(N)832 1170 y Fq(n)877 1158 y Fy(\()p Fv(T)12 b Fy(\))33 b Fw(the)g(set)g(of)h (lines)g(in)f Fy(\003\()p Fv(T)12 b Fy(\))32 b Fw(which)j(ar)l(e)f(on)f (sc)l(ale)h Fy([)p Fv(n)p Fy(])p Fw(,)g(for)g Fv(n)c Fu(\024)f Fv(n)3151 1170 y Fq(T)3203 1158 y Fw(.)49 b(One)118 1264 y(has)737 1370 y Fv(M)9 b Fy(\()p Fv(T)j Fy(\))23 b(=)1123 1292 y Fp(X)1062 1474 y Fd(v)p Fn(2)p Fq(V)14 b Fk(\()p Fq(T)9 b Fk(\))1317 1370 y Fu(j)p Fs(\027)1388 1382 y Fd(v)1435 1370 y Fu(j)23 b Fv(>)f Fy(2)1610 1336 y Fq(n)1651 1344 y Fl(T)1697 1336 y Fn(\000)p Fk(1)1786 1370 y Fv(;)184 b(N)2060 1382 y Fq(n)2105 1370 y Fy(\()p Fv(T)12 b Fy(\))22 b Fu(\024)h Fv(K)c Fy(2)2472 1336 y Fn(\000)p Fq(n)2569 1370 y Fv(M)9 b Fy(\()p Fv(T)j Fy(\))p Fv(;)406 b Fy(\(3)p Fv(:)p Fy(13\))118 1609 y Fw(with)30 b(the)g(same)h(c)l(onstant)e Fv(K)35 b Fw(as)30 b(in)g(\(3.9\).)118 1786 y(Pr)l(o)l(of.)37 b Fy(The)23 b(\014rst)f(b)r(ound)h(in)g(\(3.13\))f(can)h(b)r(e)g(pro)n(v)n(ed)e(b)n (y)i Fw(r)l(e)l(ductio)i(ad)h(absur)l(dum)d Fy(as)f(in)h(Ref.)g([23],)g (while)118 1892 y(the)30 b(pro)r(of)f(of)h(the)g(second)g(one)f(is)h (based)f(on)g(the)i(same)e(argumen)n(t)f(used)i(for)f(pro)n(ving)g (Lemma)g(3)h(\(cf.)118 1998 y(Ref.)e([23],)f(App)r(endix)i(A3,)e(for)g (further)h(details\).)p 3384 1990 V 189 2175 a(Another)g(di\013erence)h (with)g(resp)r(ect)f(to)h(Ref.)g([23])e(relies)h(in)h(discussing)f(the) h(c)n(hange)e(of)i(scale)e(of)i(the)118 2282 y(lines)e(when)g(p)r (erforming)f(the)h(cancellations)e(inside)i(the)g(families)g Fu(F)2316 2294 y Fq(T)2368 2282 y Fy(,)g(when)g(lo)r(oking)f(for)g(b)r (ounds)h(on)118 2388 y(the)h(en)n(tries)f(of)h(the)g(matrices)e Fu(M)1197 2358 y Fk([)p Fq(n)p Fk(])1280 2388 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\).)118 2565 y Ft(Lemma)34 b(5.)41 b Fw(Assume)31 b(that)g(the)h(pr)l(op)l(agators)h Fv(g)1688 2535 y Fk([)p Fq(p)p Fk(])1764 2565 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))31 b Fw(c)l(an)h(b)l(e)f (uniformly)i(b)l(ounde)l(d)e(for)i(al)t(l)f Fy(0)26 b Fu(\024)118 2672 y Fv(p)d Fu(\024)g Fv(n)18 b Fu(\000)g Fy(1)29 b Fw(as)1196 2696 y Fp(\014)1196 2746 y(\014)1196 2796 y(\014)1224 2791 y Fv(g)1267 2757 y Fk([)p Fq(p)p Fk(])1342 2791 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))1629 2696 y Fp(\014)1629 2746 y(\014)1629 2796 y(\014)1680 2791 y Fu(\024)23 b Fv(K)1839 2803 y Fk(1)1875 2791 y Fv(C)1940 2756 y Fn(\000)p Fk(2)1934 2814 y(0)2030 2791 y Fy(\()p Fv(\015)2110 2757 y Fn(\003)2105 2812 y Fq(p)2148 2791 y Fy(\))2180 2757 y Fn(\000)p Fq(K)2288 2765 y Fj(2)2324 2791 y Fv(;)866 b Fy(\(3)p Fv(:)p Fy(14\))118 2953 y Fw(for)33 b(some)f Fv(p)p Fw(-indep)l(endent)g(c)l(onstants)f Fv(K)1439 2965 y Fk(1)1508 2953 y Fw(and)i Fv(K)1743 2965 y Fk(2)1779 2953 y Fw(.)46 b(Assume)31 b(also)i(that)f Fv(")f Fw(is)h(smal)t(l)h(enough.)46 b(Then,)118 3059 y(with)30 b(the)g(notations)g(\(3.3\),)i(one)e(has)975 3151 y Fp(\015)975 3201 y(\015)975 3250 y(\015)1021 3246 y Fu(M)1121 3212 y Fk([)p Fq(n)p Fk(])1121 3267 y Fq(\013\013)1211 3246 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))1498 3151 y Fp(\015)1498 3201 y(\015)1498 3250 y(\015)1568 3246 y Fu(\024)22 b Fv(B)t Fy(e)1759 3212 y Fn(\000)p Fq(\024)1850 3220 y Fj(1)1883 3212 y Fk(2)1916 3187 y Fl(n)1974 3246 y Fy(min)q Fu(f)p Fv(")2194 3212 y Fk(2)2230 3246 y Fv(;)14 b("x)2353 3212 y Fk(2)2390 3246 y Fu(g)p Fv(;)978 3334 y Fp(\015)978 3383 y(\015)978 3433 y(\015)1024 3429 y Fu(M)1124 3386 y Fk([)p Fq(n)p Fk(])1124 3454 y Fq(\013\014)1211 3429 y Fy(\()p Fv(x)p Fy(;)g Fv(";)g Fs(!)s Fy(\))1498 3334 y Fp(\015)1498 3383 y(\015)1498 3433 y(\015)1568 3429 y Fu(\024)22 b Fv(B)t Fy(e)1759 3395 y Fn(\000)p Fq(\024)1850 3403 y Fj(1)1883 3395 y Fk(2)1916 3370 y Fl(n)1974 3429 y Fy(min)q Fu(f)p Fv(")2194 3395 y Fk(2)2230 3429 y Fv(;)14 b(")2306 3395 y Fk(3)p Fq(=)p Fk(2)2410 3429 y Fu(j)p Fv(x)p Fu(jg)p Fv(;)980 3516 y Fp(\015)980 3566 y(\015)980 3616 y(\015)1026 3612 y Fu(M)1126 3569 y Fk([)p Fq(n)p Fk(])1126 3637 y Fq(\014)s(\014)1211 3612 y Fy(\()p Fv(x)p Fy(;)g Fv(";)g Fs(!)s Fy(\))1498 3516 y Fp(\015)1498 3566 y(\015)1498 3616 y(\015)1568 3612 y Fu(\024)22 b Fv(B)t Fy(e)1759 3577 y Fn(\000)p Fq(\024)1850 3585 y Fj(1)1883 3577 y Fk(2)1916 3552 y Fl(n)1960 3612 y Fv(")1999 3577 y Fk(2)2036 3612 y Fv(;)3213 3429 y Fy(\(3)p Fv(:)p Fy(15\))118 3794 y Fw(for)31 b(suitable)f Fv(n)p Fw(-indep)l(endent)g(c)l(onstants)f Fv(B)34 b Fw(and)c Fv(\024)1763 3806 y Fk(1)1800 3794 y Fw(.)118 3971 y(Pr)l(o)l(of.)48 b Fy(Again)30 b(w)n(e)g(only)h(discuss)f(the)h (di\013erences)f(with)i(resp)r(ect)e(to)h(Ref.)g([23].)45 b(First)31 b(w)n(e)f(sho)n(w)g(that)118 4078 y(no)h(cancellation)g(is)g (needed)h(for)f(self-energy)f(clusters)h Fv(T)43 b Fy(with)32 b Fv(C)2261 4090 y Fk(0)2299 4078 y Fv(\015)2347 4047 y Fn(\003)2342 4104 y Fq(n)p Fk(\()p Fq(M)6 b Fk(\()p Fq(T)j Fk(\)\))2638 4078 y Fu(\024)29 b Fy(4)p Fu(j)p Fs(!)24 b Fu(\001)d Fs(\027)6 b Fu(j)p Fy(,)32 b(if)g Fs(\027)37 b Fy(is)32 b(the)118 4184 y(momen)n(tum)40 b(\015o)n(wing)e(through)h(the)h(en)n(tering)e(line)i(of)f Fv(T)12 b Fy(.)71 b(Note)40 b(that)f(w)n(e)g(can)g(extract)g(from)g (the)118 4290 y(self-energy)d(v)-5 b(alue)37 b(a)g(factor)f(e)1126 4260 y Fn(\000)p Fq(\024)1217 4268 y Fj(0)1250 4260 y Fq(M)6 b Fk(\()p Fq(T)j Fk(\))p Fq(=)p Fk(4)1530 4290 y Fu(\024)38 b Fy(e)1670 4260 y Fn(\000)p Fq(\024)1761 4268 y Fj(0)1793 4260 y Fk(2)1826 4235 y Fl(n)p Fj(\()p Fl(M)5 b Fj(\()p Fl(T)j Fj(\)\))2057 4260 y Fq(=)p Fk(8)2128 4290 y Fy(.)66 b(If)38 b(w)n(e)f(set)g(2)2623 4260 y Fn(\000)p Fq(n)2733 4290 y Fy(log)14 b(1)p Fv(=\013)2991 4260 y Fn(\003)2991 4311 y Fq(n)3075 4290 y Fy(=)39 b Fv(a)3223 4302 y Fq(n)3268 4290 y Fy(,)h(w)n(e)118 4406 y(ha)n(v)n(e)c(lim)434 4418 y Fq(n)p Fn(!1)625 4406 y Fv(a)669 4418 y Fq(n)753 4406 y Fy(=)h(0)g(\(b)r(ecause)f Fv(B)1349 4375 y Fn(\003)1426 4406 y Fv(<)h Fu(1)p Fy(\),)j(hence)c (for)g Fs(!)28 b Fu(\001)c Fs(\027)42 b Fy(small)37 b(enough)f(e)2865 4375 y Fn(\000)p Fq(\024)2956 4383 y Fj(0)2988 4375 y Fk(2)3021 4350 y Fl(n)p Fj(\()p Fl(M)5 b Fj(\()p Fl(T)i Fj(\)\))3251 4375 y Fq(=)p Fk(8)3361 4406 y Fu(\024)118 4512 y Fy(\()p Fv(C)209 4524 y Fk(0)247 4512 y Fv(\015)295 4482 y Fn(\003)290 4539 y Fq(n)p Fk(\()p Fq(M)f Fk(\()p Fq(T)j Fk(\)\))557 4512 y Fy(\))589 4481 y Fq(\024)628 4489 y Fj(0)660 4481 y Fq(=)p Fk(8)p Fq(a)763 4492 y Fl(n)p Fj(\()p Fl(M)c Fj(\()p Fl(T)i Fj(\)\))1021 4512 y Fu(\024)22 b Fv(C)1173 4482 y Fk(2)1167 4532 y(0)1211 4512 y Fy(\()p Fv(\015)1291 4482 y Fn(\003)1286 4539 y Fq(n)p Fk(\()p Fq(M)6 b Fk(\()p Fq(T)j Fk(\)\))1553 4512 y Fy(\))1585 4482 y Fk(2)1645 4512 y Fu(\024)23 b Fy(16)p Fu(j)p Fs(!)d Fu(\001)f Fs(\027)6 b Fu(j)2039 4482 y Fk(2)2076 4512 y Fy(.)189 4618 y(Hence)25 b(w)n(e)g(need)g(the)h (cancellations)d(only)i(for)g(self-energy)e(clusters)i Fv(T)36 b Fy(with)25 b Fv(C)2713 4630 y Fk(0)2751 4618 y Fv(\015)2799 4588 y Fn(\003)2794 4645 y Fq(n)p Fk(\()p Fq(M)6 b Fk(\()p Fq(T)j Fk(\)\))3084 4618 y Fv(>)22 b Fy(4)p Fu(j)p Fs(!)16 b Fu(\001)d Fs(\027)6 b Fu(j)118 4724 y Fy(if)27 b Fs(\027)33 b Fy(is)26 b(the)h(momen)n(tum)g(\015o)n (wing)f(through)g(the)h(en)n(tering)e(line)i(of)g Fv(T)12 b Fy(.)35 b(In)27 b(suc)n(h)f(a)h(case)e(one)i(can)f(reason)1730 4924 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 118 555 a Fy(as)24 b(follo)n(ws.)35 b(F)-7 b(or)23 b(an)n(y)h(line)g Fv(`)f Fu(2)g Fy(\003\()p Fv(T)12 b Fy(\))24 b(and)g(for)g(an)n(y)g Fv(n)e Fu(\024)h Fv(n)1962 567 y Fq(`)2018 555 y Fy(one)h(has,)h(b)n(y)f(the)g(Diophan)n (tine)h(conditions)118 692 y(\(3.8\),)j Fu(jj)p Fs(!)21 b Fu(\001)d Fs(\027)562 662 y Fk(0)556 716 y Fq(`)599 692 y Fu(j)h(\000)724 591 y Fp(q)p 807 591 335 4 v 101 x Fv(\025)p 807 705 49 4 v -43 x Fk([)p Fq(n)p Fk(])855 716 y Fq(j)938 692 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))p Fu(j)23 b(\025)f Fv(C)1333 704 y Fk(0)1371 692 y Fv(\015)1419 662 y Fn(\003)1414 722 y Fq(n)p Fk(\()p Fo(\027)1524 702 y Fj(0)1519 744 y Fl(`)1556 722 y Fk(\))1586 692 y Fy(,)28 b(if)g Fs(\027)1767 662 y Fk(0)1761 716 y Fq(`)1832 692 y Fy(is)f(de\014ned)h(as)1243 940 y Fs(\027)1297 905 y Fk(0)1291 960 y Fq(`)1357 940 y Fy(=)1509 861 y Fp(X)1455 1030 y Fc(w)p Fe(2)p Fl(V)12 b Fj(\()p Fl(T)7 b Fj(\))1498 1080 y Fc(w)p Fe(\024)p Fc(v)1706 940 y Fu(j)p Fs(\027)1777 952 y Fd(v)1824 940 y Fu(j)p Fv(;)180 b(`)23 b Fu(\021)f Fv(`)2230 952 y Fd(v)2277 940 y Fv(:)913 b Fy(\(3)p Fv(:)p Fy(16\))118 1273 y(On)19 b(the)h(other)f(hand)h(one)f (has)g Fu(j)p Fs(\027)1153 1243 y Fk(0)1147 1297 y Fq(`)1190 1273 y Fu(j)k(\024)g Fv(M)9 b Fy(\()p Fv(T)j Fy(\),)20 b(so)f(that)h Fv(C)1907 1285 y Fk(0)1945 1273 y Fv(\015)1993 1243 y Fn(\003)1988 1303 y Fq(n)p Fk(\()p Fo(\027)2098 1283 y Fj(0)2093 1326 y Fl(`)2130 1303 y Fk(\))2183 1273 y Fu(\025)j Fv(C)2330 1285 y Fk(0)2367 1273 y Fv(\015)2415 1243 y Fn(\003)2410 1300 y Fq(n)p Fk(\()p Fq(M)6 b Fk(\()p Fq(T)j Fk(\)\))2700 1273 y Fv(>)23 b Fy(4)p Fu(j)p Fs(!)5 b Fu(\001)r Fs(\027)h Fu(j)p Fy(.)34 b(Therefore)118 1391 y(w)n(e)27 b(can)h(b)r(ound)226 1620 y(2)282 1499 y Fp(\014)282 1549 y(\014)282 1599 y(\014)282 1649 y(\014)309 1620 y Fu(j)p Fs(!)22 b Fu(\001)c Fs(\027)509 1586 y Fk(0)503 1640 y Fq(`)546 1620 y Fu(j)g(\000)670 1515 y Fp(q)p 753 1515 335 4 v 105 x Fv(\025)p 753 1633 49 4 v 802 1577 a Fk([)p Fq(n)p Fk(])802 1643 y Fq(j)885 1620 y Fy(\()p Fv(";)c Fs(!)s Fy(\))1088 1499 y Fp(\014)1088 1549 y(\014)1088 1599 y(\014)1088 1649 y(\014)1138 1620 y Fu(\025)1226 1499 y Fp(\014)1226 1549 y(\014)1226 1599 y(\014)1226 1649 y(\014)1254 1620 y Fu(j)p Fs(!)21 b Fu(\001)d Fs(\027)1447 1632 y Fq(`)1479 1620 y Fu(j)h(\000)1604 1515 y Fp(q)p 1687 1515 335 4 v 105 x Fv(\025)p 1687 1633 49 4 v -43 x Fk([)p Fq(n)p Fk(])1735 1643 y Fq(j)1818 1620 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))2021 1499 y Fp(\014)2021 1549 y(\014)2021 1599 y(\014)2021 1649 y(\014)2071 1620 y Fu(\025)2169 1564 y Fy(1)p 2169 1601 42 4 v 2169 1677 a(2)2234 1499 y Fp(\014)2234 1549 y(\014)2234 1599 y(\014)2234 1649 y(\014)2262 1620 y Fu(j)p Fs(!)21 b Fu(\001)e Fs(\027)2462 1586 y Fk(0)2456 1640 y Fq(`)2499 1620 y Fu(j)f(\000)2623 1515 y Fp(q)p 2706 1515 335 4 v 105 x Fv(\025)p 2706 1633 49 4 v 2755 1577 a Fk([)p Fq(n)p Fk(])2755 1643 y Fq(j)2837 1620 y Fy(\()p Fv(";)c Fs(!)s Fy(\))3040 1499 y Fp(\014)3040 1549 y(\014)3040 1599 y(\014)3040 1649 y(\014)3082 1620 y Fv(:)108 b Fy(\(3)p Fv(:)p Fy(17\))118 1869 y(This)26 b(implies)g(the)g(follo)n(wing.)35 b(When)27 b(considering)d(a)i(family)g Fu(F)2162 1881 y Fq(T)2214 1869 y Fy(,)g(a)f(line)h Fv(`)d Fu(2)g Fy(\003\()p Fv(T)12 b Fy(\))26 b(with)g(momen)n(tum)118 1975 y Fs(\027)166 1987 y Fq(`)226 1975 y Fy(can)i(b)r(e)g(on)g(a)g(scale)f([)p Fv(n)950 1987 y Fq(`)982 1975 y Fy(])h(suc)n(h)g(that)g Fv(g)1444 1945 y Fk([)p Fq(n)1504 1954 y Fl(`)1533 1945 y Fk(])1556 1975 y Fy(\()p Fs(!)21 b Fu(\001)e Fs(\027)1765 1945 y Fk(0)1759 1999 y Fq(`)1802 1975 y Fy(;)14 b Fv(";)g Fs(!)s Fy(\))23 b(=)h(0.)38 b(But)28 b(in)g(suc)n(h)g(a)g(case)f(it)i (is)e(obtained,)118 2082 y(b)n(y)h(shifting)h(the)g(external)f(lines)h (of)f Fv(T)12 b Fy(,)28 b(from)h(a)f(line)g(with)i(non-v)-5 b(anishing)27 b(propagator,)f(that)j(is)g(from)118 2188 y(a)c(line)g(for)f(whic)n(h)h(\(3.6\))g(holds.)35 b(Then,)26 b(ev)n(en)e(if)i(the)f(b)r(ounds)g(\(3.6\))g(can)f(fail)h(to)g(hold,)g (one)g(still)g(obtains)118 2294 y(b)r(ounds)k(of)f(the)g(same)g(form)g (with)h(the)f(only)g(di\013erence)h(that)f Fv(\014)2172 2264 y Fk(2)2238 2294 y Fy(is)g(replaced)g(with)g Fv(\014)2891 2264 y Fk(2)2929 2294 y Fv(=)p Fy(4)f(in)i(the)g(\014rst)118 2401 y(line)i(and)g(with)g(4)p Fv(\014)728 2370 y Fk(2)796 2401 y Fy(in)g(the)g(second)f(line.)47 b(In)30 b(particular)g(for)g Fv(\014)j Fy(=)28 b(1)p Fv(=)p Fy(4)h(the)i(inequalities)f(\(3.10\))g (and)118 2507 y(\(3.12\))d(are)g(still)g(satis\014ed)h(for)f(all)g (self-energy)f(clusters)h(in)h(the)g(family)g Fu(F)2486 2519 y Fq(T)2538 2507 y Fy(.)189 2613 y(This)35 b(sho)n(ws)f(that)h (one)g(can)g(reason)e(in)i(Ref.)h([23])e(to)h(deduce)g(the)h(b)r(ounds) f(\(3.15\),)h(whic)n(h)f(are)f(of)118 2719 y(algebraic)f(nature,)j(and) e(are)f(due)i(to)f(symmetry)g(prop)r(erties)g(of)g(the)h(self-energy)e (matrices,)j(that)e(is)118 2826 y Fu(M)218 2796 y Fk([)p Fn(\024)p Fq(n)p Fk(])p Fq(T)401 2826 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))23 b(=)g Fu(M)899 2796 y Fk([)p Fn(\024)p Fq(n)p Fk(])1033 2826 y Fy(\()p Fu(\000)p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))27 b(and)h Fu(M)1674 2796 y Fk([)p Fn(\024)p Fq(n)p Fk(])p Fn(y)1839 2826 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))23 b(=)f Fu(M)2336 2796 y Fk([)p Fn(\024)p Fq(n)p Fk(])2471 2826 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\).)p 3384 2818 42 42 v 118 3003 a Ft(Lemma)34 b(6.)41 b Fw(Assume)31 b(that)g(the)h(pr)l(op)l(agators)h Fv(g)1688 2973 y Fk([)p Fq(p)p Fk(])1764 3003 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))31 b Fw(c)l(an)h(b)l(e)f(uniformly)i(b)l(ounde)l(d)e(for)i (al)t(l)f Fy(0)26 b Fu(\024)118 3109 y Fv(p)d Fu(\024)g Fv(n)14 b Fu(\000)g Fy(1)25 b Fw(as)j(in)g(\(3.14\),)i(for)e(some)g Fv(p)p Fw(-indep)l(endent)g(c)l(onstants)f Fv(K)2263 3121 y Fk(1)2327 3109 y Fw(and)h Fv(K)2557 3121 y Fk(2)2594 3109 y Fw(.)38 b(Assume)26 b(also)j(that)f Fv(")f Fw(is)118 3216 y(smal)t(l)32 b(enough.)41 b(The)32 b(matric)l(es)f Fu(M)1264 3185 y Fk([)p Fn(\024)p Fq(n)p Fk(])1398 3216 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))31 b Fw(ar)l(e)g(di\013er)l(entiable)h(in)f Fv(x)g Fw(in)f(the)h(sense)g(of) g(Whitney,)118 3322 y(and)f(for)h(al)t(l)g Fv(x)578 3292 y Fn(0)602 3322 y Fv(;)14 b(x)30 b Fw(wher)l(e)g(they)g(ar)l(e)g (de\014ne)l(d)g(one)g(has)348 3446 y Fp(\015)348 3496 y(\015)348 3546 y(\015)394 3541 y Fu(M)494 3507 y Fk([)p Fn(\024)p Fq(n)p Fk(])628 3541 y Fy(\()p Fv(x)707 3507 y Fn(0)731 3541 y Fy(;)14 b Fv(";)g Fs(!)s Fy(\))k Fu(\000)g(M)1140 3507 y Fk([)p Fn(\024)p Fq(n)p Fk(])1275 3541 y Fy(\()p Fv(x)p Fy(;)c Fv(";)g Fs(!)s Fy(\))k Fu(\000)g Fv(@)1707 3553 y Fq(x)1749 3541 y Fu(M)1849 3507 y Fk([)p Fn(\024)p Fq(n)p Fk(])1984 3541 y Fy(\()p Fv(x)p Fy(;)c Fv(";)g Fs(!)s Fy(\))g(\()p Fv(x)2364 3507 y Fn(0)2406 3541 y Fu(\000)k Fv(x)p Fy(\))2568 3446 y Fp(\015)2568 3496 y(\015)2568 3546 y(\015)2638 3541 y Fy(=)23 b Fv(")2765 3507 y Fk(2)2816 3541 y Fv(o)p Fy(\()p Fu(j)p Fv(x)2958 3507 y Fn(0)3000 3541 y Fu(\000)18 b Fv(x)p Fu(j)p Fy(\))p Fv(;)348 3629 y Fp(\015)348 3678 y(\015)348 3728 y(\015)394 3724 y Fv(@)438 3736 y Fq(x)480 3724 y Fu(M)580 3690 y Fk([)p Fn(\024)p Fq(n)p Fk(])714 3724 y Fy(\()p Fv(x)p Fy(;)c Fv(";)g(!)s Fy(\))993 3629 y Fp(\015)993 3678 y(\015)993 3728 y(\015)1063 3724 y Fu(\024)22 b Fv(B)t(")1256 3690 y Fk(2)1293 3724 y Fv(;)1897 b Fy(\(3)p Fv(:)p Fy(18\))118 3944 y Fw(for)31 b(a)g(suitable)f(p)l(ositive)i(c)l(onstant)d Fv(B)t Fw(.)40 b(Mor)l(e)l(over)31 b(for)g(al)t(l)g Fv(j)e Fy(=)23 b(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(d)30 b Fw(and)h(for)g(a)f (suitable)h(c)l(onstant)118 4050 y Fv(A)f Fw(one)g(has)1330 4061 y Fp(\014)1330 4111 y(\014)1330 4161 y(\014)1358 4156 y Fv(@)1402 4168 y Fq(x)1443 4156 y Fv(\025)1491 4113 y Fk([)p Fn(\024)p Fq(n)p Fk(])1491 4180 y Fq(j)1627 4156 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))1914 4061 y Fp(\014)1914 4111 y(\014)1914 4161 y(\014)1964 4156 y Fu(\024)23 b Fv(A")2153 4122 y Fk(2)2190 4156 y Fv(;)1000 b Fy(\(3)p Fv(:)p Fy(19\))118 4335 y Fw(wher)l(e)31 b(the)e(derivative)j(is)e(in)g(the)g(sense)g(of)g(Whitney.)118 4512 y(Pr)l(o)l(of.)62 b Fy(The)35 b(pro)r(of)g(of)g(\(3.18\))f(can)h (b)r(e)h(p)r(erformed)f(as)f(in)i(Ref.)f([23];)k(cf.)60 b(in)35 b(particular)f(Section)i(6.)118 4618 y(Then)21 b(prop)r(ert)n(y)f(\(3.19\))g(follo)n(ws)g(from)h(general)e(prop)r (erties)h(of)h(Hermitian)g(matrices.)34 b(One)21 b(can)f(refer)g(to)118 4724 y(Ref.)h([23],)g(App)r(endix)h(A4,)f(in)g(the)g(case)e(in)i(whic)n (h)f(the)h(eigen)n(v)-5 b(alues)19 b Fv(a)2297 4736 y Fq(i)2345 4724 y Fy(are)h(all)g(distinct.)35 b(Otherwise)20 b(one)1730 4924 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 118 555 a Fy(can)27 b(apply)h(the)g(results)f (on)g(non-analytic)f(Hermitian)i(matrices)f(discussed)g(in)h(Ref.)g ([29],)f(Chapter)g(2,)118 662 y(Section)22 b(6:)33 b(one)22 b(can)f(rely)g(on)h(Rellic)n(h's)f(theorem)h([39])f(to)g(deduce)h (di\013eren)n(tiabilit)n(y)g(of)f(the)h(eigen)n(v)-5 b(alues)118 768 y(and)28 b(on)f(Lidski)-9 b(\025)-32 b(\020's)26 b(theorem)h([33])g(to)g(obtain)h(a)f(b)r(ound)h(on)f(the)h (deriv)-5 b(ativ)n(e.)p 3384 760 42 42 v 118 945 a Ft(Lemma)34 b(7.)41 b Fw(Assume)31 b(that)g(the)h(pr)l(op)l(agators)h Fv(g)1688 915 y Fk([)p Fq(p)p Fk(])1764 945 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))31 b Fw(c)l(an)h(b)l(e)f (uniformly)i(b)l(ounde)l(d)e(for)i(al)t(l)f Fy(0)26 b Fu(\024)118 1052 y Fv(p)h Fu(\024)g Fv(n)20 b Fu(\000)f Fy(1)32 b Fw(as)g(in)g(\(3.14\),)j(for)e(some)f Fv(p)p Fw(-indep)l(endent)h(c)l(onstants)e Fv(K)2320 1064 y Fk(1)2388 1052 y Fw(and)i Fv(K)2623 1064 y Fk(2)2660 1052 y Fw(.)45 b(Assume)31 b(also)i(that)f Fv(")118 1167 y Fw(is)f(smal)t(l)h(enough.)42 b(The)32 b(self-ener)l(gies)f Fv(\025)p 1388 1180 49 4 v 1437 1124 a Fk([)p Fq(p)p Fk(])1437 1190 y Fq(j)1513 1167 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))30 b Fw(satisfy)i(for)g(al)t(l)g Fy(0)24 b Fu(\024)g Fv(p)h Fu(\024)g Fv(n)30 b Fw(and)h(al)t(l)h Fy(1)24 b Fu(\024)h Fv(j)30 b Fu(\024)24 b Fv(d)31 b Fw(the)118 1274 y(closeness)g(pr)l(op)l(erty)1083 1298 y Fp(\014)1083 1348 y(\014)1083 1398 y(\014)1111 1394 y Fv(\025)p 1111 1407 V -44 x Fk([)p Fq(p)p Fk(])1159 1417 y Fq(j)1235 1394 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))k Fu(\000)g Fv(\025)p 1539 1407 V 1588 1350 a Fk([)p Fq(p)p Fn(\000)p Fk(1])1588 1417 y Fq(j)1749 1394 y Fy(\()p Fv(";)c Fs(!)s Fy(\))1952 1298 y Fp(\014)1952 1348 y(\014)1952 1398 y(\014)2002 1394 y Fu(\024)23 b Fv(A)p Fy(e)2189 1359 y Fn(\000)p Fq(\024)2280 1367 y Fj(1)2312 1359 y Fk(2)2345 1334 y Fl(p)2384 1394 y Fv(")2423 1359 y Fk(2)3213 1394 y Fy(\(3)p Fv(:)p Fy(20\))118 1563 y Fw(and)30 b(one)g(has)896 1574 y Fp(\014)896 1624 y(\014)896 1674 y(\014)923 1669 y Fv(\025)971 1626 y Fk([)p Fq(p)p Fk(])971 1693 y Fq(j)1048 1669 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))1335 1574 y Fp(\014)1335 1624 y(\014)1335 1674 y(\014)1386 1669 y Fu(\024)22 b Fv(A)14 b Fy(min)q Fu(f)p Fv(")1769 1635 y Fk(2)1805 1669 y Fv(;)g("x)1928 1635 y Fk(2)1966 1669 y Fu(g)p Fv(;)183 b(j)28 b Fy(=)22 b(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(r)n(;)566 b Fy(\(3)p Fv(:)p Fy(21\))118 1839 y Fw(for)31 b(a)f(suitable)g(p)l(ositive)h(c)l(onstant)e Fv(A)p Fw(.)118 2016 y(Pr)l(o)l(of.)39 b Fy(The)28 b(pro)r(of)f(can)h (b)r(e)g(p)r(erformed)f(as)h(in)g(Ref.)g([23],)f(b)n(y)h(using)f(the)h (b)r(ounds)g(\(3.15\))f(and)h(the)g(fact)118 2122 y(that)g(the)g (matrices)f Fu(M)874 2092 y Fk([)p Fn(\024)p Fq(n)p Fk(])1008 2122 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))28 b(are)f(Hermitian)g(\(see)h(Ref.)g([23],)f(Lemma)g(2\).)118 2300 y Ft(Lemma)34 b(8.)41 b Fw(Assume)31 b(that)g(the)h(pr)l(op)l (agators)h Fv(g)1688 2269 y Fk([)p Fq(p)p Fk(])1764 2300 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))31 b Fw(c)l(an)h(b)l(e)f(uniformly)i(b)l(ounde)l(d)e(for)i(al)t(l)f Fy(0)26 b Fu(\024)118 2406 y Fv(p)h Fu(\024)g Fv(n)20 b Fu(\000)f Fy(1)32 b Fw(as)g(in)g(\(3.14\),)j(for)e(some)f Fv(p)p 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Fy(\()p Fv(";)c Fs(!)s Fy(\))2698 2636 y Fp(\014)2698 2686 y(\014)2698 2736 y(\014)2740 2731 y Fv(:)450 b Fy(\(3)p Fv(:)p Fy(22\))118 2968 y Fw(The)32 b(same)g(holds)h(if)f Fv(g)842 2938 y Fk([)p Fq(n)p Fk(])924 2968 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))26 b(=)f(0)31 b Fw(but)f(\(3.6\))j(ar)l(e)e(satis\014e)l (d)h(with)g Fv(\014)2444 2938 y Fk(2)2513 2968 y Fw(r)l(eplac)l(e)l(d)g (with)g Fv(\014)3062 2938 y Fk(2)3099 2968 y Fv(=)p Fy(4)f Fw(in)g(the)118 3074 y(\014rst)e(line)h(and)h(with)f Fy(4)p Fv(\014)887 3044 y Fk(2)954 3074 y Fw(in)f(the)h(se)l(c)l(ond)g (line.)118 3251 y(Pr)l(o)l(of.)63 b Fy(The)36 b(inequalit)n(y)f (\(3.22\))g(can)g(b)r(e)h(pro)n(v)n(ed)e(b)n(y)h(induction)h(on)g Fv(n)p Fy(.)61 b(F)-7 b(or)35 b Fv(n)h Fy(=)g(0)f(it)h(is)g(trivially) 118 3358 y(satis\014ed)21 b(as)g(the)h(matrix)g Fu(M)1029 3327 y Fk([0])1103 3358 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))22 b(do)r(es)f(not)h(dep)r(end)g(on)g Fv(x)p Fy(.)35 b(Let)22 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y Fn(\003)1568 4011 y Fq(m)1627 4019 y Fj(0)1673 3817 y Fp(\023)1734 3834 y Fk(2)p Fq(k)1822 3934 y Fy(exp)1963 3792 y Fp( )2028 3934 y Fv(K)6 b Fu(j)p Fs(\027)g Fu(j)2292 3855 y Fp(X)2219 4031 y Fq(n)p Fk(=)p Fq(m)2370 4039 y Fj(0)2402 4031 y Fk(+1)2532 3878 y Fy(1)p 2510 3915 87 4 v 2510 3991 a(2)2552 3967 y Fq(n)2620 3934 y Fy(log)2774 3878 y(1)p 2751 3915 89 4 v 2751 3991 a Fv(\015)2799 3967 y Fn(\003)2794 4011 y Fq(n)2849 3792 y Fp(!)2929 3934 y Fv(;)3213 3787 y Fy(\(3)p Fv(:)p Fy(28\))118 4219 y(and)30 b(one)g(can)g(c)n(ho)r(ose) f Fv(m)931 4231 y Fk(0)995 4219 y Fy(=)f Fv(m)1161 4231 y Fk(0)1198 4219 y Fy(\()p Fv(\024)1278 4231 y Fk(0)1315 4219 y Fy(\),)j(so)f(the)h(last)e(exp)r(onen)n(tial)h(is)g(less)g(than) g(e)2734 4189 y Fq(\024)2773 4197 y Fj(0)2806 4189 y Fn(j)p Fo(\027)5 b Fn(j)p Fq(=)p Fk(4)2959 4219 y Fy(.)45 b(By)30 b(making)118 4326 y(use)h(of)f(the)h(b)r(ound)767 4263 y Fp(Q)846 4351 y Fd(v)o Fn(2)p Fq(V)15 b Fk(\()p Fq(\022)r Fk(\))1090 4326 y Fy(e)1127 4295 y Fn(\000)p Fq(\024)p Fn(j)p Fo(\027)1276 4312 y Fd(v)1322 4295 y Fn(j)1374 4326 y Fu(\024)28 b Fy(e)1504 4295 y Fn(\000)p Fq(\024)p Fn(j)p Fo(\027)5 b Fn(j)1681 4326 y Fy(,)32 b(this)f(pro)r(duces)f(an)g(o)n(v)n(erall)f(factor)h(e)2918 4295 y Fn(\000)p Fq(\024)3009 4303 y Fj(0)3041 4295 y Fn(j)p Fo(\027)5 b Fn(j)p Fq(=)p Fk(2)3194 4326 y Fy(.)46 b(This)118 4441 y(completes)27 b(the)h(pro)r(of,)f(and)h(it)g(giv)n(es) e Fv(")1371 4453 y Fk(0)1431 4441 y Fy(=)d Fv(O)r Fy(\()p Fv(C)1681 4411 y Fk(2)1675 4462 y(0)1719 4441 y Fy(\()p Fv(\015)1799 4411 y Fn(\003)1794 4461 y Fq(m)1853 4469 y Fj(0)1890 4441 y Fy(\))1922 4411 y Fk(2)1960 4441 y Fy(\).)p 3384 4433 42 42 v 189 4618 a(Note)j(that)h(for)e(Diophan)n (tine)i(v)n(ectors)d(satisfying)i(the)g(b)r(ound)h(\(2.8\))f(one)g(has) f Fv(m)2775 4630 y Fk(0)2836 4618 y Fy(=)d Fv(\034)i(O)r Fy(\(log)15 b(1)p Fv(=\024)3334 4630 y Fk(0)3370 4618 y Fy(\),)118 4724 y(and)28 b(one)f(obtains)g Fv(")762 4736 y Fk(0)822 4724 y Fy(=)22 b Fv(O)r Fy(\()p Fv(C)1071 4694 y Fk(2)1065 4745 y(0)1110 4724 y Fy(\),)28 b(for)f(\014xed)h Fv(\034)37 b Fy(and)27 b Fv(C)1814 4736 y Fk(0)1852 4724 y Fy(.)1730 4924 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 189 555 a Fy(If)29 b(w)n(e)f(are)f(in)n (terested)h(in)g(studying)g(the)h(conserv)-5 b(ation)27 b(of)h(a)g(maximal)f(torus)h(with)g(rotation)g(v)n(ector)118 662 y Fs(!)g Fy(satisfying)c(the)h(Bryuno)f(condition)h Fv(B)t Fy(\()p Fs(!)s Fy(\))f Fv(<)e Fu(1)p Fy(,)k(w)n(e)f(can)f(use)h (directly)g(the)g(sequence)g Fu(f)p Fv(\015)3083 674 y Fq(n)3127 662 y Fy(\()p Fs(!)s Fy(\))p Fu(g)3296 632 y Fn(1)3296 682 y Fq(n)p Fk(=0)118 768 y Fy(for)34 b(the)h(m)n (ultiscale)f(decomp)r(osition,)h(without)g(in)n(tro)r(ducing)f(a)f (further)i(sequence)e Fu(f)p Fv(\015)2945 738 y Fn(\003)2940 789 y Fq(n)2985 768 y Fu(g)3027 738 y Fn(1)3027 789 y Fq(n)p Fk(=0)3156 768 y Fy(.)57 b(Then)118 874 y(the)28 b(result)f(stated)h(in)g(Lemma)f(9)g(holds)h(with)g Fv(\015)1654 886 y Fq(m)1713 894 y Fj(0)1749 874 y Fy(\()p Fs(!)s Fy(\))g(replacing)e Fv(\015)2307 844 y Fn(\003)2302 895 y Fq(m)2361 903 y Fj(0)2398 874 y Fy(.)189 981 y(An)k(imp)r(ortan)n(t)g (remark)e(is)i(that,)h(in)f(the)g(case)f(of)h(p)r(erturbations)f(whic)n (h)h(are)e(trigonometric)h(p)r(oly-)118 1087 y(nomials)c(of)g(degree)f Fv(N)34 b Fy(in)26 b(the)f(b)r(ound)h(\(3.9\))f(one)g(can)g(b)r(ound)g Fv(M)9 b Fy(\()p Fv(\022)r Fy(\))24 b Fu(\024)e Fv(k)s(N)9 b Fy(,)26 b(and)f(as)g(consequence)f(the)118 1193 y(pro)r(duct)k(of)f (propagators)e(in)j(\(3.28\))e(can)i(b)r(e)g(b)r(ounded)g(as)747 1359 y Fp(Y)698 1541 y Fq(`)p Fn(2)p Fk(\003\()p Fq(\022)r Fk(\))915 1342 y Fp(\014)915 1392 y(\014)915 1442 y(\014)943 1438 y Fv(g)986 1395 y Fk([)p Fq(n)1046 1404 y Fl(`)1074 1395 y Fk(])983 1463 y Fq(`)1097 1342 y Fp(\014)1097 1392 y(\014)1097 1442 y(\014)1148 1438 y Fu(\024)22 b Fy(exp)1376 1296 y Fp( )1442 1438 y Fy(2)p Fv(K)6 b(N)j(k)1725 1334 y Fn(1)1698 1359 y Fp(X)1695 1535 y Fq(n)p Fk(=0)1867 1382 y Fy(1)p 1844 1419 87 4 v 1844 1495 a(2)1886 1471 y Fq(n)1955 1438 y Fy(log)2178 1382 y(1)p 2086 1419 226 4 v 2086 1495 a Fv(\013)2139 1507 y Fq(n)2184 1495 y Fy(\()p Fs(!)s Fy(\))2321 1296 y Fp(!)2410 1438 y Fy(=)23 b(e)2535 1404 y Fk(4)p Fq(N)6 b(k)q(B)s Fk(\()p Fo(!)r Fk(\))2822 1438 y Fv(;)368 b Fy(\(3)p Fv(:)p Fy(29\))118 1716 y(whic)n(h)23 b(implies)f Fv(")666 1728 y Fk(0)726 1716 y Fy(=)h Fv(O)r Fy(\(e)948 1686 y Fn(\000)p Fk(4)p Fq(N)6 b(B)s Fk(\()p Fo(!)r Fk(\))1252 1716 y Fy(\).)35 b(W)-7 b(e)23 b(can)g(compare)e(this)i(result)f(with)h(the)g(one)f (found)h(in)g(Ref.)g([25])118 1822 y(for)c(maximal)f(tori,)j(where)d(a) h(b)r(ound)g(of)h(this)f(kind)g(with)h(the)f(factor)g(4)f(replaced)h (with)g(the)h(lik)n(ely)e(optimal)118 1929 y(2)25 b(w)n(as)f(obtained.) 36 b(With)26 b(the)f(tec)n(hniques)g(describ)r(ed)g(in)h(this)f(pap)r (er)g(some)f(further)h(w)n(ork)f(is)h(necessary)118 2035 y(in)f(order)e(to)h(reac)n(h)f(the)i(factor)e(2;)i(cf.)36 b(for)23 b(instance)g(Ref.)h([2].)35 b(On)23 b(the)h(other)f(hand)g(an) 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y(write)d Fs(!)f Fy(=)e Fv(\013)p Fs(\027)6 b Fv(=)p Fu(j)p Fs(\027)g Fu(j)20 b Fy(+)g Fs(\014)s Fy(,)31 b(with)g Fs(\014)23 b Fu(\001)e Fs(\027)34 b Fy(=)28 b(0.)45 b(Then)31 b(w)n(e)f(de\014ne)h Fv(\013)p Fy(\()p Fv(t)p Fy(\),)i Fv(t)28 b Fu(2)h Fy([)p Fu(\000)p Fy(1)p Fv(;)14 b Fy(1],)30 b(suc)n(h)g(that)h Fv(F)12 b Fy(\()p Fv(t)p Fy(\))29 b(=)1730 4924 y(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 118 586 a Fv(\013)p Fy(\()p Fv(t)p Fy(\))p Fu(j)p Fs(\027)6 b Fu(j)21 b(\006)470 479 y Fp(q)p 553 479 579 4 v 107 x Fv(\025)p 553 599 49 4 v 602 543 a Fk([)p Fq(n)p Fk(])602 609 y Fq(i)684 586 y Fy(\()p Fv(";)14 b Fy(\()p Fv(\013)p Fy(\()p Fv(t)p Fy(\))p Fv(;)g Fs(\014)s Fy(\)\))28 b(=)e Fv(tC)1338 598 y Fk(0)1376 586 y Fv(\015)1424 556 y Fn(\003)1419 613 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))1559 586 y Fy(,)30 b(so)f(that)h(d)p Fv(F)7 b(=)p Fy(d)p Fv(t)27 b Fy(=)f Fu(j)p Fs(\027)6 b Fu(j)p Fy(\(1)19 b(+)g Fv(O)r Fy(\()2614 526 y Fu(p)p 2685 526 39 4 v 2685 586 a Fv(")o Fy(\)\)d)p 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y(sets)j(is)f(2)p Fv(C)473 1281 y Fk(0)510 1269 y Fv(\015)558 1239 y Fn(\003)553 1296 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))725 1269 y Fy(instead)32 b(of)g Fv(C)1174 1281 y Fk(0)1211 1269 y Fv(\015)1259 1239 y Fn(\003)1254 1296 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))1394 1269 y Fy(.)50 b(By)31 b(the)i(closeness)d(prop)r(ert)n(y)h (of)g(Lemma)h(7,)h(there)e(is)h(some)118 1376 y Fv(n)168 1388 y Fk(1)205 1376 y Fy(\()p Fs(\027)6 b Fy(\))33 b(=)g Fv(O)r Fy(\(log)15 b(log)f(1)p Fv(=\015)926 1345 y Fn(\003)921 1402 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))1060 1376 y Fy(\))34 b(suc)n(h)f(that)h(all)f(the)h(sets)g Fv(I)1980 1388 y Fq(n)2025 1376 y Fy(\()p Fv(i;)14 b Fs(\027)6 b Fy(\))34 b(fall)g(inside)f Fv(J)2677 1391 y Fq(n)2718 1399 y Fj(1)2751 1391 y Fk(\()p Fo(\027)5 b Fk(\))2883 1376 y Fy(for)33 b Fv(n)g Fu(\025)g Fv(n)3247 1388 y Fk(1)3284 1376 y Fy(\()p Fs(\027)6 b Fy(\).)118 1491 y(Therefore)24 b(for)g(all)h Fs(\027)j Fu(2)c Fr(Z)943 1450 y Fq(r)943 1512 y Fn(\003)982 1491 y Fy(,)h(all)g Fv(i)d Fy(=)h Fv(r)16 b Fy(+)d(1)p Fv(;)h(:)g(:)g(:)e(;)i(d)p Fy(,)26 b(and)e(all)h Fv(n)e Fu(\024)f Fv(n)2211 1503 y Fk(1)2249 1491 y Fy(\()p Fs(\027)6 b Fy(\))25 b(w)n(e)f(ha)n(v)n(e)g(to)g(exclude)h(all)g(v)-5 b(alues)118 1598 y(of)30 b Fs(!)f Fu(2)e Fy(\012)446 1568 y Fk([)p Fq(n)p Fn(\000)p Fk(1])643 1598 y Fy(whic)n(h)j(fall)g (inside)g(the)g(set)f Fv(J)1586 1610 y Fq(n)1632 1598 y Fy(\()p Fv(i;)14 b Fs(\027)6 b Fy(\);)30 b(w)n(e)f(refer)g(to)h(Ref.) g([21],)g(Section)f(7,)h(for)f(details.)118 1704 y(Note)h(that)g Fs(!)f Fu(2)e Ff(B)755 1716 y Fq(r)822 1704 y Fy(implies)j Fv(n)1156 1716 y Fk(1)1193 1704 y Fy(\()p Fs(\027)6 b Fy(\))27 b Fu(\024)f Fv(C)6 b(n)p Fy(\()p Fs(\027)g Fy(\),)31 b(for)e(some)g(constan)n(t)g Fv(C)6 b Fy(.)44 b(Hence)30 b(w)n(e)f(can)h(b)r(ound)g(the)118 1810 y(measure)d(of)g(the)h(set)g (of)f(excluded)h(v)-5 b(alues)27 b(b)n(y)h(a)f(constan)n(t)g(times)416 2100 y(const)p Fv(:)721 1997 y Fq(d)678 2022 y Fp(X)643 2198 y Fq(i)p Fk(=)p Fq(r)r Fk(+1)848 2100 y Fv(C)907 2112 y Fk(0)979 2022 y Fp(X)958 2201 y Fo(\027)5 b Fn(2)p Fm(Z)1086 2180 y Fl(r)1133 1990 y Fq(n)1174 1998 y Fj(1)1206 1990 y Fk(\()p Fo(\027)g Fk(\))1157 2022 y Fp(X)1154 2197 y Fq(n)p Fk(=1)1325 2027 y Fv(\015)1373 1997 y Fn(\003)1368 2054 y Fq(n)p Fk(\()p Fo(\027)g Fk(\))p 1325 2081 183 4 v 1367 2157 a Fu(j)p Fs(\027)h Fu(j)1541 2100 y(\024)23 b Fy(const)p Fv(:)13 b(sC)1953 2112 y Fk(0)2034 1997 y Fn(1)2007 2022 y Fp(X)2004 2197 y Fq(n)p Fk(=0)2143 2100 y Fy(2)2185 2066 y Fq(n)p Fk(\()p Fq(r)r Fn(\000)p Fk(1\))2399 2100 y Fv(\015)2447 2066 y Fn(\003)2442 2121 y Fq(n)2502 2100 y Fy(log)h(log)g(1)p Fv(=\015)2876 2066 y Fn(\003)2871 2121 y Fq(n)1541 2384 y Fu(\024)23 b Fy(const)p Fv(:)13 b(sC)1953 2396 y Fk(0)2034 2280 y Fn(1)2007 2305 y Fp(X)2004 2481 y Fq(n)p Fk(=0)2143 2384 y Fv(n)p Fy(2)2235 2350 y Fq(n)p Fk(\()p Fq(r)r Fn(\000)p Fk(1\))2449 2384 y Fv(\015)2497 2350 y Fn(\003)2492 2405 y Fq(n)2537 2384 y Fv(;)3213 2234 y Fy(\(3)p Fv(:)p Fy(32\))118 2658 y(whic)n(h)28 b(is)f(b)r(ounded)h(prop)r(ortionally)e(to)h Fv(C)1483 2670 y Fk(0)1549 2658 y Fy(b)n(y)g(Lemma)g(2.)189 2766 y(Analogously)42 b(one)g(discusses)h(the)h(other)e(conditions)h(in)h (\(3.8\).)83 b(Simply)44 b(one)f(de\014nes)g Fv(F)12 b Fy(\()p Fv(t)p Fy(\))50 b(=)118 2908 y Fv(\013)p Fy(\()p Fv(t)p Fy(\))p Fu(j)p Fs(\027)6 b Fu(j)11 b(\006)452 2801 y Fp(q)p 535 2801 579 4 v 107 x Fv(\025)p 535 2921 49 4 v 584 2865 a Fk([)p Fq(n)p Fk(])584 2931 y Fq(i)666 2908 y Fy(\()p Fv(";)j Fy(\()p Fv(\013)p Fy(\()p Fv(t)p Fy(\))p Fv(;)g Fs(\014)s Fy(\)\))e Fu(\006)1200 2807 y Fp(q)p 1283 2807 579 4 v 101 x Fv(\025)p 1283 2921 49 4 v -43 x Fk([)p Fq(n)p Fk(])1331 2931 y Fq(j)1414 2908 y Fy(\()p Fv(";)i Fy(\()p Fv(\013)p Fy(\()p Fv(t)p Fy(\))p Fv(;)g Fs(\014)s Fy(\)\))24 b(=)f Fv(tC)2061 2920 y Fk(0)2098 2908 y Fv(\015)2146 2878 y Fn(\003)2141 2935 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))2281 2908 y Fy(,)25 b(so)e(that)h(again)f(one)g(has)h(d)p Fv(F)7 b(=)p Fy(d)p Fv(t)23 b Fy(=)118 3028 y Fu(j)p Fs(\027)6 b Fu(j)p Fy(\(1)18 b(+)g Fv(O)r Fy(\()490 2968 y Fu(p)p 560 2968 39 4 v 560 3028 a Fv(")p Fy(\)\)d)p Fv(\013=)p Fy(d)p Fv(t)28 b Fy(=)g Fv(C)1060 3040 y Fk(0)1097 3028 y Fv(\015)1145 2998 y Fn(\003)1140 3055 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))1280 3028 y Fy(,)28 b(and)f(one)h(can)f(pro)r(ceed)g(as)g(b)r (efore.)p 3384 3020 42 42 v 189 3206 a(T)-7 b(o)31 b(complete)g(the)g (pro)r(of)g(of)g(Theorem)g(1)f(w)n(e)h(ha)n(v)n(e)f(to)h(pro)n(v)n(e)f (the)h(last)g(assertion)f(ab)r(out)h(maximal)118 3312 y(tori,)c(that)h(is)g(that)f(for)h Fv(r)d Fy(=)e Fv(d)28 b Fy(most)f(of)h(phase)f(space)g(is)g(\014lled)h(b)n(y)g(in)n(v)-5 b(arian)n(t)26 b(tori.)189 3420 y(W)-7 b(e)40 b(summarise)e(what)h(w)n (e)g(ha)n(v)n(e)f(found)i(so)e(far.)72 b(The)39 b(in)n(v)-5 b(arian)n(t)38 b(tori)h(are)f(determined)i(b)n(y)f(the)118 3526 y(corresp)r(onding)26 b(rotation)h(v)n(ectors)g Fs(!)s Fy(.)38 b(F)-7 b(or)28 b Fs(!)f Fu(2)d Fy(\012)1754 3538 y Fn(\003)1792 3526 y Fy(\()p Fv(C)1883 3538 y Fk(0)1921 3526 y Fy(\))k(w)n(e)g(can)g(parameterise)e(the)j(in)n(v)-5 b(arian)n(t)27 b(torus)118 3632 y(with)h(rotation)f(v)n(ector)f Fs(!)k Fy(as)1216 3647 y Fp(\032)1293 3710 y Fs(\013)23 b Fy(=)f Fs( )g Fy(+)c Ft(a)p Fy(\()p Fs( )s Fv(;)c Fs(!)s Fv(;)g(")p Fy(\))p Fv(;)1293 3809 y Ft(A)23 b Fy(=)f Fs(!)g Fy(+)c(\()p Fs(!)j Fu(\001)d Fv(@)1838 3827 y Fo( )1896 3809 y Fy(\))c Ft(a)p Fy(\()p Fs( )s Fv(;)g Fs(!)s Fv(;)g(")p Fy(\))p Fv(;)3213 3764 y Fy(\(3)p Fv(:)p Fy(33\))118 3979 y(with)34 b Fs( )h Fu(2)e Fr(T)559 3938 y Fq(r)596 3979 y Fy(.)54 b(Moreo)n(v)n(er)30 b(the)k(function)f Ft(a)h Fy(is)f(analytic)g(in)g Fs( )k Fy(\(as)c(the)g(F)-7 b(ourier)32 b(co)r(e\016cien)n(ts)h(deca)n(y)118 4086 y(exp)r(onen)n(tially\),)27 b(while)h(it)g(is)g(de\014ned)g(only)f(on)g (a)g(Can)n(torian)f(set)i(of)f(v)-5 b(alues)28 b Fs(!)s Fy(.)189 4193 y(F)-7 b(or)30 b(eac)n(h)g(v)-5 b(alue)31 b(of)g Fs( )j Fy(w)n(e)c(can)h(consider)f(the)h(map)g Fs(!)g Fu(7!)d Ft(A)p Fy(\()p Fs(!)s Fy(\),)k(de\014ned)g(in)f (\(3.33\).)46 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Fn(\003)1059 799 y Fy(\()p Fv(C)1150 811 y Fk(0)1188 799 y Fy(\)\))d(=)1364 686 y Fp(Z)1410 875 y Fn(A)1463 858 y Fl(c)1463 891 y Fe(\003)1498 875 y Fk(\()p Fq(C)1572 883 y Fj(0)1604 875 y Fk(\))1648 799 y Fy(d)p Ft(A)f Fy(=)1877 686 y Fp(Z)1923 875 y Fk(\012)1970 858 y Fl(c)1970 891 y Fe(\003)2005 875 y Fk(\()p Fq(C)2079 883 y Fj(0)2111 875 y Fq(;R)p Fn(j)p Fq(")p Fn(j)p Fk(\))2296 799 y Fy(d)p Fs(!)16 b Fu(j)p Fy(det)e Fv(@)2614 811 y Fo(!)2669 799 y Ft(A)p Fu(j)g Fv(;)412 b Fy(\(3)p Fv(:)p Fy(34\))118 1016 y(pro)n(vided)29 b(that)h(the)h(Jacobian)d(in)i(the)h (last)f(in)n(tegral)e(is)i(w)n(ell)g(de\014ned)g(\(that)h(is)f(the)g (map)g Fs(!)g Fu(!)d Ft(A)p Fy(\()p Fs(!)s Fy(\))118 1122 y(is)i(smo)r(oth)g(enough,)h(at)f(least)g(in)h(the)f(sense)g(of)g (Whitney\))i(and)e(is)g(uniformly)g(b)r(ounded.)43 b(This)29 b(turns)118 1229 y(out)f(to)f(b)r(e)h(the)g(case,)f(as)g(the)h(follo)n (wing)e(result)i(sho)n(ws.)118 1406 y Ft(Lemma)j(12.)36 b Fw(The)30 b(solutions)f(of)h(the)g(e)l(quations)f(of)h(motion)g Ft(h)p Fy(\()p Fs( )s Fv(;)14 b Fs(\014)2352 1418 y Fk(0)2389 1406 y Fv(;)g Fs(!)s Fv(;)g(")p Fy(\))29 b Fw(ar)l(e)g(di\013er)l (entiable)i(in)f Fs(!)118 1512 y Fw(in)g(the)g(sense)g(of)g(Whitney)g (for)h Fs(!)26 b Fu(2)d Fy(\012)1366 1524 y Fn(\003)1404 1512 y Fy(\()p Fv(C)1495 1524 y Fk(0)1533 1512 y Fv(;)14 b(R)q Fu(j)p Fv(")p Fu(j)p Fy(\))p Fw(.)118 1689 y(Pr)l(o)l(of.)70 b Fy(The)39 b(pro)r(of)e(\(for)h(an)n(y)g(v)-5 b(alue)38 b(of)g Fv(r)43 b Fu(\024)d Fv(d)p Fy(,)i(not)c(necessarily)e Fv(r)44 b Fy(=)c Fv(d)p Fy(\))f(can)f(b)r(e)g(p)r(erformed)g(as)118 1796 y(for)d(Lemma)h(10,)h(with)g(the)f(only)f(di\013erence)h(that)g (no)n(w)g(w)n(e)f(ha)n(v)n(e)g(to)h(deal)f(with)i(the)f(renormalised) 118 1902 y(expansion)30 b(for)g Fv(h)685 1914 y Fo(\027)5 b Fq(;\015)820 1902 y Fy(instead)31 b(of)f(the)h(matrices)f Fu(M)1789 1872 y Fk([)p Fn(\024)p Fq(n)p Fk(])1923 1902 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\),)32 b(hence)f(with)g (trees)f(instead)g(of)g(self-)118 2008 y(energy)f(clusters.)43 b(The)30 b(condition)g Fv(d)p Fy(\()p Fs(!)s Fv(;)14 b(@)5 b Fy(\012\))27 b Fu(\025)g Fv(R)q Fu(j)p Fv(")p Fu(j)i Fy(yields)h(that)g(the)h(actions)e(v)-5 b(ariables)29 b Ft(A)h Fy(remain)118 2114 y(in)e(\012)g(for)f(all)g(v)-5 b(alues)27 b(of)h Fs( )s Fy(.)p 3384 2106 42 42 v 189 2292 a(As)23 b(a)f(consequence)f(w)n(e)h(can)g(b)r(ound)h(the)g (Jacobian)e(in)i(\(3.34\))e(b)n(y)h(using)h(\(3.33\),)f(whic)n(h)h(giv) n(es)e Fv(@)3211 2304 y Fo(!)3265 2292 y Ft(A)j Fy(=)118 2398 y Fb(1)-27 b Fy(1)24 b(+)h Fv(@)329 2410 y Fo(!)383 2398 y Fy(\()p Fs(!)j Fu(\001)c Fv(@)594 2415 y Fo( )651 2398 y Fy(\))14 b Ft(a)p Fy(\()p Fs( )s Fv(;)g Fs(!)s Fv(;)g(")p Fy(\),)40 b(and)d(Lemma)g(12,)h(whic)n(h)f(assures)e(that)i (the)h(last)e(deriv)-5 b(ativ)n(e)36 b(\(in)i(the)118 2504 y(sense)30 b(of)h(Whitney\))g(is)g(b)r(ounded)g(prop)r(ortionally) d(to)j Fv(")p Fy(.)45 b(Therefore)29 b(w)n(e)i(can)f(b)r(ound)h(meas)o (\()p Fu(A)3193 2474 y Fq(c)3193 2525 y Fn(\003)3232 2504 y Fy(\()p Fv(C)3323 2516 y Fk(0)3361 2504 y Fy(\)\))118 2611 y(prop)r(ortionally)18 b(to)i Fv(C)808 2623 y Fk(0)865 2611 y Fy(b)n(y)g(Lemma)f(11,)i(and)f(b)n(y)f(taking)g Fv(C)1957 2623 y Fk(0)2018 2611 y Fy(=)k Fv(O)r Fy(\()2203 2540 y Fp(p)p 2287 2540 85 4 v 2287 2611 a Fu(j)p Fv(")p Fu(j)p Fy(\))d(\(whic)n(h)g(is)g(allo)n(w)n(ed)e(b)n(y)i(Lemma)118 2717 y(9\),)29 b(w)n(e)f(obtain)g(the)h(last)f(assertion)f(of)h (Theorem)g(1.)39 b(Cf.)h(also)27 b(Refs.)i([11])f(and)g([37],)g(where)g (the)h(usual)118 2823 y(Diophan)n(tine)d(conditions)g(w)n(ere)f (considered)g(in)h(the)g(analytic)g(and)g(di\013eren)n(tiable)f(case,)h (resp)r(ectiv)n(ely)-7 b(.)549 3071 y Fz(4.)50 b(Fixing)38 b(the)f(rotation)h(v)m(ector:)49 b(pro)s(of)38 b(of)g(Theorem)g(2)118 3248 y Fy(In)d(the)g(follo)n(wing)f(w)n(e)g(assume)g(that)i Fs(!)h Fy(is)e(\014xed,)h(and)f(that)g(it)g(satis\014es)f(the)h(Bryuno) f(Diophan)n(tine)118 3355 y(condition)27 b Fv(B)t Fy(\()p Fs(!)s Fy(\))d Fv(<)f Fu(1)p Fy(,)k(with)h Fv(B)t Fy(\()p Fs(!)s Fy(\))g(de\014ned)g(in)g(\(2.4\).)37 b(Set)28 b Fv(\015)2132 3367 y Fq(n)2200 3355 y Fy(=)23 b Fv(C)2353 3319 y Fn(\000)p Fk(1)2347 3377 y(0)2442 3355 y Fv(\013)2495 3367 y Fq(n)2540 3355 y Fy(\()p Fs(!)s Fy(\).)189 3472 y(Let)29 b Fv(")d Fu(2)g Fy(\()p Fv(")556 3484 y Fk(0)593 3472 y Fv(=)p Fy(4)p Fv(;)14 b(")753 3484 y Fk(0)789 3472 y Fy(])29 b(and)g(set)h Fv(\025)1184 3429 y Fk([0])1184 3497 y Fq(d)1285 3472 y Fy(=)25 b Fv("a)1458 3484 y Fq(s)1522 3472 y Fy(and)k Fv(")1724 3484 y Fk(0)1761 3472 y Fv(a)1805 3484 y Fq(s)1867 3472 y Fy(=)c(\003)2015 3484 y Fk(0)2052 3472 y Fy(.)42 b(De\014ne)29 b Fv(n)2425 3484 y Fk(0)2488 3472 y Fu(2)d Fr(N)j Fy(suc)n(h)g(that)h Fv(C)3093 3484 y Fk(0)3130 3472 y Fv(\015)3173 3484 y Fq(n)3214 3492 y Fj(0)3247 3484 y Fk(+1)3361 3472 y Fv(<)118 3578 y Fy(2)160 3514 y Fu(p)p 229 3514 95 4 v 64 x Fy(\003)287 3590 y Fk(0)347 3578 y Fu(\024)22 b Fv(C)493 3590 y Fk(0)531 3578 y Fv(\015)574 3590 y Fq(n)615 3598 y Fj(0)652 3578 y Fy(.)37 b(W)-7 b(e)28 b(set)1233 3708 y Fv(\015)1281 3674 y Fn(\003)1276 3729 y Fq(n)1344 3708 y Fy(=)1432 3616 y Fp(n)1501 3652 y Fv(\015)1544 3664 y Fq(n)1589 3652 y Fv(;)390 b(n)23 b(<)g(n)2213 3664 y Fk(0)2250 3652 y Fy(,)1501 3751 y Fv(\015)1544 3763 y Fq(n)1589 3751 y Fy(2)1631 3721 y Fn(\000)p Fq(n)p Fk(\()p Fq(r)r Fk(+1\))1896 3751 y Fv(;)83 b(n)23 b Fu(\025)g Fv(n)2213 3763 y Fk(0)2250 3751 y Fy(,)3255 3708 y(\(4)p Fv(:)p Fy(1\))118 3892 y(and,)j(b)n(y)g(using)g(the)g(sequence)f Fu(f)p Fv(\015)1204 3862 y Fn(\003)1199 3913 y Fq(n)1244 3892 y Fu(g)1286 3862 y Fn(1)1286 3913 y Fq(n)p Fk(=0)1415 3892 y Fy(,)h(w)n(e)g(pro)r(ceed)f(as)g(in)h(Section)g(3,)g(for)g (constructing)f(the)h(m)n(ulti-)118 3998 y(scale)j(decomp)r(osition)g (of)h(the)g(propagators.)40 b(Though,)30 b(w)n(e)g(de\014ne)g(\001)2377 3968 y Fk([)p Fq(n)p Fk(])2460 3998 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))26 b(=)h(\001)2934 3968 y Fk([0])3009 3998 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))30 b(for)118 4105 y Fv(n)23 b Fu(\024)g Fv(n)329 4117 y Fk(0)366 4105 y Fy(.)189 4211 y(The)28 b(main)f(di\013erence)h(is)f(that)h(w)n(e)f(shall)g(need)h(the)g(follo) n(wing)f(Diophan)n(tine)g(conditions:)949 4326 y Fp(\014)949 4376 y(\014)949 4426 y(\014)949 4475 y(\014)977 4446 y Fs(!)21 b Fu(\001)d Fs(\027)24 b Fu(\006)1254 4335 y Fp(q)p 1338 4335 335 4 v 1338 4446 a Fv(\025)p 1338 4459 49 4 v -43 x Fk([)p Fq(n)p Fk(])1386 4469 y Fq(i)1469 4446 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))1672 4326 y Fp(\014)1672 4376 y(\014)1672 4426 y(\014)1672 4475 y(\014)1722 4446 y Fu(\025)23 b Fv(C)1869 4458 y Fk(0)1907 4446 y Fv(\015)1955 4412 y Fn(\003)1950 4469 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))2089 4446 y Fv(;)949 4558 y Fp(\014)949 4608 y(\014)949 4658 y(\014)949 4708 y(\014)977 4679 y Fs(!)21 b Fu(\001)d Fs(\027)24 b Fu(\006)1254 4568 y Fp(q)p 1338 4568 335 4 v 1338 4679 a Fv(\025)p 1338 4692 49 4 v -43 x Fk([)p Fq(n)p Fk(])1386 4702 y Fq(i)1469 4679 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))k Fu(\006)1773 4573 y Fp(q)p 1856 4573 335 4 v 106 x Fv(\025)p 1856 4692 49 4 v 1905 4636 a Fk([)p Fq(n)p Fk(])1905 4702 y Fq(j)1987 4679 y Fy(\()p Fv(";)c Fs(!)s Fy(\))2190 4558 y Fp(\014)2190 4608 y(\014)2190 4658 y(\014)2190 4708 y(\014)2241 4679 y Fu(\025)23 b Fv(C)2388 4691 y Fk(0)2425 4679 y Fv(\015)2473 4644 y Fn(\003)2468 4701 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))3255 4563 y Fy(\(4)p Fv(:)p Fy(2\))1730 4924 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 118 555 a Fy(for)32 b(all)h Fv(i;)14 b(j)36 b Fy(=)31 b(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(d)p Fy(,)34 b(for)e(all)h Fs(\027)k Fu(2)32 b Fr(Z)1414 514 y Fq(r)1414 576 y Fn(\003)1485 555 y Fy(suc)n(h)g(that)h Fv(n)p Fy(\()p Fs(\027)6 b Fy(\))32 b Fu(\025)f Fv(n)2208 567 y Fk(0)2278 555 y Fy(and)h(for)g(all)h Fv(n)e Fu(\025)g Fv(n)2924 567 y Fk(0)2961 555 y Fy(.)53 b(W)-7 b(e)33 b(do)f(not)118 662 y(imp)r(ose)27 b(an)n(y)e(conditions)h(lik)n(e)h (\(4.2\))f(for)g Fv(n)d Fu(\024)f Fv(n)1633 674 y Fk(0)1671 662 y Fy(,)k(b)r(ecause)h(for)f(suc)n(h)g(scales)f(one)i(has)f Fu(j)p Fs(!)19 b Fu(\001)d Fs(\027)6 b Fu(j)23 b Fv(>)g Fy(2)3239 598 y Fu(p)p 3307 598 95 4 v 3307 662 a Fy(\003)3365 674 y Fk(0)3402 662 y Fy(,)118 778 y(so)k(that)g(w)n(e)g(can)g(b)r (ound)h Fu(j)p Fy(\()p Fs(!)20 b Fu(\001)e Fs(\027)6 b Fy(\))1191 747 y Fk(2)1246 778 y Fu(\000)17 b Fv(\025)p 1328 791 49 4 v 1377 734 a Fk([)p Fq(n)p Fk(])1377 801 y Fq(i)1460 778 y Fy(\()p Fv(";)d Fs(!)s Fy(\))p Fu(j)27 b Fy(with)h(\()p Fs(!)20 b Fu(\001)e Fs(\027)6 b Fy(\))2141 747 y Fk(2)2178 778 y Fv(=)p Fy(2)27 b(for)f(all)h Fv(i)c Fy(=)g(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(d)p Fy(.)37 b(In)27 b(the)h(same)118 884 y(w)n(a)n(y)22 b(w)n(e)i(ha)n(v)n(e)e(excluded)h (in)h(\(4.2\))f(the)h(v)-5 b(alues)23 b(of)h Fs(\027)29 b Fu(2)23 b Fr(Z)1902 843 y Fq(r)1902 904 y Fn(\003)1964 884 y Fy(suc)n(h)g(that)g Fv(n)p Fy(\()p Fs(\027)6 b Fy(\))24 b Fu(\024)e Fv(n)2651 896 y Fk(0)2688 884 y Fy(.)36 b(Hence,)24 b(at)g(the)g(price)118 990 y(of)30 b(adding)g(a)g(factor)f(2)844 960 y Fk(2)911 990 y Fy(in)h(the)h(b)r (ound)f(of)g(eac)n(h)g(propagator,)e(w)n(e)h(can)h(con\014ne)g(ourselv) n(es)e(to)i(imp)r(ose)118 1096 y(\(4.3\))d(only)h(for)f Fs(\027)33 b Fy(suc)n(h)27 b(that)h Fv(n)p Fy(\()p Fs(\027)6 b Fy(\))23 b Fu(\025)g Fv(n)1403 1108 y Fk(0)1468 1096 y Fy(and)k(for)g Fv(n)c Fu(\025)g Fv(n)1967 1108 y Fk(0)2004 1096 y Fy(.)189 1203 y(Then)28 b(w)n(e)f(can)g(pro)n(v)n(e)f(the)i (follo)n(wing)f(result.)118 1380 y Ft(Lemma)32 b(13.)k Fw(Cal)t(l)31 b Fv(N)861 1392 y Fq(n)906 1380 y Fy(\()p Fv(\022)r Fy(\))f Fw(the)g(set)g(of)g(lines)g(in)g Fy(\003\()p Fv(\022)r Fy(\))g Fw(which)i(ar)l(e)e(on)g(sc)l(ale)g Fy([)p Fv(n)p Fy(])p Fw(.)39 b(One)29 b(has)937 1553 y Fv(N)1004 1565 y Fq(n)1049 1553 y Fy(\()p Fv(\022)r Fy(\))24 b Fu(\024)e Fv(K)e Fy(2)1398 1519 y Fn(\000)p Fq(n)1494 1553 y Fv(M)9 b Fy(\()p Fv(\022)r Fy(\))p Fv(;)184 b(M)9 b Fy(\()p Fv(\022)r Fy(\))24 b(=)2256 1474 y Fp(X)2202 1656 y Fd(v)p Fn(2)p Fq(V)14 b Fk(\()p Fq(\022)r Fk(\))2443 1553 y Fu(j)p Fs(\027)2514 1565 y Fd(v)2560 1553 y Fu(j)p Fv(;)649 b Fy(\(4)p Fv(:)p Fy(3\))118 1802 y Fw(for)31 b(a)f(suitable)g(c)l(onstant)f Fv(K)6 b Fw(.)38 b(One)29 b(c)l(an)h(take)g Fv(K)f Fy(=)22 b(2)p Fw(.)118 1979 y(Pr)l(o)l(of.)38 b Fy(The)25 b(pro)r(of)g(pro)r(ceeds)g(exactly)g(as)f (for)h(Lemma)h(3,)f(with)h(the)g(only)f(di\013erence)g(that)h(w)n(e)f (ha)n(v)n(e)f(to)118 2086 y(deal)30 b(in)h(a)g(di\013eren)n(t)f(w)n(a)n (y)g(with)h(the)g(lines)g(on)f(scales)g Fv(n)e(<)g(n)2076 2098 y Fk(0)2143 2086 y Fy(and)j(those)f(with)h(scales)f Fv(n)e Fu(\025)g Fv(n)3176 2098 y Fk(0)3213 2086 y Fy(.)46 b(The)118 2192 y(same)27 b(w)n(as)g(done)g(in)h(Ref.)g([23].)p 3384 2184 42 42 v 189 2369 a(In)g(the)g(same)f(w)n(a)n(y)f(the)i(follo) n(wing)f(result)g(is)g(pro)n(v)n(ed.)118 2546 y Ft(Lemma)32 b(14.)k Fw(Cal)t(l)31 b Fv(N)861 2558 y Fq(n)906 2546 y Fy(\()p Fv(T)12 b Fy(\))29 b Fw(the)h(set)f(of)i(lines)f(in)g Fy(\003\()p Fv(T)12 b Fy(\))29 b Fw(which)i(ar)l(e)f(on)g(sc)l(ale)h Fy([)p Fv(n)p Fy(])p Fw(.)38 b(One)29 b(has)704 2720 y Fv(M)9 b Fy(\()p Fv(T)j Fy(\))22 b(=)1089 2641 y Fp(X)1029 2823 y Fd(v)o Fn(2)p Fq(V)15 b Fk(\()p Fq(T)9 b Fk(\))1283 2720 y Fu(j)p Fs(\027)1354 2732 y Fd(v)1401 2720 y Fu(j)23 b Fv(>)g Fy(2)1577 2685 y Fq(n)1618 2693 y Fl(T)1663 2685 y Fn(\000)p Fk(1)1753 2720 y Fv(;)183 b(N)2026 2732 y Fq(n)2071 2720 y Fy(\()p Fv(T)12 b Fy(\))23 b Fu(\024)f Fv(K)e Fy(2)2439 2685 y Fn(\000)p Fq(n=)p Fk(2)2602 2720 y Fv(M)9 b Fy(\()p Fv(T)j Fy(\))p Fv(;)415 b Fy(\(4)p Fv(:)p Fy(4\))118 2969 y Fw(for)31 b(a)f(suitable)g(c)l(onstant)f Fv(K)6 b Fw(.)189 3146 y Fy(Then)28 b(Lemma)f(5)g(is)h(replaced)e(with) j(the)f(follo)n(wing)e(one.)118 3323 y Ft(Lemma)44 b(15.)67 b Fw(Assume)39 b(that)g(the)h(pr)l(op)l(agators)h Fv(g)1804 3293 y Fk([)p Fq(p)p Fk(])1879 3323 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))40 b Fw(c)l(an)f(b)l(e)h(uniformly)g(b)l (ounde)l(d)g(for)g(al)t(l)118 3429 y Fy(0)24 b Fu(\024)g Fv(p)g Fu(\024)f Fv(n)c Fu(\000)g Fy(1)29 b Fw(as)i(\(3.14\),)i(for)e (some)g Fv(p)p Fw(-indep)l(endent)f(c)l(onstant)g Fv(K)6 b Fw(.)40 b(Assume)29 b(also)j(that)e Fv(")3077 3441 y Fk(0)3145 3429 y Fw(is)g(smal)t(l)118 3536 y(enough.)39 b(With)30 b(the)g(notations)g(\(3.3\))h(one)f(has)946 3622 y Fp(\015)946 3672 y(\015)946 3721 y(\015)992 3717 y Fu(M)1092 3683 y Fk([)p Fq(n)p Fk(])1092 3738 y Fq(\013\013)1182 3717 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))1469 3622 y Fp(\015)1469 3672 y(\015)1469 3721 y(\015)1538 3717 y Fu(\024)23 b Fv(B)t Fy(e)1730 3683 y Fn(\000)p Fq(\024)1821 3691 y Fj(1)1853 3683 y Fk(2)1886 3658 y Fl(n=)p Fj(2)2004 3717 y Fy(min)p Fu(f)p Fv(")2223 3683 y Fk(2)2260 3717 y Fv(;)14 b("x)2383 3683 y Fk(2)2420 3717 y Fu(g)p Fv(;)948 3804 y Fp(\015)948 3854 y(\015)948 3904 y(\015)994 3900 y Fu(M)1094 3857 y Fk([)p Fq(n)p Fk(])1094 3925 y Fq(\013\014)1182 3900 y Fy(\()p Fv(x)p Fy(;)g Fv(";)g Fs(!)s Fy(\))1469 3804 y Fp(\015)1469 3854 y(\015)1469 3904 y(\015)1538 3900 y Fu(\024)23 b Fv(B)t Fy(e)1730 3866 y Fn(\000)p Fq(\024)1821 3874 y Fj(1)1853 3866 y Fk(2)1886 3841 y Fl(n=)p Fj(2)2004 3900 y Fy(min)p Fu(f)p Fv(")2223 3866 y Fk(2)2260 3900 y Fv(;)14 b(")2336 3866 y Fk(3)p Fq(=)p Fk(2)2440 3900 y Fu(j)p Fv(x)p Fu(jg)p Fv(;)951 3987 y Fp(\015)951 4037 y(\015)951 4087 y(\015)997 4083 y Fu(M)1097 4039 y Fk([)p Fq(n)p Fk(])1097 4108 y Fq(\014)s(\014)1182 4083 y Fy(\()p Fv(x)p Fy(;)g Fv(";)g Fs(!)s Fy(\))1469 3987 y Fp(\015)1469 4037 y(\015)1469 4087 y(\015)1538 4083 y Fu(\024)23 b Fv(B)t Fy(e)1730 4048 y Fn(\000)p Fq(\024)1821 4056 y Fj(1)1853 4048 y Fk(2)1886 4023 y Fl(n=)p Fj(2)1990 4083 y Fv(")2029 4048 y Fk(2)2066 4083 y Fv(;)3255 3900 y Fy(\(4)p Fv(:)p Fy(5\))118 4264 y Fw(for)31 b(al)t(l)g Fv(n)22 b Fu(2)i Fr(N)29 b Fw(and)i(for)f(suitable)g Fv(n)p Fw(-indep)l(endent)g(c)l(onstants)f Fv(B)34 b Fw(and)c Fv(\024)2421 4276 y Fk(1)2458 4264 y Fw(.)189 4441 y Fy(Therefore)g(w)n(e)g(can)h(pro)n(v)n(e)e(the)j(follo)n(wing)e (estimates.)47 b(The)31 b(pro)r(of)f(is)h(the)h(same)e(as)g(for)h (Lemma)g(6,)118 4547 y(as)c(one)g(easily)g(realizes)f(that)i(it)g(w)n (orks)e(for)h(\014xed)h(v)-5 b(alues)27 b(of)h Fv(")f Fy(and)g Fs(!)s Fy(.)118 4724 y Ft(Lemma)44 b(16.)68 b Fw(L)l(et)40 b Fs(!)i Fw(satisfy)f(the)f(Diophantine)h(c)l(ondition)f (\(1.4\))h(and)g(assume)e(that)h Fv(")f Fw(is)h(smal)t(l)1730 4924 y Fy(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 118 555 a Fw(enough,)28 b(say)e Fu(j)p Fv(")p Fu(j)d Fv(<)f(")799 567 y Fk(0)836 555 y Fw(.)38 b(Then)26 b(the)g(series)g(\(3.7\))h(admits)g(the)f(b)l(ound) f Fu(j)p Fv(h)2385 567 y Fo(\027)5 b Fq(;\015)2491 555 y Fu(j)23 b Fv(<)f(A)14 b Fy(e)2737 525 y Fn(\000)p Fq(\024)p Fn(j)p Fo(\027)5 b Fn(j)2915 555 y Fu(j)p Fv(")p Fu(j)25 b Fw(for)i(suitable)118 662 y(p)l(ositive)k(c)l(onstants)e Fv(\024)h Fw(and)g Fv(A)p Fw(.)39 b(One)29 b(has)i Fv(")1513 674 y Fk(0)1573 662 y Fy(=)22 b Fv(O)r Fy(\()p Fv(C)1822 632 y Fk(2)1816 682 y(0)1861 662 y Fy(\()p Fv(\015)1941 632 y Fn(\003)1936 682 y Fq(m)1995 690 y Fj(0)2031 662 y Fy(\))2063 632 y Fk(2)2101 662 y Fy(\))30 b Fw(with)g Fv(m)2416 674 y Fk(0)2483 662 y Fw(dep)l(ending)h(on)f Fs(!)i Fw(and)f Fv(\024)3290 674 y Fk(0)3327 662 y Fw(.)189 839 y Fy(With)f(resp)r(ect)g(to)f(Section)h(3)f(the)h(\014rst)f (di\013erences)h(app)r(ear)e(when)i(dealing)f(with)h(Whitney)g(exten-) 118 945 y(sions)22 b(of)h(the)g(matrices)f Fu(M)973 915 y Fk([)p Fn(\024)p Fq(n)p Fk(])1107 945 y Fy(\()p Fv(x)p Fy(;)i Fv(";)14 b Fs(!)s Fy(\):)34 b(indeed)23 b(no)n(w)f Fs(!)j Fy(is)e(assumed)f(to)h(b)r(e)g(\014xed,)g(while)g Fv(")g Fy(is)f(the)i(free)118 1052 y(parameter.)47 b(W)-7 b(e)32 b(de\014ne)f Fu(E)993 1064 y Fq(n)1034 1072 y Fj(0)1100 1052 y Fu(\021)e Fy(\()p Fv(")1265 1064 y Fk(0)1302 1052 y Fv(=)p Fy(4)p Fv(;)14 b(")1462 1064 y Fk(0)1498 1052 y Fy(])31 b(and)h(for)e Fv(n)g(>)f(n)2072 1064 y Fk(0)2109 1052 y Fy(,)j(recursiv)n(ely)-7 b(,)31 b Fu(E)2643 1064 y Fq(n)2717 1052 y Fy(=)e Fu(E)2855 1064 y Fq(n)p Fn(\000)p Fk(1)3006 1052 y Fu(n)20 b(E)3119 1021 y Fq(o)3112 1072 y(n)3157 1052 y Fy(,)33 b(where)118 1158 y Fu(E)169 1128 y Fq(o)162 1178 y(n)236 1158 y Fy(is)c(the)g(set)g(of)f(v)-5 b(alues)29 b(of)g Fv(")24 b Fu(2)i(E)1224 1170 y Fq(n)1298 1158 y Fy(suc)n(h)i(that)h(the)h(conditions)e(\(4.2\))g(are)g (violated.)40 b(W)-7 b(e)29 b(de\014ne)g(also)118 1264 y Fu(E)162 1276 y Fn(\003)223 1264 y Fy(=)23 b Fu(\\)366 1234 y Fn(1)366 1285 y Fq(n)p Fk(=)p Fq(n)499 1293 y Fj(0)536 1264 y Fu(E)580 1276 y Fq(n)625 1264 y Fy(.)118 1441 y Ft(Lemma)32 b(17.)k Fw(F)-6 b(or)30 b(al)t(l)g Fv(n)23 b Fu(\025)g Fy(0)29 b Fw(and)h(al)t(l)h Fv(";)14 b(")1516 1411 y Fn(0)1562 1441 y Fu(2)23 b(E)1684 1453 y Fq(n)1759 1441 y Fw(one)30 b(has)302 1510 y Fp(\015)302 1559 y(\015)302 1609 y(\015)348 1605 y Fu(M)448 1571 y Fk([)p Fn(\024)p Fq(n)p Fk(])582 1605 y Fy(\()p Fv(x)p Fy(;)14 b Fv(")737 1571 y Fn(0)761 1605 y Fv(;)g Fs(!)s Fy(\))k Fu(\000)g(M)1094 1571 y Fk([)p Fn(\024)p Fq(n)p Fk(])1229 1605 y Fy(\()p Fv(x)p Fy(;)c Fv(";)g Fs(!)s Fy(\))k Fu(\000)g Fv(@)1661 1617 y Fq(")1697 1605 y Fu(M)1797 1571 y Fk([)p Fn(\024)p Fq(n)p Fk(])1931 1605 y Fy(\()p Fv(x)p Fy(;)c Fv(";)g Fs(!)s Fy(\))g(\()p Fv(")2303 1571 y Fn(0)2345 1605 y Fu(\000)k Fv(")p Fy(\))2499 1510 y Fp(\015)2499 1559 y(\015)2499 1609 y(\015)2568 1605 y Fy(=)23 b Fv(")14 b(o)p Fy(\()p Fu(j)p Fv(")2843 1571 y Fn(0)2885 1605 y Fu(\000)k Fv(")p Fu(j)p Fy(\))p Fv(;)302 1692 y Fp(\015)302 1742 y(\015)302 1792 y(\015)348 1788 y Fv(@)392 1800 y Fq(")428 1788 y Fu(M)528 1753 y Fk([)p Fn(\024)p Fq(n)p Fk(])662 1788 y Fy(\()p Fv(x)p Fy(;)c Fv(";)g Fs(!)s Fy(\))949 1692 y Fp(\015)949 1742 y(\015)949 1792 y(\015)1018 1788 y Fu(\024)23 b Fv(B)t(;)3255 1696 y Fy(\(4)p Fv(:)p Fy(6\))118 1952 y Fw(and,)31 b(as)f(a)g(c)l(onse)l (quenc)l(e,)646 2115 y Fv(B)713 2081 y Fn(0)759 2115 y Fu(\024)847 2020 y Fp(\014)847 2070 y(\014)847 2119 y(\014)875 2115 y Fv(@)919 2127 y Fq(")954 2115 y Fv(\025)p 954 2128 49 4 v 1003 2072 a Fk([)p Fq(n)p Fk(])1003 2138 y Fq(j)1085 2115 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))1288 2020 y Fp(\014)1288 2070 y(\014)1288 2119 y(\014)1339 2115 y Fu(\024)23 b Fv(B)t(;)183 b(r)22 b Fy(+)c(1)k Fu(\024)h Fv(j)28 b Fu(\024)22 b Fv(d;)646 2298 y(B)713 2264 y Fn(0)759 2298 y Fu(\024)847 2202 y Fp(\014)847 2252 y(\014)847 2302 y(\014)875 2298 y Fv(@)919 2310 y Fq(")968 2206 y Fp(\020)1018 2298 y Fv(\025)p 1018 2311 V -43 x Fk([)p Fq(n)p Fk(])1066 2321 y Fq(i)1149 2298 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))k Fu(\006)g Fv(\025)p 1453 2311 V 1502 2255 a Fk([)p Fq(n)p Fk(])1502 2321 y Fq(j)1584 2298 y Fy(\()p Fv(";)c Fs(!)s Fy(\))1787 2206 y Fp(\021)1837 2202 y(\014)1837 2252 y(\014)1837 2302 y(\014)1887 2298 y Fu(\024)23 b Fv(B)t(;)184 b(r)21 b Fy(+)d(1)23 b Fu(\024)f Fv(j)28 b(<)23 b(i)g Fu(\024)f Fv(d;)3255 2207 y Fy(\(4)p Fv(:)p Fy(7\))118 2467 y Fw(for)31 b(suitable)f(p)l(ositive)h(c)l(onstants)e Fv(B)34 b Fw(and)c Fv(B)1541 2436 y Fn(0)1565 2467 y Fw(.)118 2644 y(Pr)l(o)l(of.)58 b Fy(The)34 b(pro)r(of)f(can)h(b)r(e)h(p)r(erformed)e(as)g(for)h(Lemma) g(10,)h(with)f(the)h(parameter)d Fv(")i Fy(no)n(w)f(pla)n(ying)118 2750 y(the)42 b(role)e(of)i(the)f(parameters)f Fs(!)s Fy(.)78 b(W)-7 b(e)42 b(do)f(not)g(giv)n(e)g(the)g(details,)k(whic)n (h,)g(ho)n(w)n(ev)n(er,)e(ha)n(v)n(e)d(b)r(een)118 2856 y(w)n(ork)n(ed)35 b(out)i(in)g(Ref.)g([23].)64 b(Again)36 b(the)i(upp)r(er)f(b)r(ound)g(\(4.7\))f(follo)n(ws)g(from)h(\(4.6\);)k (cf.)65 b(analogous)118 2963 y(commen)n(ts)33 b(in)g(the)h(pro)r(of)f (of)g(Lemma)g(6.)53 b(T)-7 b(o)33 b(obtain)g(the)h(lo)n(w)n(er)d(b)r (ound)j(one)f(has)g(to)g(use)g(also)f(that)118 3069 y Fv(\025)166 3081 y Fq(j)202 3069 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))25 b(=)f Fv(a)647 3081 y Fq(j)682 3069 y Fv(")19 b Fy(+)f Fv(O)r Fy(\()p Fv(")959 3039 y Fk(2)997 3069 y Fy(\),)30 b(with)f Fv(a)1316 3081 y Fq(j)1375 3069 y Fu(6)p Fy(=)c(0)j(and)h Fv(a)1742 3081 y Fq(i)1794 3069 y Fu(6)p Fy(=)24 b Fv(a)1927 3081 y Fq(j)1991 3069 y Fy(for)k Fv(i;)14 b(j)29 b Fy(=)c Fv(r)d Fy(+)c(1)p Fv(;)c(:)g(:)g(:)f(;)h(d)p Fy(;)30 b(again)d(cf.)41 b(Ref.[23])118 3175 y(for)27 b(details.)p 3384 3167 42 42 v 189 3352 a(Lemma)h(17)g(implies)h(that)g(the)g(matrices)f Fu(M)1640 3322 y Fk([)p Fn(\024)p Fq(n)p Fk(])1774 3352 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))29 b(can)f(b)r(e)h (extended)g(in)g(\(0)p Fv(;)14 b(")2959 3364 y Fk(0)2996 3352 y Fy(\))29 b(to)g(smo)r(oth)118 3459 y Fv(C)183 3429 y Fk(1)252 3459 y Fy(functions)j(\(Whitney)h(extensions\).)48 b(Again)31 b(a)g(closeness)f(prop)r(ert)n(y)h(of)g(the)h (self-energies,)f(whic)n(h)118 3565 y(reads)1030 3576 y Fp(\014)1030 3626 y(\014)1030 3675 y(\014)1058 3671 y Fv(\025)p 1058 3684 49 4 v -43 x Fk([)p Fq(n)p Fk(])1106 3694 y Fq(j)1189 3671 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))k Fu(\000)g Fv(\025)p 1493 3684 V -43 x Fk([)p Fq(n)p Fn(\000)p Fk(1])1541 3694 y Fq(j)1709 3671 y Fy(\()p Fv(";)c Fs(!)s Fy(\))1912 3576 y Fp(\014)1912 3626 y(\014)1912 3675 y(\014)1963 3671 y Fu(\024)22 b Fv(B)t Fy(e)2154 3637 y Fn(\000)p Fq(\024)2245 3645 y Fj(1)2278 3637 y Fk(2)2311 3612 y Fl(n=)p Fj(2)2414 3671 y Fv(")2453 3637 y Fk(2)2490 3671 y Fv(;)742 b Fy(\(4)p Fv(:)p Fy(8\))118 3829 y(follo)n(ws)33 b(from)h(Lemma)f(15.)56 b(As)34 b(b)r(efore)f(the)i(b)r(ounds)f (\(4.8\))f(can)h(b)r(e)g(impro)n(v)n(ed)f(for)h(the)g(\014rst)g Fv(r)i Fy(self-)118 3945 y(energies,)e(and)g(giv)n(e)f Fu(j)p Fv(\025)879 3902 y Fk([)p Fq(n)p Fk(])879 3968 y Fq(j)963 3945 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))p Fu(j)34 b(\024)f Fv(A)14 b Fy(min)p Fu(f)p Fv(")1700 3915 y Fk(2)1737 3945 y Fv(;)g("x)1860 3915 y Fk(2)1897 3945 y Fu(g)p Fy(,)35 b Fv(j)k Fy(=)33 b(1)p Fv(;)14 b(:)g(:)g(:)f(r)r Fy(.)57 b(What)34 b(really)f(c)n(hanges)g(with)118 4051 y(resp)r(ect)27 b(to)g(the)h(previous)e(case)g(is)h(the)h (estimate)f(of)h(the)f(set)g(of)h(allo)n(w)n(ed)d(v)-5 b(alues)27 b(of)h Fv(")p Fy(,)f(whic)n(h)g(explains)118 4157 y(wh)n(y)40 b(w)n(e)f(ha)n(v)n(e)g(required)g(the)h(stronger)e (condition)i(on)g Fv(@)2027 4127 y Fk(2)2022 4187 y Fo(\014)2072 4157 y Fv(f)2113 4169 y Fi(0)2154 4157 y Fy(\()p Fs(\014)2241 4169 y Fk(0)2279 4157 y Fy(\))g(that)g(its)g(eigen)n(v)-5 b(alues)39 b(are)g(non-)118 4264 y(degenerate.)d(The)27 b(follo)n(wing)g(result)g(holds.)118 4441 y Ft(Lemma)j(18.)35 b Fw(The)29 b(L)l(eb)l(esgue)g(me)l(asur)l(e)e(of)j(the)e(set)g Fy(\(0)p Fv(;)14 b(")1953 4453 y Fk(0)1990 4441 y Fy(\))h Fu(n)g(E)2145 4411 y Fn(\003)2211 4441 y Fw(is)29 b(b)l(ounde)l(d)f(pr) l(op)l(ortional)t(ly)k(to)c(some)118 4547 y(value)i Fv(G)p Fy(\()p Fv(")468 4559 y Fk(0)506 4547 y Fy(\))p Fw(,)g(with)h Fv(G)p Fy(\()p Fv(")p Fy(\))23 b(=)g Fv(o)p Fy(\()p Fv(")p Fy(\))p Fw(.)118 4724 y(Pr)l(o)l(of.)45 b Fy(As)30 b(in)g(the)g(pro)r (of)f(of)h(Lemma)g(12)f(w)n(e)g(start)g(with)i(the)f(\014rst)f (conditions)h(in)g(\(4.2\).)43 b(By)29 b(setting)1730 4924 y(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 118 580 a Fv(")29 b Fy(=)h Fv(")p Fy(\()p Fv(t)p Fy(\),)j(with)f Fv(t)d Fu(2)h Fy([)p Fu(\000)p Fy(1)p Fv(;)14 b Fy(1],)32 b(and)f(de\014ning)g Fv(F)12 b Fy(\()p Fv(t)p Fy(\))31 b(=)e Fs(!)24 b Fu(\001)d Fs(\027)26 b Fu(\006)2148 479 y Fp(q)p 2231 479 429 4 v 101 x Fv(\025)p 2231 593 49 4 v -43 x Fk([)p Fq(n)p Fk(])2279 603 y Fq(j)2362 580 y Fy(\()p Fv(")p Fy(\()p Fv(t)p Fy(\))p Fv(;)14 b Fs(!)s Fy(\))30 b(=)f Fv(tC)2872 592 y Fk(0)2910 580 y Fv(\015)2958 550 y Fn(\003)2953 607 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))3093 580 y Fy(,)33 b(one)e(has)118 741 y Fu(j)p Fy(d)p Fv(F)7 b(=)p Fy(d)p Fv(t)p Fu(j)23 b Fy(=)g Fu(j)p Fv(@)566 753 y Fq(")602 640 y Fp(q)p 685 640 335 4 v 101 x Fv(\025)p 685 754 49 4 v -43 x Fk([)p Fq(n)p Fk(])733 765 y Fq(j)816 741 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))p Fu(j)g(j)p Fy(d)p Fv("=)p Fy(d)p Fv(t)p Fu(j)22 b Fy(=)h Fv(C)1474 753 y Fk(0)1511 741 y Fv(\015)1559 711 y Fn(\003)1554 768 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))1694 741 y Fy(,)26 b(so)d(that,)j(b)n(y)e(using)g(that)h Fv(\025)p 2545 754 V -43 x Fk([)p Fq(n)p Fk(])2593 765 y Fq(j)2676 741 y Fy(\()p Fv(";)14 b Fs(!)s Fy(\))23 b(=)g Fv(a)3034 753 y Fq(j)3068 741 y Fv(")12 b Fy(+)g Fv(O)r Fy(\()p Fv(")3332 711 y Fk(2)3370 741 y Fy(\),)118 861 y(one)36 b(\014nds)g Fu(j)p Fy(d)p Fv("=)p Fy(d)p Fv(t)p Fu(j)i(\024)f Fv(B)t(C)1001 873 y Fk(0)1038 801 y Fu(p)p 1108 801 39 4 v 1108 861 a Fv(")o(\015)1194 831 y Fn(\003)1189 888 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))1366 861 y Fy(for)35 b(some)h(constan)n(t)f Fv(B)t Fy(.)64 b(Again,)38 b(for)d(\014xed)i Fs(\027)42 b Fy(and)36 b Fv(i)p Fy(,)i(b)n(y)e(the)118 967 y(closeness)28 b(prop)r(ert)n(y)f(of)i(the)h(self-energies,)d(w)n (e)i(can)f(imp)r(ose)h(only)g(the)g(conditions)f(corresp)r(onding)f(to) 118 1074 y(the)33 b(scales)f(up)h(to)g Fv(n)785 1086 y Fk(1)822 1074 y Fy(\()p Fs(\027)6 b Fy(\))32 b(=)g Fv(O)r Fy(\(log)15 b(log)f(1)p Fv(=\015)1541 1043 y Fn(\003)1536 1100 y Fq(n)p Fk(\()p Fo(\027)5 b Fk(\))1675 1074 y Fy(\),)35 b(at)d(the)i(price)e(of)h(enlarging)e(the)j(sets)e(of)h(excluded)118 1197 y(v)-5 b(alues)28 b(\(b)n(y)h(a)f(factor)g(2\).)40 b(Hence)29 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Fv(\015)2548 3294 y Fn(\003)2543 3349 y Fq(n)2589 3329 y Fy(\))2621 3294 y Fn(\000)p Fk(2)2710 3329 y Fu(j)p Fv(x)2780 3294 y Fn(0)2822 3329 y Fu(\000)k Fv(x)p Fu(j)p Fv(;)194 b Fy(\()p Fv(A)p Fy(1)p Fv(:)p Fy(5\))118 3468 y Fw(for)31 b(a)f(suitable)g(p)l(ositive)h(c)l(onstant) e Fy(\010)p Fw(.)118 3645 y(Pr)l(o)l(of.)39 b Fy(W)-7 b(e)28 b(can)f(write)788 3833 y Fu(j)p Fv(\037)863 3845 y Fq(n)908 3833 y Fy(\()p Fv(x)987 3799 y Fn(0)1011 3833 y Fy(\))19 b Fu(\000)f Fv(\037)1197 3845 y Fq(n)1242 3833 y Fy(\()p Fv(x)p Fy(\))p Fu(j)24 b(\024)f Fv(\014)1539 3799 y Fn(\000)p Fk(2)1628 3833 y Fy(\()p Fv(\015)1708 3799 y Fn(\003)1703 3854 y Fq(n)1749 3833 y Fy(\))1781 3799 y Fn(\000)p Fk(2)1884 3833 y Fu(j)p Fv(x)1954 3799 y Fn(0)1996 3833 y Fu(\000)18 b Fv(x)p Fu(j)2164 3720 y Fp(Z)2247 3741 y Fk(1)2210 3909 y(0)2298 3833 y Fy(d)p Fv(t)c(@)2432 3845 y Fq(x)2474 3833 y Fv(\037)p Fy(\()p Fv(x)p Fy(\()p Fv(t)p Fy(\)\))p Fv(;)438 b Fy(\()p Fv(A)p Fy(1)p Fv(:)p Fy(6\))118 4018 y(where)27 b Fv(x)p Fy(\()p Fv(t)p Fy(\))d(=)f Fv(\014)662 3988 y Fn(\000)p Fk(2)751 4018 y Fy(\()p Fv(\015)831 3988 y Fn(\003)826 4038 y Fq(n)872 4018 y Fy(\))904 3988 y Fn(\000)p Fk(2)993 4018 y Fy(\()p Fv(x)c Fy(+)f Fv(t)p Fy(\()p Fv(x)1283 3988 y Fn(0)1326 4018 y Fu(\000)g Fv(x)p Fy(\)\))29 b(and)e Fu(j)p Fv(@)1777 4030 y Fq(x)1819 4018 y Fv(\037)p Fy(\()p Fv(x)p Fy(\()p Fv(t)p Fy(\)\))p Fu(j)e(\024)d Fy(const)p Fv(:)p 3384 4010 42 42 v 189 4195 a Fy(By)32 b(noting)h(that)f Fv( )826 4207 y Fq(n)903 4195 y Fy(=)f(1)21 b Fu(\000)h Fv(\037)1201 4207 y Fq(n)1246 4195 y Fy(,)34 b(w)n(e)e(see)g(that)h (Lemma)g(A1)f(yields)h(the)g(same)f(b)r(ounds)g(as)g(\(A1.6\))118 4301 y(also)27 b(if)h(w)n(e)f(replace)g Fv(\037)817 4313 y Fq(n)889 4301 y Fy(with)h Fv( )1132 4313 y Fq(n)1178 4301 y Fy(.)118 4479 y Ft(Lemma)33 b(A2.)38 b Fw(F)-6 b(or)31 b Fs(!)s Fv(;)14 b Fs(!)962 4448 y Fn(0)1009 4479 y Fu(2)25 b Fy(\012)1149 4491 y Fn(\003)1187 4479 y Fy(\()p Fv(C)1278 4491 y Fk(0)1316 4479 y Fy(\))30 b Fw(assume)h(that)f(the)h(b)l(ounds)f(\(3.28\))i(hold)g(for)g(al)t(l)f Fv(n)2986 4448 y Fn(0)3034 4479 y Fu(\024)24 b Fv(n)p Fw(.)40 b(Then)118 4585 y(one)30 b(has)135 4724 y Fu(k)p Fv(g)220 4690 y Fk([)p Fq(n)p Fk(])302 4724 y Fy(\()p Fs(!)397 4690 y Fn(0)439 4724 y Fu(\001)18 b Fs(\027)6 b Fy(;)14 b Fv(";)g Fs(!)710 4690 y Fn(0)732 4724 y Fy(\))19 b Fu(\000)f Fv(g)909 4690 y Fk([)p Fq(n)p Fk(])991 4724 y Fy(\()p Fs(!)k Fu(\001)c Fs(\027)6 b Fy(;)14 b Fv(";)g Fs(!)s Fy(\))k Fu(\000)g Fv(@)1553 4736 y Fo(!)1607 4724 y Fv(g)1650 4690 y Fk([)p Fq(n)p Fk(])1733 4724 y Fy(\()p Fs(!)j Fu(\001)e Fs(\027)6 b Fy(;)14 b Fv(";)g Fs(!)s Fy(\))g(\()p Fs(!)2259 4690 y Fn(0)2299 4724 y Fu(\000)k Fs(!)s Fy(\))p Fu(k)23 b Fy(=)f(\()p Fv(\015)2709 4690 y Fn(\003)2704 4745 y Fq(n)2750 4724 y Fy(\))2782 4690 y Fn(\000)p Fq(\016)2870 4724 y Fu(j)p Fs(\027)6 b Fu(j)14 b Fv(o)p Fy(\()p Fu(j)p Fs(!)3142 4690 y Fn(0)3184 4724 y Fu(\000)k Fs(!)s Fu(j)p Fy(\))p Fv(;)1730 4924 y Fy(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 135 555 a Fu(k)p Fv(@)221 567 y Fo(!)275 555 y Fv(g)318 521 y Fk([)p Fq(n)p Fk(])400 555 y Fy(\()p Fs(!)22 b Fu(\001)c Fs(\027)6 b Fy(;)14 b Fv(";)g Fs(!)s Fy(\))p Fu(k)22 b(\024)h Fv(D)15 b Fy(\()p Fv(\015)1133 521 y Fn(\003)1128 576 y Fq(n)1174 555 y Fy(\))1206 521 y Fn(\000)p Fq(\016)1295 555 y Fu(j)p Fs(\027)6 b Fu(j)p Fv(;)1774 b Fy(\()p Fv(A)p Fy(1)p Fv(:)p Fy(7\))118 740 y Fw(for)31 b(suitable)f(p)l(ositive)h(c)l (onstants)e Fv(D)j Fw(and)e Fv(\016)s Fw(.)118 917 y(Pr)l(o)l(of.)39 b Fy(By)27 b(using)g(the)h(de\014nition)g(\(3.5\))f(w)n(e)h(ha)n(v)n(e) 321 1101 y Fv(g)364 1067 y Fk([)p Fq(n)p Fk(])447 1101 y Fy(\()p Fs(!)542 1067 y Fn(0)583 1101 y Fu(\001)19 b Fs(\027)6 b Fy(;)14 b Fv(";)g Fs(!)855 1067 y Fn(0)877 1101 y Fy(\))19 b Fu(\000)f Fv(g)1054 1067 y Fk([)p Fq(n)p Fk(])1136 1101 y Fy(\()p Fs(!)k Fu(\001)c Fs(\027)6 b Fy(;)14 b Fv(";)g Fs(!)s Fy(\))22 b(=)h(\011)1728 1113 y Fq(n)1773 1101 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)1959 1067 y Fn(0)1981 1101 y Fy(\))p Fv(D)2084 1067 y Fn(\000)p Fk(1)2082 1122 y Fq(n)2174 1101 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)2360 1067 y Fn(0)2383 1101 y Fy(\))k Fu(\000)g Fy(\011)2581 1113 y Fq(n)2626 1101 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)s Fy(\))p Fv(D)2915 1067 y Fn(\000)p Fk(1)2913 1122 y Fq(n)3004 1101 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)s Fy(\))511 1233 y(=)22 b Fu(\000)p Fy(\011)728 1245 y Fq(n)773 1233 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)s Fy(\))p Fv(D)1062 1198 y Fn(\000)p Fk(1)1060 1253 y Fq(n)1150 1233 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)1336 1198 y Fn(0)1359 1233 y Fy(\))g(\()q Fv(D)1507 1245 y Fq(n)1552 1233 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)1738 1198 y Fn(0)1760 1233 y Fy(\))19 b Fu(\000)f Fv(D)1963 1245 y Fq(n)2008 1233 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)s Fy(\)\))g Fv(D)2343 1198 y Fn(\000)p Fk(1)2341 1253 y Fq(n)2432 1233 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)s Fy(\))672 1424 y(+)794 1320 y Fq(n)755 1345 y Fp(X)756 1521 y Fq(p)p Fk(=0)889 1424 y Fy(\011)954 1436 y Fq(n;p)1052 1424 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)1238 1389 y Fn(0)1261 1424 y Fv(;)g Fs(!)s Fy(\))p Fv(D)1464 1389 y Fn(\000)p Fk(1)1462 1444 y Fq(n)1553 1424 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)1739 1389 y Fn(0)1762 1424 y Fy(\))p Fv(;)1375 b Fy(\()p Fv(A)p Fy(1)p Fv(:)p Fy(8\))118 1684 y(where)27 b(w)n(e)g(can)h(write)341 1869 y Fv(D)410 1881 y Fq(n)455 1869 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)641 1834 y Fn(0)664 1869 y Fy(\))k Fu(\000)g Fv(D)866 1881 y Fq(n)911 1869 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)s Fy(\))2063 b(\()p Fv(A)p Fy(1)p Fv(:)p Fy(9\))530 2000 y(=)23 b(\()p Fs(!)713 1966 y Fn(0)754 2000 y Fu(\001)c Fs(\027)24 b Fy(+)18 b Fs(!)j Fu(\001)d Fs(\027)6 b Fy(\)\(\()p Fs(!)1286 1966 y Fn(0)1328 2000 y Fu(\000)18 b Fs(!)s Fy(\))h Fu(\001)f Fs(\027)6 b Fy(\))19 b Fu(\000)f(M)1854 1966 y Fk([)p Fn(\024)p Fq(n)p Fk(])1988 2000 y Fy(\()p Fs(!)2083 1966 y Fn(0)2125 2000 y Fu(\001)g Fs(\027)6 b Fy(;)14 b Fv(";)g Fs(!)2396 1966 y Fn(0)2418 2000 y Fy(\))19 b(+)f Fu(M)2652 1966 y Fk([)p Fn(\024)p Fq(n)p Fk(])2786 2000 y Fy(\()p Fs(!)k Fu(\001)c Fs(\027)6 b Fy(;)14 b Fv(";)g Fs(!)s Fy(\))118 2184 y(so)27 b(that)h(w)n(e)f (obtain)130 2369 y Fv(D)199 2381 y Fq(n)244 2369 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)430 2334 y Fn(0)453 2369 y Fy(\))k Fu(\000)g Fv(D)655 2381 y Fq(n)700 2369 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)s Fy(\))2233 b(\()p Fv(A)p Fy(1)p Fv(:)p Fy(10\))319 2518 y(=)407 2426 y Fp(\020)456 2518 y Fy(2)p Fs(\027)6 b Fy(\()p Fs(!)21 b Fu(\001)d Fs(\027)6 b Fy(\))19 b Fu(\000)f Fv(@)938 2530 y Fo(!)992 2518 y Fu(M)1092 2484 y Fk([)p Fn(\024)p Fq(n)p Fk(])1226 2518 y Fy(\()p Fs(!)k Fu(\001)c Fs(\027)6 b Fv(;)14 b(";)g Fs(!)s Fy(\))1643 2426 y Fp(\021)1710 2518 y Fu(\001)19 b Fy(\()p Fs(!)1847 2484 y Fn(0)1889 2518 y Fu(\000)f Fs(!)s Fy(\))g(+)g(\(\()p Fs(!)2295 2484 y Fn(0)2337 2518 y Fu(\000)g Fs(!)s Fy(\))h Fu(\001)f Fs(\027)6 b Fy(\))2661 2484 y Fk(2)2717 2518 y Fy(+)18 b Fv(")2839 2484 y Fk(2)2876 2518 y Fu(j)p Fs(\027)6 b Fu(j)14 b Fv(o)p Fy(\()p Fu(j)p Fs(!)3148 2484 y Fn(0)3189 2518 y Fu(\000)k Fs(!)s Fu(j)p Fy(\))p Fv(;)118 2721 y Fy(b)n(y)27 b(the)h(assumed)f(estimate)h(\(3.28\).)189 2827 y(If)g(\011)337 2839 y Fq(n)382 2827 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)568 2797 y Fn(0)591 2827 y Fy(\))23 b Fu(6)p Fy(=)g(0)k(w)n(e)g (can)g(b)r(ound)h(the)g(last)f(sum)h(in)g(\(A1.8\))g(b)n(y)550 2967 y Fq(n)511 2992 y Fp(X)512 3168 y Fq(p)p Fk(=0)645 3071 y Fy(\011)710 3083 y Fq(n;p)808 3071 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)994 3037 y Fn(0)1017 3071 y Fv(;)g Fs(!)s Fy(\))1163 3001 y Fp(\015)1163 3050 y(\015)1209 3071 y Fv(D)1280 3037 y Fn(\000)p Fk(1)1278 3092 y Fq(n)1369 3071 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)1555 3037 y Fn(0)1578 3071 y Fy(\))1610 3001 y Fp(\015)1610 3050 y(\015)1679 3071 y Fu(\024)23 b Fy(const)p Fv(:)14 b Fy(\()p Fv(\015)2074 3037 y Fn(\003)2069 3092 y Fq(n)2114 3071 y Fy(\))2146 3037 y Fn(\000)p Fk(2)2289 2967 y Fq(n)2249 2992 y Fp(X)2250 3168 y Fq(p)p Fk(=0)2369 3071 y Fy(\()p Fv(\015)2449 3037 y Fn(\003)2444 3092 y Fq(p)2487 3071 y Fy(\))2519 3037 y Fn(\000)p Fk(2)2623 3071 y Fu(j)p Fs(\027)6 b Fu(j)14 b(j)p Fs(!)2823 3037 y Fn(0)2864 3071 y Fu(\000)k Fs(!)s Fu(j)1679 3293 y(\024)23 b Fy(const)p Fv(:)14 b Fy(\()p Fv(\015)2074 3258 y Fn(\003)2069 3313 y Fq(n)2114 3293 y Fy(\))2146 3258 y Fn(\000)p Fk(5)2235 3293 y Fu(j)p Fs(\027)6 b Fu(j)14 b(j)p Fs(!)2435 3258 y Fn(0)2476 3293 y Fu(\000)19 b Fs(!)s Fu(j)p Fv(;)482 b Fy(\()p Fv(A)p Fy(1)p Fv(:)p Fy(11\))118 3477 y(where)22 b(w)n(e)f(ha)n(v)n(e)g(used)h(Lemma)g(A1)g(to)f(b)r(ound)i(\011)1667 3489 y Fq(n;p)1766 3477 y Fy(\()p Fs(\027)6 b Fv(;)14 b Fs(!)1952 3447 y Fn(0)1974 3477 y Fv(;)g Fs(!)s Fy(\),)23 b(and)f(\(A1.4\))g(to)g(p)r(erform)g(the)g(sum)g(o)n(v)n(er)118 3583 y Fv(p)h Fy(=)g(0)p Fv(;)14 b(:)g(:)g(:)f(;)h(n)p Fy(.)36 b(Note)27 b(that)g(in)g(order)e(to)h(pro\014tably)g(use)h(the)g (b)r(ound)g(\(A1.5\))f(w)n(e)h(ha)n(v)n(e)e(to)h(use)h(that)g(the)118 3690 y(b)r(ounds)34 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b(other)g(p)r(ositiv)n(e)g(constan)n(t) f(\(again)h(the)g(pro)r(of)g(pro)r(ceeds)g(as)f(for)h(Lemma)g(9,)h(but) g(with)g Fv(\024)f Fy(replaced)118 1636 y(with)34 b Fv(\024=)p Fy(2\).)52 b(By)33 b(writing)g Fu(A)p Fy(\()p Fs(!)1141 1606 y Fn(0)1164 1636 y Fy(\))g(=)e Fu(A)p Fy(\()p Fs(!)s Fy(\))23 b(+)f(\()p Fu(A)p Fy(\()p Fs(!)1821 1606 y Fn(0)1845 1636 y Fy(\))g Fu(\000)g(A)p Fy(\()p Fs(!)s Fy(\)\))34 b(one)e(can)h(iterate)g(the)g(construction)118 1743 y(ab)r(o)n(v)n(e)23 b(for)g(the)h(di\013erence)g Fu(A)p Fy(\()p Fs(!)1141 1713 y Fn(0)1164 1743 y Fy(\))11 b Fu(\000)g(A)p Fy(\()p Fs(!)s Fy(\).)36 b(The)24 b(only)f(di\013erence)h(with)g(resp)r(ect)g (to)f(the)h(previous)f(case)118 1849 y(is)28 b(that)f(no)n(w)g(the)h (factor)f Fv(\024=)p Fy(2)g(is)g(replaced)g(with)h Fv(\024=)p Fy(4.)189 1956 y(All)33 b(the)h(other)e(terms)h(of)f(the)i(double)f (sum)g(in)g(\(A1.14\))f(can)h(b)r(e)g(discussed)g(in)g(a)f(similar)g(w) n(a)n(y)-7 b(,)34 b(b)n(y)118 2063 y(relying)j(once)h(more)f(on)g 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Fs(!)590 4630 y Fn(\003)653 4618 y Fy(in)n(to)g(\()p Fs(!)s Fv(;)14 b Ft(0)p Fy(\))23 b Fu(2)h Fr(R)1197 4577 y Fq(r)1247 4618 y Fu(\002)13 b Fr(R)1389 4577 y Fq(s)1425 4618 y Fy(,)25 b(where)g Fs(!)j Fy(has)c(rationally)g(indep)r(enden)n(t)i (comp)r(onen)n(ts,)f(then)118 4724 y(the)j(action)f(v)-5 b(ariables)26 b(\()p Ft(A)p Fv(;)14 b Ft(B)p Fy(\))29 b(are)d(mixed)i(together)f(and)g(also)g(terms)g(of)h(the)g(form)f Fv(A)2892 4736 y Fq(i)2920 4724 y Fv(B)2983 4736 y Fq(j)3045 4724 y Fy(app)r(ear.)1730 4924 y(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 189 555 a Fy(Though,)23 b(it)h(is)f(easy)f(to)h (extend)g(the)h(analysis)d(to)i(suc)n(h)g(a)g(case.)34 b(And)24 b(with)g(a)e(little)i(further)f(w)n(ork,)g(w)n(e)118 662 y(can)e(consider)g(also)g(Hamiltonians)g(with)h(an)n(y)f(unp)r (erturb)r(ed)i(Hamiltonian)e Fu(H)2597 674 y Fk(0)2656 662 y Fy(satisfying)h(a)f(con)n(v)n(exit)n(y)118 768 y(prop)r(ert)n(y)38 b(\(so)h(that)g(the)g(eigen)n(v)-5 b(alues)38 b(of)h(the)g(matrix)g(det)14 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Fu(H)1418 3846 y Fk(0)1456 3834 y Fy(\()p Ft(A)p Fv(;)14 b Ft(B)p Fy(\))19 b(+)f Fv("@)1882 3846 y Fi(B)1939 3834 y Fv(f)9 b Fy(\()p Ft(A)p Fv(;)14 b Ft(B)p Fv(;)g Fs(\013)p Fv(;)g Fs(\014)s Fy(\))p Fv(:)3192 3677 y Fy(\()p Fv(A)p Fy(2)p Fv(:)p Fy(4\))118 4028 y(The)26 b(main)g(di\013erence)f(with)i (resp)r(ect)e(to)h(the)g(analysis)f(in)h(Sections)f(3)g(and)h(4)f(is)h (that)g(the)g(propagators)118 4134 y(are)32 b(of)g(the)h(form)f (\(3.4\),)i(with)f(\()p Fv(x)1211 4104 y Fk(2)1270 4134 y Fu(\000)22 b(M)1457 4104 y Fk([)p Fn(\024)p Fq(n)p Fk(])1591 4134 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g(!)s Fy(\)\))1902 4104 y Fn(\000)p Fk(1)2024 4134 y Fy(replaced)32 b(with)h(\()p Fv(ix)22 b Fu(\000)f(M)2867 4104 y Fk([)p Fn(\024)p Fq(n)p Fk(])3002 4134 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g(!)s Fy(\)\))3313 4104 y Fn(\000)p Fk(1)3402 4134 y Fy(,)118 4240 y(where)27 b(w)n(e)g(can)h(write)f Fu(M)945 4210 y Fk([)p Fn(\024)p Fq(n)p Fk(])1079 4240 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g(!)s Fy(\))24 b(=)e Fu(C)h 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Fu(\021)c(N)1104 525 y Fk([)p Fn(\024)p Fq(n)p Fk(])1239 555 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g(!)s Fy(\))28 b(is)h(suc)n(h)f (that,)h(b)n(y)f(extracting)f(the)i(dominan)n(t)f(order,)f(and)118 662 y(setting)h Fv(x)23 b Fy(=)g(0,)k(one)g(has)724 792 y Fu(N)804 758 y Fk([)p Fn(\024)p Fq(n)p Fk(])939 792 y Fy(\(0;)14 b Fv(";)g Fs(!)s Fy(\))23 b(=)f Fu(N)1411 758 y Fk([0])1486 792 y Fy(\(0;)14 b Fv(";)g Fs(!)s Fy(\))k(+)g Fv(O)r Fy(\()p Fv(")2005 758 y Fk(2)2043 792 y Fy(\))p Fv(;)784 1075 y Fu(N)864 1041 y Fk([0])939 1075 y Fy(\(0;)c Fv(";)g Fs(!)s Fy(\))23 b(=)1331 859 y Fp(0)1331 1005 y(B)1331 1054 y(B)1331 1108 y(@)1573 917 y Fy(0)391 b(0)236 b(0)205 b(0)1418 1016 y Fu(\000)p Fv("@)1566 1034 y Fo(\014)1615 1016 y Fv(@)1659 1028 y Fi(A)1720 1016 y Fv(f)91 b Fu(\000)p Fv("@)2000 1034 y Fo(\014)2050 1016 y Fv(@)2094 1028 y Fi(B)2151 1016 y Fv(f)h Fy(0)102 b Fu(\000)p Fv("@)2581 986 y Fk(2)2576 1045 y Fo(\014)2626 1016 y Fv(f)1497 1134 y("@)1585 1104 y Fk(2)1580 1157 y Fi(A)1640 1134 y Fv(f)199 b("@)1963 1146 y Fi(A)2023 1134 y Fv(@)2067 1146 y Fi(B)2124 1134 y Fv(f)119 b Fy(0)83 b Fv("@)2492 1146 y Fi(A)2552 1134 y Fv(@)2596 1152 y Fo(\014)2646 1134 y Fv(f)1447 1234 y("@)1530 1246 y Fi(B)1587 1234 y Fv(@)1631 1246 y Fi(A)1691 1234 y Fv(f)200 b("@)2020 1204 y Fk(2)2015 1257 y Fi(B)2072 1234 y Fv(f)171 b Fy(0)84 b Fv("@)2493 1246 y Fi(B)2550 1234 y Fv(@)2594 1251 y Fo(\014)2644 1234 y Fv(f)2709 859 y Fp(1)2709 1005 y(C)2709 1054 y(C)2709 1108 y(A)2796 1075 y Fv(;)3192 1005 y Fy(\()p Fv(A)p Fy(2)p Fv(:)p Fy(6\))118 1349 y(whereas)37 b(the)h(other)f (terms)g(dep)r(ending)h(on)g Fv(x)g Fy(whic)n(h)g(are)e(not)i (negligible)f(with)h(resp)r(ect)g(with)g(the)118 1455 y(dominan)n(t)27 b(ones)g(app)r(ear)g(as)710 1719 y Fu(N)790 1676 y Fk([)p Fn(\024)p Fq(n)p Fk(])778 1741 y(1)925 1719 y Fy(\()p Fv(x)p Fy(;)14 b Fv(";)g Fs(!)s Fy(\))23 b Fu(\021)1323 1527 y Fp(0)1323 1673 y(B)1323 1726 y(@)1409 1570 y Fv(O)r Fy(\()p Fv(")1545 1539 y Fk(2)1583 1570 y Fv(x)p Fy(\))84 b Fv(O)r 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b(can)g(b)r(ound)h(the)f(propagators)d(in)k(terms)f(of)g(the)h (eigen)n(v)-5 b(alues)21 b(of)h(a)g(suitable)g(symplectic)h(matrix)e Fv(S)5 b Fy(:)118 2318 y(for)27 b(the)h(latter,)f(b)r(esides)g Fv(d)g Fy(harmless)g(eigen)n(v)-5 b(alues)26 b(of)h(order)f(1)h(\(in)h Fv(")f Fy(and)g Fv(x)p Fy(\))h(there)f(are)f Fv(r)k Fy(eigen)n(v)-5 b(alues)118 2434 y(prop)r(ortional)24 b(to)h Fv(x)740 2404 y Fk(2)777 2434 y Fy(,)h(while)g(the)g(other)e Fv(s)i Fy(eigen)n(v)-5 b(alues)24 b(are)g(of)h(the)h(form)f Fv(x)2498 2404 y Fk(2)2549 2434 y Fu(\000)14 b Fv(\025)2676 2391 y Fk([0])2676 2457 y Fq(j)2751 2434 y Fy(\()p Fv(x;)g(";)g Fs(!)s Fy(\))g(+)g Fv(O)r Fy(\()p Fv("x)p Fy(\))g(+)118 2563 y Fv(O)r Fy(\()p Fv(")254 2533 y Fk(2)292 2563 y Fy(\))27 b(,)h(with)f Fv(\025)638 2520 y Fk([0])638 2586 y Fq(j)713 2563 y Fy(\()p Fv(x;)14 b(";)g Fs(!)s Fy(\))24 b(=)e Fv("a)1194 2575 y Fq(j)s Fn(\000)p Fq(r)1313 2563 y Fy(\()p Fs(!)s Fy(\),)28 b Fv(j)g Fy(=)23 b Fv(r)d Fy(+)d(1)p Fv(;)d(:)g(:)g(:)f(;)h(d)p 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Fy(\(2002\),)g(no.)36 b(3,)27 b(421{460.)118 1223 y([18])36 b(G.)28 b(Galla)n(v)n(otti,)e(G.) i(Gen)n(tile,)g(A.)g(Giuliani,)g Fw(F)-6 b(r)l(actional)31 b(Lindste)l(dt)f(series)p Fy(,)e(Preprin)n(t,)f(2005.)118 1393 y([19])36 b(G.)28 b(Galla)n(v)n(otti,)e(G.)i(Gen)n(tile,)g(A.)g (Giuliani,)g(w)n(ork)e(in)i(preparation.)118 1563 y([20])36 b(G.)19 b(Gen)n(tile,)i Fw(Diagr)l(ammatic)i(te)l(chniques)f(in)f(p)l (erturb)l(ations)h(the)l(ory,)i(and)e(applic)l(ations)p Fy(,)h(Symmetry)284 1669 y(and)31 b(p)r(erturbation)f(theory)g(\(Rome,) i(1998\),)f(59{78,)e(W)-7 b(orld)31 b(Sci.)47 b(Publishing,)31 b(Riv)n(er)f(Edge,)h(NJ,)284 1775 y(1999.)118 1929 y([21])36 b(G.)g(Gen)n(tile,)h Fw(Quasi-p)l(erio)l(dic)i(solutions)e(for)h (two-level)g(systems)p Fy(,)f(Comm.)60 b(Math.)h(Ph)n(ys.)e Ft(242)284 2035 y Fy(\(2003\),)27 b(no.)36 b(1-2,)27 b(221-250.)118 2209 y([22])36 b(G.)29 b(Gen)n(tile,)g Fw(Pur)l(e)h(p)l(oint)h(sp)l(e)l(ctrum)e(for)j(two-level)f(systems)f 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