Content-Type: multipart/mixed; boundary="-------------0306030547403" This is a multi-part message in MIME format. ---------------0306030547403 Content-Type: text/plain; name="03-253.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-253.comments" 2000 Mathematics Subject Classification: Primary 46J10, 43A60; Secondary 35P05. ---------------0306030547403 Content-Type: text/plain; name="03-253.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-253.keywords" Abelian C*-algebra, locally compact group, filter, Gelfand spectrum, asymptotic independence, syndetic, replete, minimal function, vanishing oscillation. ---------------0306030547403 Content-Type: application/postscript; name="marius.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="marius.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: marius.dvi %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMSY8 CMBX10 CMR10 CMMI10 CMSY7 CMBX12 CMR8 %%+ CMSY10 CMTI10 MSBM10 EUFM7 EUFM10 CMEX10 CMSY6 CMR9 CMMI8 MSBM7 %%+ CMMI12 MSAM7 MSAM10 CMR6 CMMI6 CMTI8 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips marius.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.03:1919 %%BeginProcSet: texc.pro %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 70 /F put dup 71 /G put dup 77 /M put dup 78 /N put dup 82 /R put dup 83 /S put dup 85 /U put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163B3DA4F9700DE5807F164169FBB 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 71 /G put dup 114 /r put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % 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 (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 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 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 70 /F put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 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 4 /squaresolid put dup 123 /complement put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM7 %!PS-AdobeFont-1.1: MSAM7 2.1 %%CreationDate: 1992 Oct 17 08:30:29 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM7 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 123 /complement put readonly def /FontBBox{0 -576 1569 1076}readonly def /UniqueID 5032012 def currentdict end currentfile eexec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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. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 67 /C put dup 71 /G put dup 73 /I put dup 76 /L put dup 81 /Q put dup 82 /R put dup 83 /S put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 10 /circlemultiply put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 41 /arrowdblright put dup 44 /arrowdblboth put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 74 /J put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (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 78 /N put dup 82 /R put 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: 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 34 /epsilon put dup 39 /phi1 put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 70 /F put dup 71 /G put dup 72 /H put dup 77 /M put dup 83 /S put dup 88 /X put dup 90 /Z put dup 101 /e put dup 105 /i put dup 106 /j put dup 110 /n put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 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%!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % 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 (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 34 /quotedblright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 52 /four put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 64 /at put dup 65 /A put dup 66 /B put dup 69 /E put dup 73 /I put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 92 /quotedblleft put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % 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 (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 80 /summationtext put dup 82 /integraltext put dup 99 /hatwider put dup 100 /hatwidest put dup 103 /tildewidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 36 /subsetornotdbleql put dup 67 /C put dup 78 /N put dup 82 /R 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 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % 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 (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 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 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB0B4E70F212EC976D65099D84E0D37A7A771C3101D6AD26A0513378F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 67 /C put dup 70 /F put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 108 /l put dup 114 /r put dup 115 /s put dup 116 /t put dup 121 /y put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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 (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 14 /delta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 106 /j put dup 108 /l put dup 110 /n put dup 114 /r put dup 115 /s put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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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 (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 72 /H put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 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 3 /asteriskmath put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 65 /A put dup 66 /B put dup 70 /F put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % 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 (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 21 /breve put dup 44 /comma put dup 48 /zero put dup 50 /two put dup 51 /three put dup 74 /J put dup 77 /M put dup 97 /a put dup 101 /e put dup 105 /i put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (marius.dvi) @start /Fa 134[35 5[29 30 16[36 26[46 70[{}5 66.4176 /CMTI8 rf /Fb 221[46 34[{}1 99.6264 /CMMI12 rf /Fc 185[39 70[{}1 49.8132 /EUFM7 rf /Fd 185[73 70[{}1 119.552 /EUFM10 rf /Fe 141[29 42[47 71[{}2 49.8132 /CMMI6 rf /Ff 139[24 24 24 43[39 19[30 30 30 4[47 43[{}8 49.8132 /CMR6 rf /Fg 132[45 118[71 4[{}2 90.9091 /MSAM10 rf /Fh 132[35 123[{}1 66.4176 /MSAM7 rf /Fi 184[78 61 70[{}2 99.6264 /EUFM10 rf /Fj 134[59 59 1[59 62 44 44 46 59 62 56 62 93 31 2[31 62 56 34 51 62 50 62 54 10[85 2[62 9[42 20[56 56 56 56 2[31 37 32[62 12[{}32 99.6264 /CMBX12 rf /Fk 252[42 3[{}1 83.022 /CMSY10 rf /Fl 188[84 67[{}1 119.552 /CMMI12 rf /Fm 165[44 7[48 3[48 78[{}3 66.4176 /MSBM7 rf /Fn 133[33 35 40 3[25 33 4[43 3[29 24 3[33 10[48 1[58 4[43 5[68 4[58 55 45 4[53 2[55 35 1[20 19[46 4[33 34[{}22 66.4176 /CMMI8 rf /Fo 134[41 41 55 1[43 30 30 30 2[38 43 64 21 2[21 43 1[23 34 43 34 1[38 4[38 8[43 57 1[52 60 1[70 3[28 3[52 2[54 58 60 5[21 2[38 38 1[38 2[38 38 1[21 26 21 9[38 34[{}39 74.7198 /CMR9 rf /Fp 207[18 44[32 2[48{}3 49.8132 /CMSY6 rf /Fq 150[43 43 131 2[131 91 16[43 1[96 62[54 54 12[38 38 42 42{}13 90.9091 /CMEX10 rf /Fr 170[59 1[75 75 3[76 95 5[71 56 70[{}7 90.9091 /EUFM10 rf /Fs 170[45 13[54 42 70[{}3 66.4176 /EUFM7 rf /Ft 165[61 7[66 3[66 10[66 30[71 36[{}5 90.9091 /MSBM10 rf /Fu 128[46 4[37 44 42 60 42 49 30 37 38 42 46 46 51 74 23 42 28 28 46 42 28 42 46 42 42 46 4[47 4[91 68 68 65 51 66 70 62 70 68 82 57 1[48 35 68 70 59 62 69 65 64 68 5[28 28 7[46 46 2[28 33 28 2[37 37 28 4[47 14[46 46 3[80 53 51 56 11[{}67 90.9091 /CMTI10 rf /Fv 141[76 3[45 2[45 25 35 35 45 45 9[61 61 16[62 1[77 1[65 1[70 48 60 73 5[45 1[51 51 0 0 3[61 91 4[91 2[91 7[91 5[71 71 1[71 2[71 71 2[71 2[45 3[71 7[71 25 71{}38 90.9091 /CMSY10 rf /Fw 134[37 4[27 28 28 5[20 10[35 3[20 1[20 20[46 2[51 16[35 35 35 4[55 1[27 27 40[{}16 66.4176 /CMR8 rf /Fx 134[71 3[75 52 53 55 1[75 67 75 112 37 2[37 75 67 41 61 75 60 75 65 14[100 101 4[81 2[50 1[106 3[97 1[102 13[67 67 67 3[45 37 44[{}31 119.552 /CMBX12 rf /Fy 133[42 45 52 4[43 41 3[55 1[27 1[37 3[45 42 47 39 39 48 6[62 53 75 86 53 62 53 56 69 1[58 69 73 88 62 2[40 76 71 58 2[65 69 68 48 1[71 45 71 25 25 18[59 4[42 1[59 57 2[40 52 1[52 2[55 53 2[43 3[40 1[51 12[{}53 90.9091 /CMMI10 rf /Fz 128[45 45 3[40 48 48 66 48 51 35 36 36 48 51 45 51 76 25 48 28 25 51 45 28 40 51 40 51 45 1[25 1[25 45 25 1[68 1[93 68 68 66 51 67 71 62 71 68 83 57 71 47 33 68 71 59 62 69 66 64 68 3[71 1[25 25 45 45 45 45 45 45 45 45 45 45 45 25 30 25 71 45 35 35 25 4[45 11[45 45 45 45 4[76 51 51 53 8[71 2[{}88 90.9091 /CMR10 rf /FA 134[55 55 76 1[58 41 41 43 1[58 52 58 87 29 55 32 29 58 2[48 58 46 58 51 3[29 1[29 6[73 58 78 1[71 2[99 63 82 2[82 2[69 80 76 8[29 52 52 52 52 52 52 52 52 52 52 1[29 33[58 12[{}47 90.9091 /CMBX10 rf /FB 252[34 3[{}1 58.1154 /CMSY7 rf /FC 188[59 67[{}1 83.022 /CMMI10 rf /FD 134[44 44 60 44 46 32 33 33 1[46 42 46 69 23 2[23 46 42 25 37 46 37 46 42 3[23 1[23 3[85 34[42 2[42 2[23 28 23 31[46 12[{}31 83.022 /CMR10 rf /FE 139[37 38 39 14[42 53 46 31[72 65[{}7 83.022 /CMBX10 rf /FF 149[20 2[35 35 31[50 3[46 56 14[47 71 19 14[71 29[35 2[55{}12 66.4176 /CMSY8 rf /FG 138[54 38 38 38 2[49 54 4[27 3[43 3[49 19[89 2[50 22[49 49 1[49 3[27 22[49 21[{}16 99.6264 /CMR12 rf /FH 136[96 2[51 52 51 1[73 66 73 1[36 2[36 73 66 1[58 73 58 73 66 14[98 2[103 5[47 2[86 2[96 1[99 19[43 2[66 42[{}24 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 921 846 a FH(On)44 b(Ab)t(elian)g(C*-Algebras)f (whic)l(h)h(are)857 1029 y(Indep)t(enden)l(t)i(with)d(Resp)t(ect)g(to)g (a)g(Filter)1583 1281 y FG(Marius)33 b(M\025)-49 b(an)m(toiu)2268 1245 y FF(\003)1657 1485 y FG(June)33 b(3,)f(2003)1743 1920 y FE(Abstract)435 2072 y FD(W)-7 b(e)27 b(examine)g(extensions)g (of)g(the)g(concept)g(of)h(asymptotic)e(indep)r(endence,)i(in)n(tro)r 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Fz(,)f(i.e.)51 b(it)34 b(is)f(non-v)m(oid,)i(it) f(do)s(es)f(not)h(con)m(tain)83 2869 y Fy(X)k Fz(and)30 b(it)g(is)h(stable)g(under)e(\014nite)h(unions)g(and)f(subsets.)83 3069 y FA(De\014nition)35 b(1.2)46 b Fu(F)-7 b(or)33 b(any)f Fy(')26 b Fv(2)f Fy(B)5 b(C)i Fu(,)30 b Fy(")c(>)f Fz(0)32 b Fu(and)g Fy(\025)25 b Fv(2)g Ft(C)31 b Fu(we)h(set)f Fy(M)2506 3083 y Fn(")2543 3069 y Fz(\()p Fy(')p Fz(;)15 b Fy(\025)p Fz(\))27 b(:=)e Fv(f)p Fy(x)h Fv(2)f Fy(X)33 b Fv(j)25 b(j)p Fy(')p Fz(\()p Fy(x)p Fz(\))19 b Fv(\000)f Fy(\025)p Fv(j)25 b Fy(<)83 3181 y(")p Fv(g)33 b Fu(and)h(we)f(c)-5 b(al)5 b(l)33 b(it)g Fz(an)f Fy(")p Fu(-)p Fz(lev)m(el)g(set)p Fu(.)249 3381 y Fz(Clearly)h Fy(M)655 3395 y Fn(")692 3381 y Fz(\()p Fy(')p Fz(;)15 b Fy(\025)p Fz(\))34 b(is)e(op)s(en)g(in) g Fy(X)39 b Fz(and)32 b(it)h(is)f(non)m(v)m(oid)h(for)f(an)m(y)h Fy(")28 b(>)h Fz(0)j(i\013)h Fy(\025)f Fz(b)s(elongs)g(to)h(the)g (range)83 3494 y(of)e Fy(')p Fz(.)83 3694 y FA(Lemma)k(1.3)46 b Fu(F)-7 b(or)33 b Fy(')26 b Fv(2)f Fy(B)5 b(C)39 b Fu(and)33 b Fy(\025)25 b Fv(2)g Ft(C)p Fu(,)32 b(the)h(fol)5 b(lowing)34 b(c)-5 b(onditions)34 b(ar)-5 b(e)34 b(e)-5 b(quivalent:)249 3807 y(\(i\))33 b Fy(\025)25 b Fv(2)g Fy(R)617 3822 y Fs(F)663 3807 y Fz(\()p Fy(')p Fz(\))p Fu(.)249 3920 y(\(ii\))33 b(Ther)-5 b(e)33 b(exists)g(an)g(ultr)-5 b(a\014lter)35 b Fr(U)d Fu(on)h Fy(X)7 b Fu(,)33 b(\014ner)g(than)h Fr(F)p Fu(,)d(such)i(that)h Fz(lim)2872 3935 y Fs(U)2936 3920 y Fy(')26 b Fz(=)f Fy(\025)p Fu(.)249 4032 y(\(iii\))33 b(F)-7 b(or)33 b(any)h Fy(")25 b(>)g Fz(0)p Fu(,)33 b Fy(M)1141 4046 y Fn(")1178 4032 y Fz(\()p Fy(')p Fz(;)15 b Fy(\025)p Fz(\))37 b Fy(=)-55 b Fv(2)25 b Fr(G)p Fu(.)249 4232 y Fz(The)30 b(pro)s(of)g(is)g(a)h(simple)f(exercice)i(in)e(top)s (ology)-8 b(.)249 4345 y(Three)21 b(simple)h(examples)g(are)g(of)g (distinguished)f(in)m(terest)i(and)e(will)h(b)s(e)f(met)h(in)g (subsequen)m(t)f(sections:)249 4458 y Fu(The)31 b(trivial)g(\014lter)e Fr(F)968 4472 y Fw(tr)1052 4458 y Fz(=)c Fv(f)p Fy(X)7 b Fv(g)30 b Fz(will)e(b)s(e)g(mainly)g(considered)g(for)g (illustrational)i(purp)s(oses.)39 b(F)-8 b(or)29 b(an)m(y)83 4571 y(unital)j Fy(C)419 4538 y FF(\003)458 4571 y Fz(-subalgebra)g Fv(A)g Fz(of)g Fy(B)5 b(C)i Fz(,)32 b(one)g(has)g Fv(A)1763 4538 y Fs(F)1805 4546 y Ff(tr)1888 4571 y Fz(=)c Fv(f)p Fz(0)p Fv(g)p Fz(,)2212 4548 y(~)2181 4571 y Fv(A)2254 4538 y Fs(F)2296 4546 y Ff(tr)2380 4571 y Fz(=)g Ft(C)j Fz(and)g Fv(A)2827 4586 y Fs(F)2869 4594 y Ff(tr)2953 4571 y Fz(=)d Fv(A)p Fz(.)45 b(The)31 b(seminorm)83 4684 y Fv(k)26 b(\001)f(k)249 4699 y Fs(F)291 4707 y Ff(t)q Fe(r)385 4684 y Fz(coincides)31 b(with)f(the)h(usual)f(norm)f Fv(k)d(\001)g(k)k Fz(and)g(one)h(has)f Fy(R)2380 4699 y Fs(F)2422 4707 y Ff(tr)2478 4684 y Fz(\()p Fy(')p Fz(\))d(=)e Fy(R)q Fz(\()p Fy(')p Fz(\))31 b(for)f(an)m(y)h Fy(')p Fz(.)249 4797 y(Of)h(direct)g(in)m(terest)h(will)g(b)s(e)e Fu(the)k(standar)-5 b(d)37 b(\014lter)c Fr(F)2064 4811 y Fw(st)2123 4797 y Fz(,)g(comp)s(osed)f(of)g(the)g(complemen)m(ts)h (of)g(all)g(the)83 4910 y(relativ)m(ely)28 b(compact)e(subsets)f(of)g Fy(X)7 b Fz(.)39 b(Then)24 b(w)m(e)i(ha)m(v)m(e)h Fv(A)2026 4877 y Fs(F)2068 4885 y Ff(st)2149 4910 y Fz(=)e Fv(A)10 b(\\)g Fy(C)2464 4924 y Fw(0)2528 4910 y Fz(and)2730 4887 y(~)2699 4910 y Fv(A)2772 4877 y Fs(F)2814 4885 y Ff(st)2896 4910 y Fz(=)25 b Fv(A)10 b(\\)g Fy(C)d Fz(\()3288 4887 y(_)3253 4910 y Fy(X)f Fz(\))25 b(\(the)h(alge-)83 5023 y(bra)d(of)h(all)g(functions)f(in)g Fv(A)g Fz(whic)m(h)h(can)g(b)s (e)e(extended)i(con)m(tin)m(uously)g(to)h(the)e(one-p)s(oin)m(t)i (compacti\014cation)83 5135 y(of)37 b Fy(X)7 b Fz(\).)60 b(W)-8 b(e)38 b(shall)f(call)h(resp)s(ectiv)m(ely)g(the)f(ob)5 b(jects)37 b Fv(k)f Fy(')g Fv(k)2153 5150 y Fs(F)2195 5158 y Ff(st)2253 5135 y Fv(\021k)f Fy(')i Fv(k)2545 5149 y Fw(asy)2686 5135 y Fz(and)f Fy(R)2938 5150 y Fs(F)2980 5158 y Ff(st)3037 5135 y Fz(\()p Fy(')p Fz(\))h Fv(\021)e Fy(R)3378 5149 y Fw(asy)3483 5135 y Fz(\()p Fy(')p Fz(\))j Fu(the)83 5248 y(asymptotic)d(seminorm)d Fz(and)f Fu(the)i(asymptotic)i (r)-5 b(ange)32 b Fz(of)f Fy(')26 b Fv(2)g Fy(B)5 b(C)i Fz(.)41 b(The)30 b(complex)i(n)m(um)m(b)s(er)e Fy(\025)g Fz(is)h(in)g(the)83 5361 y(asymptotic)h(range)f(of)g Fy(')g Fz(i\013)g(for)f(an)m(y)i Fy(")26 b(>)f Fz(0)31 b(the)g(lev)m(el)i(set)e Fy(M)2249 5375 y Fn(")2286 5361 y Fz(\()p Fy(')p Fz(;)15 b Fy(\025)p Fz(\))32 b(is)f(not)g(relativ)m (ely)i(compact.)43 b(This)83 5474 y(example)28 b(motiv)-5 b(ated)28 b(our)f(w)m(ork)g(and)g(man)m(y)g(of)g(the)h(results)e(b)s (elo)m(w)i(are)f(in)m(teresting)i(and)d(non-trivial)i(for)83 5587 y Fr(F)d Fz(=)g Fr(F)316 5601 y Fw(st)375 5587 y Fz(.)1903 5836 y(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 249 482 a Fz(Occasionally)34 b(w)m(e)f(shall)f (meet)h Fu(the)i(F)-7 b(r)n(\023)-44 b(echet)34 b(\014lter)p Fz(,)g(comp)s(osed)e(of)g(the)g(complemen)m(ts)h(of)g(the)f(\014nite)83 594 y(subsets)d(of)i Fy(X)7 b Fz(;)30 b(it)h(is)f(con)m(tained)h(in)f (the)h(standard)e(\014lter.)40 b(One)30 b(has)g Fr(F)2535 608 y Fw(F)-6 b(r)2632 594 y Fz(=)24 b Fr(F)2783 608 y Fw(st)2872 594 y Fz(if)31 b Fy(X)37 b Fz(is)30 b(discrete.)41 b(In)30 b(the)83 707 y(opp)s(osite)36 b(case,)h(in)e(whic)m(h)g(the)h (single)g(\014nite)f(op)s(en)g(set)g(is)h Fv(;)p Fz(,)h(w)m(e)f(ha)m(v) m(e)g(ob)m(viously)g Fv(k)e(\001)f(k)3223 722 y Fs(F)3265 733 y Ff(F)-5 b(r)3333 707 y Fz(=)p Fv(k)33 b(\001)h(k)i Fz(and)83 820 y Fy(R)152 835 y Fs(F)194 846 y Ff(F)-5 b(r)286 820 y Fz(=)25 b Fy(R)q Fz(.)40 b(There)29 b(are)h(also)h(man)m (y)e(in)m(termediary)h(cases,)h(for)f(whic)m(h)f(w)m(e)h(ha)m(v)m(e)h (nothing)e(sp)s(ecial)h(to)h(sa)m(y)-8 b(.)83 1104 y Fx(2)135 b Fd(F)p Fx(-Indep)t(endence,)44 b(the)h(General)h(Case)83 1307 y Fz(As)29 b(in)g(the)g(preceding)g(section,)i Fr(F)d Fz(will)i(b)s(e)e(a)i(\014lter)f(on)g(the)g(lo)s(cally)i(compact)f (\(non-compact\))h(space)e Fy(X)7 b Fz(.)83 1420 y(W)-8 b(e)35 b(shall)f(b)s(e)g(concerned)g(with)f(the)h Fy(C)1457 1387 y FF(\003)1496 1420 y Fz(-algebra)i Fv(C)j Fz(generated)c(b)m(y)e 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Fz(w)m(e)g(start)g(our)f(systematic)i(study)d(of)i Fr(F)p Fz(-indep)s(endence)e(b)m(y)h(giving)83 2097 y(a)31 b(criterion)g(in)f(terms)g(of)h Fy(")p Fz(-lev)m(el)i(sets.)83 2338 y Fj(2.1)112 b Fi(F)p Fj(-indep)s(endence)40 b(of)e(functions)g (and)g(algebras)83 2510 y Fz(Let)c Fv(A)e Fz(and)h Fv(B)i Fz(b)s(e)e(t)m(w)m(o)h(unital)f Fy(C)1267 2477 y FF(\003)1306 2510 y Fz(-subalgebras)g(of)h Fy(B)5 b(C)35 b Fv(\021)30 b Fy(B)5 b(C)i Fz(\()p Fy(X)g Fz(\).)49 b(W)-8 b(e)34 b(denote)g(b)m(y)e Fv(h)q(A)20 b(\001)g(B)s(i)33 b Fz(or)g(b)m(y)g Fv(C)83 2623 y Fz(the)g Fy(C)314 2590 y FF(\003)353 2623 y Fz(-subalgebra)g(of)g Fy(B)5 b(C)39 b Fz(generated)34 b(b)m(y)f(the)g(union)f(of)i Fv(A)e Fz(and)g Fv(B)s Fz(.)48 b(W)-8 b(e)35 b(shall)e(b)s(e)f(in)m(terested)i(in)f(the)83 2736 y(sp)s(ectrum)25 b(of)h Fv(C)623 2751 y Fs(F)695 2736 y Fz(=)f Fv(C)5 b Fy(=)p Fv(C)942 2703 y Fs(F)989 2736 y Fz(;)28 b(w)m(e)e(recall)i(that)f Fv(C)1656 2703 y Fs(F)1727 2736 y Fv(\021)e(h)q(A)19 b(\001)i(B)s(i)2095 2694 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b(decreasing.)249 4935 y(\(iii\))i(If)e Fv(;)25 b(2)g Fr(M)p Fz(,)31 b(then)f Fr(M)1126 4902 y Fw([)p Fs(F)p Fw(])1238 4935 y Fz(=)24 b Fv(;)31 b Fz(for)f(an)m(y)h Fr(F)p Fz(;)f(w)m(e)h(shall)g(t)m(ypically)h(a)m(v)m (oid)g(this)e(situation.)249 5048 y(\(iv\))h(If)f Fy(X)j Fv(2)25 b Fr(M)p Fz(,)31 b(then)f Fg({)p Fr(G)c Fv(\021)f Fz(2)1349 5015 y Fn(X)1437 5048 y Fv(n)c Fr(G)30 b Fz(con)m(tains)i Fr(M)2056 5015 y Fw([)p Fs(F)p Fw(])2142 5048 y Fz(,)e(strictly)i(in)e (general.)249 5185 y(\(v\))h Fr(M)26 b Fv(\032)615 5111 y Fq(\000)656 5185 y Fr(M)751 5152 y Fw([)p Fs(F)p Fw(])837 5111 y Fq(\001)879 5129 y Fw([)p Fs(F)p Fw(])964 5185 y Fz(,)31 b(strictly)g(in)f(general.)249 5308 y(In)38 b(the)g(sequel)h(it)g(will)g(b)s(e)f(imp)s(ortan)m(t)g(to)i(isolate)g (situations)f(in)f(whic)m(h)h Fr(M)2974 5275 y Fw([)p Fs(F)p Fw(])3098 5308 y Fz(can)g(b)s(e)e(explicitly)83 5421 y(calculated.)1903 5836 y(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 83 482 a FA(De\014nition)35 b(2.6)46 b Fu(The)33 b(family)h Fr(M)f Fu(of)f(subsets)h(of)g Fy(X)40 b Fu(wil)5 b(l)33 b(b)-5 b(e)32 b(c)-5 b(al)5 b(le)-5 b(d)34 b Fr(F)p Fu(-)p Fz(admissible)e Fu(if)249 594 y(\(i\))h Fr(M)g Fu(is)f(stable)i(under)f(sup)-5 b(ersets,)249 707 y(\(ii\))33 b(for)g(any)g Fy(G)26 b Fv(2)f Fr(G)p Fu(,)33 b(if)f Fy(S)40 b(=)-55 b Fv(2)25 b Fr(M)33 b Fu(then)g Fy(G)21 b Fv([)e Fy(S)41 b(=)-55 b Fv(2)25 b Fr(M)p Fu(.)249 920 y Fz(It)k(is)g(natural)g(to)h(assume)f (\(i\).)41 b(Condition)29 b(\(ii\))h(is)f(once)g(again)i(connected)e (to)h(our)f(image)h(that)f(\\the)83 1033 y(elemen)m(ts)k(of)f Fr(G)g Fz(should)f(b)s(e)g(small".)46 b(It)32 b(is)g(satis\014ed)g(for) f(an)m(y)h Fr(M)g Fz(if)g Fr(F)27 b Fz(=)h Fr(F)2709 1047 y Fw(tr)2767 1033 y Fz(.)45 b(But)32 b(\(ii\))h(is)f(b)m(y)f(no)h (means)83 1146 y(automatical;)h(a)e(simple)f(coun)m(terexample)i(is)f (giv)m(en)g(b)m(y)f Fr(F)25 b Fz(=)g Fr(F)2307 1160 y Fw(st)2396 1146 y Fz(and)30 b Fr(M)c Fz(=)f Fv(f)p Fy(X)7 b Fv(g)p Fz(.)83 1358 y FA(Lemma)35 b(2.7)46 b 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Fv(\032)c Fg({)3186 1836 y Fq(\020)3241 1937 y Fr(M)3336 1904 y Fh({)3376 1836 y Fq(\021)3460 1937 y Fz(without)83 2067 y(an)m(y)31 b(assumption)f(on)g Fr(M)h Fz(and)e Fr(F)p Fz(.)249 2198 y(\(b\))e(Assume)f(no)m(w)h(that)h Fr(M)f Fz(is)g Fr(F)p Fz(-admissible,)g(let)h Fy(M)36 b Fv(2)24 b Fr(M)k Fz(and)e Fy(N)35 b Fv(2)25 b Fg({)2755 2097 y Fq(\020)2810 2198 y Fr(M)2905 2165 y Fh({)2945 2097 y Fq(\021)2999 2198 y Fz(,)j(i.e.)40 b Fg({)p Fy(N)46 b(=)-55 b Fv(2)25 b Fr(M)p Fz(.)39 b(One)83 2340 y(has)25 b Fy(M)36 b Fv(\032)25 b Fz(\()p Fy(M)c Fv(\\)11 b Fy(N)f Fz(\))h Fv([)g Fz(\()p Fg({)p Fy(N)f Fz(\))26 b(and)f(th)m(us)g Fy(M)c Fv(\\)11 b Fy(N)35 b Fv(2)24 b Fr(G)i Fz(w)m(ould)g(imply)f Fy(M)46 b(=)-55 b Fv(2)24 b Fr(M)p Fz(,)k(whic)m(h)d(is)h(a)g(con)m (tradiction.)83 2453 y Fg(\004)249 2566 y Fz(Note)32 b(that)f Fr(M)761 2533 y Fw([)p Fs(F)p Fw(])877 2566 y Fz(is)f(the)h(same)g(for)f(all)h(the)g(\014lters)f Fr(F)g Fz(with)g(resp)s(ect)g(to)h(whic)m(h)f Fr(M)h Fz(is)g(admissible.)83 2779 y FA(De\014nition)k(2.8)46 b Fu(L)-5 b(et)40 b Fy(')e Fv(2)e Fy(B)5 b(C)i Fu(,)40 b Fy(\025)d Fv(2)g Ft(C)i Fu(and)h Fr(M)e Fv(\032)f Fz(2)2084 2746 y Fn(X)2152 2779 y Fu(.)61 b(One)39 b(says)h(that)h Fy(')e Fz(is)f(of)g(class)i Fr(M)f Fu(at)h Fy(\025)f Fu(if)83 2891 y Fy(M)171 2905 y Fn(")208 2891 y Fz(\()p Fy(';)15 b(\025)p Fz(\))27 b Fv(2)e Fr(M)33 b Fu(for)g(al)5 b(l)33 b Fy(")26 b(>)f Fz(0)p Fu(.)249 3104 y Fz(As)34 b(a)h(simple)f(example,)i Fy(\025)31 b Fv(2)h Fy(R)q Fz(\()p Fy(')p Fz(\))j(i\013)f Fy(')h Fz(is)f(of)g(class)h Fr(M)d Fv(\021)f Fz(2)2450 3071 y Fn(X)2541 3104 y Fv(n)23 b(f;g)35 b Fz(at)g Fy(\025)p Fz(.)52 b(More)35 b(generally)-8 b(,)37 b(b)m(y)83 3217 y(Lemma)d(1.3,)j Fy(\025)32 b Fv(2)g Fy(R)835 3232 y Fs(F)881 3217 y Fz(\()p Fy(')p Fz(\))j(i\013)g Fy(')g Fz(is)f(of)h(class)g Fg({)p Fr(G)g Fz(at)g Fy(\025)p Fz(.)53 b(On)33 b(the)i(other)f(hand,)h Fy(')f Fz(is)h(of)f(class)i Fr(F)d Fz(at)i Fy(\025)g Fz(i\013)83 3330 y(lim)209 3345 y Fs(F)271 3330 y Fy(')j Fz(=)f Fy(\025)p Fz(,)i(so)f(that)g Fy(')g Fz(is)g(of)g(class)g Fr(F)1499 3344 y Fw(tr)1595 3330 y Fz(at)h Fy(\025)e Fz(i\013)h Fy(')g Fz(=)f Fy(\025)h Fz(and)f Fy(')h Fz(is)f(of)h(class)g Fr(F)2982 3344 y Fw(st)3079 3330 y Fz(at)g Fy(\025)g Fz(i\013)g Fy(')g Fz(can)g(b)s(e)83 3443 y(extended)30 b(con)m(tin)m(uously)i(on)e(the)g(one-p)s(oin)m(t)i(compacti\014cation) 2403 3420 y(_)2366 3443 y Fy(X)h Fv(\021)25 b Fy(X)j Fv([)19 b(f1g)32 b Fz(and)d Fy(')p Fz(\()p Fv(1)p Fz(\))e(=)e Fy(\025)p Fz(.)249 3556 y(The)30 b(notion)h(of)f Fr(F)p Fz(-o)m(v)m(erlapping)i(is)e(relev)-5 b(an)m(t)32 b(in)e(our)g(con)m (text)i(b)s(ecause)e(of)h(the)g(follo)m(wing)g(result:)83 3768 y FA(Prop)s(osition)36 b(2.9)46 b Fu(L)-5 b(et)37 b Fy(';)15 b( )37 b Fv(2)32 b Fy(B)5 b(C)i Fu(,)36 b Fy(\025;)15 b(\026)33 b Fv(2)f Ft(C)k Fu(and)h Fr(M)p Fy(;)15 b Fr(N)37 b Fu(two)h Fr(F)p Fu(-overlapping)f(families)g(of)g (subsets)83 3881 y(of)c Fy(X)7 b Fu(.)249 3994 y(\(a\))33 b(If)g Fy(')g Fu(is)f(of)h(class)h Fr(M)f Fu(at)g Fy(\025)f Fu(and)i Fy( )i Fu(is)c(of)h(class)h Fr(N)e Fu(at)h Fy(\026)p Fu(,)f(then)h Fy(\025\026)25 b Fv(2)g Fy(R)2852 4009 y Fs(F)2899 3994 y Fz(\()p Fy(' )s Fz(\))p Fu(.)249 4107 y(\(b\))41 b(If)f Fv(j)p Fy(')p Fv(j)i Fu(is)e(of)h(class)g Fr(M)g Fu(in)g Fv(k)f Fy(')g Fv(k)1591 4122 y Fs(F)1678 4107 y Fu(and)i Fv(j)p Fy( )s Fv(j)g Fu(is)e(of)h(class)g Fr(N)f Fu(in)h Fv(k)f Fy( )j Fv(k)2931 4122 y Fs(F)2978 4107 y Fu(,)f(then)g Fy(')f Fu(and)g Fy( )j Fu(ar)-5 b(e)83 4220 y Fr(F)p Fu(-indep)g(endent.)249 4432 y(Pr)g(o)g(of:)42 b Fz(\(a\))31 b(F)-8 b(or)31 b(an)m(y)g Fy(x)25 b Fv(2)g Fy(X)38 b Fz(one)31 b(has)334 4636 y Fv(j)p Fz(\()p Fy(' )s Fz(\)\()p Fy(x)p Fz(\))23 b Fv(\000)d Fy(\025\026)p Fv(j)25 b(\024)g(j)p Fy(')p Fz(\()p Fy(x)p Fz(\))d Fv(\000)e Fy(\025)p Fv(jj)p Fy( )s Fz(\()p Fy(x)p Fz(\))p Fv(j)i Fz(+)e Fv(j)p Fy(\025)p Fv(jj)p Fy( )s Fz(\()p Fy(x)p Fz(\))i Fv(\000)e Fy(\026)p Fv(j)25 b(\024)g Fy(C)c Fz(\()q Fv(j)p Fy(')p Fz(\()p Fy(x)p Fz(\))g Fv(\000)f Fy(\025)p 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Fv(g)p Fz(.)55 b(W)-8 b(e)36 b(also)f(use)g(the)g(notations)83 2086 y Fr(F)139 2053 y FF(\000)p Fw(1)274 2086 y Fz(:=)40 b Fv(f)p Fy(F)526 2053 y FF(\000)p Fw(1)662 2086 y Fv(j)h Fy(F)53 b Fv(2)40 b Fr(F)p Fv(g)g Fz(and)f Fr(G)1338 2053 y FF(\000)p Fw(1)1473 2086 y Fz(:=)i Fv(f)p Fy(G)1726 2053 y FF(\000)p Fw(1)1862 2086 y Fv(j)g Fy(G)g Fv(2)f Fr(G)p Fv(g)p Fz(.)69 b(Plainly)-8 b(,)43 b Fr(F)2748 2053 y FF(\000)p Fw(1)2882 2086 y Fz(also)d(is)g(a)g(\014lter)g(and)f(its)83 2199 y(asso)s(ciated)32 b(an)m(ti\014lter)f(is)g Fr(G)1050 2166 y FF(\000)p Fw(1)1144 2199 y Fz(.)249 2312 y(W)-8 b(e)26 b(shall)e(in)m(tro)s(duce)h(in)f FA([3.1])h Fz(t)m(w)m(o)g(t)m(yp)s(es)g(of)f(subsets)g(of)g Fy(X)32 b Fz(\()p Fr(G)p Fz(-syndetic)26 b(and)d Fr(G)3060 2279 y FF(\000)p Fw(1)3155 2312 y Fz(-replete\))j(whic)m(h,)83 2425 y(under)31 b(suitable)i(conditions,)h(will)f(b)s(e)f Fr(F)p Fz(-o)m(v)m(erlapping.)49 b(W)-8 b(e)34 b(also)g(consider)e(the) h(corresp)s(onding)f(classes)83 2538 y(of)40 b(functions.)68 b(As)40 b(a)g(consequence)g(of)g(the)g(results)g(in)f FA([2.2])p Fz(,)k(an)m(y)d Fr(G)p Fz(-syndetic)h(function)e(will)h(b)s (e)f Fr(F)p Fz(-)83 2651 y(indep)s(enden)m(t)28 b(with)i(resp)s(ect)f (to)h(an)m(y)g Fr(G)1466 2618 y FF(\000)p Fw(1)1561 2651 y Fz(-replete)h(one.)40 b(Unfortunately)-8 b(,)31 b(neither)e(of)h (these)g(t)m(w)m(o)h(classes)83 2764 y(of)38 b(functions)e(form)h(a)h Fy(C)969 2731 y FF(\003)1008 2764 y Fz(-algebra.)63 b(In)37 b FA([3.2])h Fz(and)f FA([3.3])h Fz(w)m(e)f(turn)g(to)h(conditions)g (stronger)f(than)g Fr(G)p Fz(-)83 2877 y(syndeticit)m(y)f(and)f Fr(G)p Fz(-repleteness,)j(whic)m(h)c(do)h(de\014ne)g Fy(C)2061 2844 y FF(\003)2100 2877 y Fz(-algebras.)56 b(W)-8 b(e)36 b(extend)f(and)g(study)f(for)h(large)83 2990 y(classes)g(of)f(\014lters)g Fr(F)f Fz(the)h(standard)f(notion)h (of)g(v)-5 b(anishing)33 b(oscillation)k(function.)50 b(The)34 b(relationship)g(of)83 3103 y Fr(G)154 3117 y Fw(st)214 3103 y Fz(-syndeticit)m(y)e(with)e(classical)i(notions)f (as)g(minimalit)m(y)g(and)f(almost)h(p)s(erio)s(dicit)m(y)g(is)g(also)g (examined.)249 3215 y(Note)26 b(that,)g(due)e(to)i(the)e(p)s(ossible)g (noncomm)m(utativit)m(y)j(of)e(the)f(group)g Fy(X)7 b Fz(,)27 b(w)m(e)e(ha)m(v)m(e)h(to)f(mak)m(e)g(c)m(hoices)83 3328 y(in)34 b(our)g(de\014nitions.)52 b(F)-8 b(or)35 b(instance,)h(m)m(y)e(syndeticit)m(y)i(ma)m(y)f(b)s(e)e(called)j (\\left)f(syndeticit)m(y")h(The)e(reader)83 3441 y(will)25 b(construct)g(easily)h(the)e(re\015ected)h(v)m(ersions.)40 b(In)23 b(certain)j(situations)f(\(only\),)i(the)e(t)m(w)m(o)h(v)m (ersions)f(de\014ne)83 3554 y(in)30 b(fact)h(the)g(same)g(ob)5 b(ject,)31 b(but)f(I)g(shall)h(not)f(insist)h(on)f(this)g(p)s(oin)m(t.) 83 3798 y Fj(3.1)112 b Fi(G)p Fj(-syndetic)39 b(and)f Fi(G)p Fj(-replete)83 3969 y FA(De\014nition)d(3.1)46 b Fu(\(a\))37 b(A)f(subset)g Fy(A)h Fu(of)g Fy(X)43 b Fu(is)37 b(c)-5 b(al)5 b(le)-5 b(d)37 b Fr(G)p Fu(-)p Fz(syndetic)g Fu(if)f(ther)-5 b(e)37 b(is)f(some)h Fy(G)c Fv(2)f Fr(G)k Fu(such)h(that)83 4082 y Fy(AG)27 b Fz(=)f Fy(X)7 b Fu(.)44 b(A)32 b(function)i Fy(')26 b Fv(2)g Fy(B)5 b(C)39 b Fu(is)33 b(c)-5 b(al)5 b(le)-5 b(d)35 b Fr(G)p Fu(-)p Fz(syndetic)c(at)j Fy(\025)26 b Fv(2)g Ft(C)33 b Fu(if)f Fy(M)2675 4096 y Fn(")2712 4082 y Fz(\()p Fy(')p Fz(;)15 b Fy(\025)p Fz(\))35 b Fu(is)e Fr(G)p Fu(-syndetic)h(for)f(al)5 b(l)83 4195 y Fy(")26 b(>)f Fz(0)p Fu(.)249 4308 y(\(b\))30 b(A)g(subset)g Fy(B)k Fu(of)c Fy(X)38 b Fu(is)30 b(c)-5 b(al)5 b(le)-5 b(d)31 b Fr(G)p Fu(-)p Fz(replete)g Fu(if)e(for)i(any)f Fy(G)c Fv(2)f Fr(G)30 b Fu(ther)-5 b(e)31 b(exists)g(an)f(element)g Fy(x)3492 4322 y Fn(G)3581 4308 y Fu(of)h Fy(X)83 4421 y Fu(such)g(that)h Fy(x)524 4435 y Fn(G)583 4421 y Fy(G)26 b Fv(\032)f Fy(B)5 b Fu(.)41 b(A)30 b(function)h Fy( )e Fv(2)c Fy(B)5 b(C)37 b Fu(is)30 b(c)-5 b(al)5 b(le)-5 b(d)33 b Fr(G)p Fu(-)p Fz(replete)c(at)j Fy(\025)25 b Fv(2)g Ft(C)30 b Fu(if)h Fy(M)3015 4435 y Fn(")3052 4421 y Fz(\()p Fy( )s Fz(;)15 b Fy(\025)p Fz(\))32 b Fu(is)f Fr(G)p Fu(-r)-5 b(eplete)83 4534 y(for)33 b(al)5 b(l)33 b Fy(")26 b(>)f Fz(0)p Fu(.)249 4746 y Fz(Since)38 b Fr(F)f Fz(and)g Fr(G)h Fz(determine)g(eac)m(h)h(other,)h(w)m(e)e(could) g(ha)m(v)m(e)h(said)f(\\)p Fr(F)p Fz(-syndetic")g(and)f(\\)p Fr(F)p Fz(-replete")83 4859 y(and)f(this)g(w)m(ould)g(ha)m(v)m(e)i(led) e(to)h(a)g(more)g(uniform)e(system)h(of)h(notations.)60 b(I)36 b(ha)m(v)m(e)h(c)m(hosen)g(the)g(presen)m(t)83 4972 y(terminology)f(b)s(ecause)e(it)g(seems)g(to)h(me)f(closer)h(to)g (the)g(substance)e(of)i(the)f(t)m(w)m(o)h(prop)s(erties.)52 b(An)m(yho)m(w,)83 5085 y(w)m(e)31 b(list)g(here)f(for)g(further)f(use) h(equiv)-5 b(alen)m(t)32 b(conditions)f(giv)m(en)g(in)f(terms)h(of)f Fr(F)p Fz(:)249 5198 y(\(a'\))i Fy(A)e Fz(is)h Fr(G)p Fz(-syndetic)g(i\013)g(there)f(exists)h Fy(F)38 b Fv(2)25 b Fr(F)30 b Fz(suc)m(h)g(that)h Fv(\\)2392 5212 y Fn(y)r FF(2)p Fn(A)2533 5198 y Fz(\()p Fy(y)s(F)13 b Fz(\))26 b(=)f Fv(;)p Fz(.)249 5311 y(\(b'\))31 b Fy(B)k Fz(is)30 b Fr(G)p Fz(-replete)i(i\013)f(for)f(an)m(y)g Fy(F)39 b Fv(2)25 b Fr(F)30 b Fz(there)g(is)g(some)h Fy(x)2312 5325 y Fn(F)2396 5311 y Fv(2)25 b Fy(X)38 b Fz(suc)m(h)30 b(that)h Fg({)p Fy(B)f Fv(\032)25 b Fy(x)3289 5325 y Fn(F)3347 5311 y Fy(F)13 b Fz(.)249 5424 y(Ob)m(viously)-8 b(,)28 b(the)e(t)m(w)m(o)i(prop)s(erties)e(ab)s(o)m(v)m(e)h(are)g (presev)m(ed)f(when)g(going)h(to)g(sup)s(ersets)e(but,)i(in)f(general,) 83 5537 y(are)36 b(not)h(stable)g(under)d(in)m(tersections.)59 b(By)37 b(replacing)g Fr(F)e Fz(with)h(a)g(\014ner)f(\014lter,)j(one)e (also)h(gets)g(a)g(ric)m(her)1903 5836 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 83 482 a Fz(an)m(ti\014lter;)32 b(therefore)f(there)f(will)h(b)s(e)f(more)g(syndetic)h(and)f(less)g (replete)i(sets.)41 b(If,)30 b(for)g(instance,)h Fr(F)f Fz(is)h(the)83 594 y(trivial)g(\014lter,)g(then)f(no)g(set)h(will)g(b)s (e)f Fr(G)p Fz(-syndetic)h(and)f(all)h(the)g(sets)g(will)f(b)s(e)g Fr(G)p Fz(-replete)i(.)249 707 y(Due)e(to)h(the)f(results)g(in)g FA([2.2])p Fz(,)h(a)f(basic)h(fact)g(will)f(b)s(e)f(the)i(follo)m(wing) g(\\dualit)m(y)g(prop)s(ert)m(y")f(\(see)h(also)83 820 y([7)q(])f(for)g(a)h(somewhat)g(related)g(setting\):)83 1031 y FA(Lemma)k(3.2)46 b Fu(\(a\))33 b Fy(A)g Fu(is)f Fr(G)p Fu(-syndetic)h(i\013)g Fg({)p Fy(A)g Fu(is)f(not)i Fr(G)2033 998 y FF(\000)p Fw(1)2127 1031 y Fu(-r)-5 b(eplete)33 b(.)249 1144 y(\(b\))g Fy(B)k Fu(is)32 b Fr(G)672 1111 y FF(\000)p Fw(1)767 1144 y Fu(-r)-5 b(eplete)33 b(i\013)f Fg({)p Fy(B)38 b Fu(is)32 b(not)i Fr(G)p Fu(-syndetic)f(.)249 1354 y(Pr)-5 b(o)g(of:)42 b Fz(The)30 b(t)m(w)m(o)h(assertions)g(are)g (equiv)-5 b(alen)m(t,)32 b(so)f(w)m(e)g(shall)f(only)h(pro)m(v)m(e)g (the)g(second)f(one:)249 1467 y Fg({)p Fy(B)35 b Fz(is)c(not)f Fr(G)p Fz(-syndetic)62 b Fv(,)30 b Fz(for)g(an)m(y)h Fy(G)26 b Fv(2)f Fr(G)p Fz(,)31 b Fv(9)p Fy(x)1985 1481 y Fn(G)2079 1467 y Fy(=)-56 b Fv(2)25 b Fz(\()p Fg({)p Fy(B)5 b Fz(\))p Fy(G)32 b Fv(,)249 1580 y(,)e Fz(for)g(an)m(y)h Fy(G)26 b Fv(2)f Fr(G)p Fz(,)31 b Fv(9)p Fy(x)1094 1594 y Fn(G)1183 1580 y Fz(suc)m(h)f(that)h Fy(x)1637 1594 y Fn(G)1696 1580 y Fy(G)1767 1547 y FF(\000)p Fw(1)1887 1580 y Fv(\032)25 b Fy(B)35 b Fv(,)30 b Fy(B)35 b Fz(is)30 b Fr(G)2474 1547 y FF(\000)p Fw(1)2569 1580 y Fz(-replete)h(.)3697 1693 y Fg(\004)249 1806 y Fz(Let)21 b(us)f(denote)h(b)m(y)g Fr(GS)g Fz(\(resp.)37 b Fr(G)1407 1773 y FF(\000)p Fw(1)1502 1806 y Fr(R)p Fz(\))21 b(the)f(family)h(of)g Fr(G)p Fz(-syndetic)h (\(resp.)37 b Fr(G)2921 1773 y FF(\000)p Fw(1)3015 1806 y Fz(-replete)22 b(\))f(subsets)f(of)83 1942 y Fy(X)7 b Fz(.)40 b(By)26 b(using)f(Lemma)h(3.2)i(and)d(the)h(notations)h(of)f FA([2.2])h Fz(w)m(e)f(can)h(write)f Fr(GS)2766 1909 y Fw([)p Fs(F)2828 1917 y Ff(tr)2880 1909 y Fw(])2929 1942 y Fz(=)f Fg({)3085 1841 y Fq(\020)3140 1942 y Fr(GS)3286 1909 y Fh({)3326 1841 y Fq(\021)3406 1942 y Fz(=)g Fr(G)3573 1909 y FF(\000)p Fw(1)3668 1942 y Fr(R)p Fz(,)83 2072 y(since)35 b Fr(GS)g Fz(is)f(ob)m(viously)h Fr(F)1048 2086 y Fw(tr)1107 2072 y Fz(-admissible)g(for)f(an)m(y)h Fr(G)p Fz(.)53 b(Remark)-5 b(ably)d(,)36 b(the)e(in)m(tersection)j(b)s (et)m(w)m(een)e(a)f Fr(G)p Fz(-)83 2185 y(syndetic)24 b(and)f(a)h Fr(G)743 2152 y FF(\000)p Fw(1)838 2185 y Fz(-replete)h(set)f(is)g(nev)m(er)g(v)m(oid.)39 b(Ho)m(w)m(ev)m(er,)28 b(in)23 b(general,)j(this)e(in)m(tersection)i(ma)m(y)e(b)s(elong)83 2298 y(to)37 b Fr(G)p Fz(.)58 b(Let)36 b(for)g(example)g Fy(X)42 b Fz(=)35 b Ft(R)g Fz(and)g Fr(F)g Fz(=)f Fr(F)1786 2265 y FF(\000)p Fw(1)1916 2298 y Fz(the)i(\014lter)g(of)g(all)h(the)f (subsets)f(of)h Ft(R)g Fz(con)m(taining)h Ft(Z)p Fz(.)83 2411 y(Then)c(\()p Fv(\000)p Fy(";)15 b(")p Fz(\))36 b(is)e(clearly)h Fr(G)p Fz(-syndetic)g(for)f(an)m(y)g Fy(")e(>)f Fz(0)j(and)g Fg({)p Ft(Z)g Fz(is)g Fr(G)2536 2378 y FF(\000)p Fw(1)2630 2411 y Fz(-replete)i(but)d(their)h(\(non-v)m (oid\))83 2524 y(in)m(tersection)e(is)f(in)f Fr(G)p Fz(.)249 2637 y(T)-8 b(o)21 b(eliminate)h(examples)e(of)h(this)f(t)m(yp)s(e,)j (w)m(e)e(mak)m(e)g(use)f(of)g(the)h(results)f(of)g FA([2.2])p Fz(.)38 b(It)21 b(will)f(b)s(e)g(con)m(v)m(enien)m(t)83 2749 y(to)31 b(imp)s(ose)f(explicit)i(assumptions:)249 2862 y FA(Hyp)s(othesis)249 2975 y Fz(\(H1\))g Fr(F)e Fz(is)g(in)m(v)-5 b(arian)m(t)32 b(under)c(left)j(translations:)42 b Fy(x)25 b Fv(2)g Fy(X)r(;)46 b(F)39 b Fv(2)24 b Fr(F)56 b Fv(\))f Fy(xF)38 b Fv(2)25 b Fr(F)p Fz(.)249 3088 y(\(H2\))32 b Fy(G)535 3102 y Fw(1)575 3088 y Fy(;)15 b(G)686 3102 y Fw(2)751 3088 y Fv(2)25 b Fr(G)56 b Fv(\))f Fy(G)1181 3102 y Fw(1)1221 3088 y Fy(G)1292 3102 y Fw(2)1358 3088 y Fv(2)24 b Fr(G)p Fz(.)249 3201 y(W)-8 b(e)28 b(note)g(that)f Fr(F)g Fz(is)g(\014ner)f(than)g(the)h(F)-8 b(r)m(\023)-43 b(ec)m(het)30 b(\014lter)d(i\013)g(for)g(ev)m(ery)h Fy(x)d Fv(2)g Fy(X)34 b Fz(there)27 b(exists)h(some)f Fy(G)f Fv(2)f Fr(G)83 3314 y Fz(suc)m(h)30 b(that)h Fy(x)25 b Fv(2)g Fy(G)p Fz(.)41 b(In)30 b(this)g(case)i(\(H1\))f(follo)m(ws)h (from)e(\(H2\).)83 3525 y FA(Theorem)35 b(3.3)46 b Fu(L)-5 b(et)32 b Fr(F)e Fu(b)-5 b(e)32 b(a)f(\014lter)h(on)g(the)g(lo)-5 b(c)g(al)5 b(ly)33 b(c)-5 b(omp)g(act,)34 b(non-c)-5 b(omp)g(act)34 b(gr)-5 b(oup)32 b Fy(X)39 b Fu(satisfying)32 b(\(H1\))83 3638 y(and)i(\(H2\).)42 b(Then:)249 3750 y(\(a\))33 b(The)g(families)g Fr(GS)h Fu(and)f Fr(G)1354 3717 y FF(\000)p Fw(1)1449 3750 y Fr(R)g Fu(ar)-5 b(e)33 b Fr(F)p Fu(-overlapping.)249 3863 y(\(b\))h(L)-5 b(et)35 b Fy(';)15 b( )32 b Fv(2)c Fy(B)5 b(C)i Fu(.)46 b(If)34 b Fv(j)p Fy(')p Fv(j)h Fu(is)f Fr(G)p Fu(-syndetic)h(at)f Fv(k)29 b Fy(')g Fv(k)2173 3878 y Fs(F)2253 3863 y Fu(and)36 b Fv(j)p Fy( )s Fv(j)f Fu(is)f Fr(G)2749 3830 y FF(\000)p Fw(1)2844 3863 y Fu(-r)-5 b(eplete)34 b(at)h Fv(k)29 b Fy( )i Fv(k)3489 3878 y Fs(F)3536 3863 y Fu(,)j(then)83 3976 y Fy(')p Fu(,)f Fy( )j Fu(ar)-5 b(e)33 b Fr(F)p Fu(-indep)-5 b(endent.)249 4187 y(Pr)g(o)g(of:)43 b Fz(W)-8 b(e)32 b(b)s(egin)e(b)m(y)h(sho)m(wing)g(that)h(the)f(class)h Fr(GS)f Fz(is)g 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b(G)215 4605 y FF(0)264 4638 y Fz(=)25 b Fy(X)7 b Fz(,)31 b(where)f Fy(G)832 4605 y FF(0)881 4638 y Fz(=)25 b Fy(G)1048 4652 y Fw(0)1108 4638 y Fv([)20 b Fz([)p Fy(x)1266 4605 y FF(\000)p Fw(1)1360 4638 y Fz(\()p Fy(GG)1537 4652 y Fw(0)1578 4638 y Fz(\)])26 b Fv(2)f Fr(G)30 b Fz(b)m(y)h(\(H1\))g(and)f (\(H2\).)249 4774 y(No)m(w,)35 b(b)m(y)f(com)m(bining)g(Lemma)f(2.7)i (with)e(Lemma)h(3.2)h(one)e(gets)i(\()p Fr(GS)p Fz(\))2796 4741 y Fw([)p Fs(F)p Fw(])2913 4774 y Fz(=)c Fg({)3075 4673 y Fq(\020)3130 4774 y Fr(GS)3276 4741 y Fh({)3316 4673 y Fq(\021)3401 4774 y Fz(=)f Fr(G)3573 4741 y FF(\000)p Fw(1)3668 4774 y Fr(R)p Fz(,)83 4910 y(that)h(is)f(a)h(strengthened)f (form)g(of)h(the)f(p)s(oin)m(t)h(\(a\).)42 b(Then)29 b(\(b\))h(follo)m(ws)i(from)e(Prop)s(osition)g(2.9.)283 b Fg(\004)249 5023 y FA(Example.)70 b Fz(As)41 b(it)f(w)m(as)h(already) g(remark)m(ed)g(b)s(efore,)h(no)f(set)f(is)h(syndetic)f(with)g(resp)s (ect)h(to)g(the)83 5135 y(trivial)36 b(\014lter)g(and)e(all)j(the)e (sets)h(are)f(replete.)57 b(So)35 b(the)g(conclusions)h(of)f(Theorem)g (3.3)i(hold)e(\(in)g(a)h(v)m(ery)83 5248 y(formal)31 b(w)m(a)m(y\),)h(as)e(w)m(ell)i(as)e(its)h(h)m(yp)s(othesis.)249 5361 y FA(Example.)63 b Fz(Our)37 b(main)h(example)g(is)g(the)h (standard)e(\014lter.)63 b(Instead)38 b(of)g Fr(G)2968 5375 y Fw(st)3028 5361 y Fz(-syndetic)h(and)e Fr(G)3678 5375 y Fw(st)3738 5361 y Fz(-)83 5474 y(replete)43 b(w)m(e)g(shall)g (sa)m(y)-8 b(,)47 b(resp)s(ectiv)m(ely)-8 b(,)48 b Fu(asymptotic)-5 b(al)5 b(ly)47 b(syndetic)c Fz(and)f Fu(asymptotic)-5 b(al)5 b(ly)47 b(r)-5 b(eplete)43 b Fz(\(b)s(oth)83 5587 y(for)37 b(sets)g(and)g(for)g(functions\),)i(although)e(this)g(is)h (not)f(standard)f(terminology)-8 b(.)63 b(In)36 b(the)i(v)m(ector)h (group)1903 5836 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 83 482 a Fy(X)36 b Fz(=)28 b Ft(R)359 449 y Fn(n)405 482 y Fz(,)33 b Ft(Z)524 449 y Fn(n)603 482 y Fz(is)f(asymptotically)j(syndetic)d(without)g(b)s(eing)g 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b(consider)e(this)h(example)g(imp)s(ortan)m(t)83 1836 y(enough)30 b(to)h(state)h(apart)83 2049 y FA(Corollary)j(3.4)46 b Fu(\(a\))30 b(The)f(interse)-5 b(ction)29 b(b)-5 b(etwe)g(en)30 b(an)f(asymptotic)-5 b(al)5 b(ly)32 b(syndetic)d(set)g(and)h(an)f (asymptot-)83 2162 y(ic)-5 b(al)5 b(ly)33 b(r)-5 b(eplete)34 b(one)f(c)-5 b(annot)34 b(b)-5 b(e)32 b(r)-5 b(elatively)34 b(c)-5 b(omp)g(act.)249 2275 y(\(b\))25 b(If)f Fv(j)p Fy(')p Fv(j)i Fu(is)e(asymptotic)-5 b(al)5 b(ly)28 b(syndetic)d(at)h Fv(k)f Fy(')h Fv(k)1935 2289 y Fa(asy)2064 2275 y Fu(and)g Fv(j)p Fy( )s Fv(j)g Fu(is)e(asymptotic)-5 b(al)5 b(ly)28 b(r)-5 b(eplete)26 b(at)f Fv(k)h Fy( )i Fv(k)3635 2289 y Fa(asy)3740 2275 y Fu(,)83 2388 y(then)33 b Fy(')g Fu(and)h Fy( )i Fu(ar)-5 b(e)33 b(asymptotic)-5 b(al)5 b(ly)36 b(indep)-5 b(endent.)249 2600 y FA(Example.)58 b Fz(T)-8 b(raditionally)g(,)40 b(a)c Fr(G)1456 2614 y Fw(F)-6 b(r)1529 2600 y Fz(-syndetic)36 b(set)h(is)f(called)i Fu(r)-5 b(elatively)39 b(dense)p Fz(.)59 b(On)35 b(the)i(real)g(line)83 2713 y Fv([)144 2727 y Fn(n)p FF(2)p Fm(Z)282 2713 y Fz(\()p Fy(n)13 b Fv(\000)g Fy(";)i(n)e Fz(+)g Fy(")p Fz(\))26 b(is)h(relativ)m(ely)i(dense)d(for)h(an)m(y)g Fy(")f(>)f Fz(0.)39 b(But)27 b Ft(Z)g Fz(or)f(ev)m(en)i Fv([)2745 2727 y Fn(n)p FF(2)p Fm(Z)2879 2709 y Fp(\003)2934 2713 y Fz(\()p Fy(n)20 b Fv(\000)g Fz(1)p Fy(=)p Fv(j)p Fy(n)p Fv(j)p Fy(;)15 b(n)21 b Fz(+)f(1)p Fy(=)p Fv(j)p Fy(n)p Fv(j)p Fz(\))83 2826 y(are)27 b(asymptotically)i(syndetic)d (without)h(b)s(eing)f(relativ)m(ely)j(dense.)39 b(The)26 b(F)-8 b(r)m(\023)-43 b(ec)m(het)29 b(\014lter)e Fr(F)3213 2840 y Fw(F)-6 b(r)3311 2826 y Fz(satis\014es)27 b(the)83 2939 y(assumptions)j(of)g(Theorem)g(3.3.)249 3052 y FA(Example.)64 b Fz(W)-8 b(e)40 b(tak)m(e)f Fy(X)46 b Fz(=)38 b Ft(R)p Fz(.)65 b(A)38 b(subset)g Fy(F)51 b Fz(of)38 b Ft(R)g Fz(is)h(called)g Fu(lar)-5 b(ge)41 b(at)e Fz(+)p Fv(1)f Fz(if)g(it)h(con)m(tains)g(an)83 3165 y(in)m(terv)-5 b(al)29 b(of)f(the)g(form)f(\()p Fy(a;)15 b Fz(+)p Fv(1)p 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b(In)32 b(fact)i Fr(F)1162 3630 y Fw(+)1221 3616 y Fz(-admissibilit)m(y)g (follo)m(ws)g(immediatly)-8 b(,)36 b(implying)d(that)g(the)h (conclusions)83 3729 y(of)d(Theorem)f(3.3)h(are)g(true)f(in)g(this)g (case.)249 3842 y FA(Example.)52 b Fz(In)33 b Fy(X)39 b Fz(=)32 b Ft(R)1127 3809 y Fw(2)1200 3842 y Fz(w)m(e)i(consider)h (the)f(an)m(ti\014lter)h Fr(G)g Fz(of)f(all)h(the)g(subsets)e(of)h Ft(R)3226 3856 y Fw(1)3297 3842 y Fz(:=)e Ft(R)22 b Fv(\002)g(f)p Fz(0)p Fv(g)p Fz(;)83 3955 y(ob)m(viously)27 b(\(H2\))h(is)e (satis\014ed,)i(but)e(not)g(\(H1\).)41 b(Then)25 b Ft(R)2019 3969 y Fw(1)2085 3955 y Fz(is)h Fr(G)2243 3922 y FF(\000)p Fw(1)2338 3955 y Fz(-replete)h(and)f(an)m(y)h(in\014nite)f(v)m(ertical) j(line)83 4068 y(is)j Fr(G)p Fz(-syndetic)g(,)g(but)f(their)g(in)m (tersection)j(is)d(a)h(singleton)h(b)s(elonging)e(to)i Fr(G)p Fz(.)44 b(It)32 b(is)f(instructiv)m(e)i(to)f(see)g(in)83 4181 y(whic)m(h)e(w)m(a)m(y)h Fr(F)p Fz(-admissibilit)m(y)h(fail.)249 4294 y FA(Remark.)76 b Fz(W)-8 b(e)44 b(ha)m(v)m(e)f(no)m(w)f(rather)g (general)h(results)f(on)g Fr(F)p Fz(-indep)s(enden)m(t)f(functions)g (on)h(lo)s(cally)83 4407 y(compact)34 b(groups.)48 b(In)32 b(order)h(to)g(get)h(also)g Fr(F)p Fz(-indep)s(enden)m(t)e Fy(C)2271 4374 y FF(\003)2310 4407 y Fz(-algebras,)j(there)e(is)g (still)h(some)f(w)m(ork)g(to)83 4520 y(do.)39 b(Unfortunately)-8 b(,)29 b(the)d(prop)s(ert)m(y)g(of)h(b)s(eing)f(of)g(a)h(certain)g (class)h Fr(M)e Fz(\(ev)m(en)i(at)f Fu(al)5 b(l)27 b Fz(the)g(resonable)g(v)-5 b(alues)83 4633 y Fy(\025)p Fz(\))31 b(do)s(es)f(not)h(ha)m(v)m(e)h(the)e(tendancy)h(to)g(force)h (functions)e(to)h(form)f Fy(C)2449 4600 y FF(\003)2488 4633 y Fz(-algebras.)42 b(As)31 b(a)g(t)m(ypical)h(example,)83 4746 y(w)m(e)j(sk)m(etc)m(h)g(brie\015y)f(a)g(case)h(in)f(whic)m(h)g(t) m(w)m(o)i(functions)e Fy(')2065 4760 y Fw(1)2139 4746 y Fz(and)f Fy(')2378 4760 y Fw(2)2452 4746 y Fz(are)i(b)s(oth)e (asymptotically)k(syndetic)83 4858 y(at)e(all)f(the)h(p)s(oin)m(ts)e 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b(course,)h(there)g(is)g(some)g(arbitrariness)f(in)g(c)m(ho)s(osing)h (to)g(v)m(erify)g(the)g(prop)s(erties)f(at)h(the)g(v)-5 b(alues)29 b(of)83 2223 y(the)i Fr(F)p Fz(-range.)43 b(The)31 b(main)g(reason)g(for)g(this)f(c)m(hoice)j(is)e(the)h(fact)g (that)f(for)g(a)g(large)h(class)g(of)f(\014lters)g Fr(F)g Fz(\(for)83 2336 y(the)j(standard)e(\014lter)i(for)f(instance\),)j(the) d(elemen)m(ts)i(of)f Fr(G)g Fz(are)g(neither)f Fr(G)p Fz(-syndetic)h(nor)f Fr(G)3316 2303 y FF(\000)p Fw(1)3411 2336 y Fz(-replete)i(.)83 2449 y(Then)26 b(functions)h(cannot)h(b)s(e)e Fr(G)p Fz(-syndetic)i(or)f Fr(G)1755 2416 y FF(\000)p Fw(1)1850 2449 y Fz(-replete)h(at)g(v)-5 b(alues)28 b(not)f(b)s (elonging)g(to)h(their)g Fr(F)p Fz(-range,)83 2562 y(th)m(us)i(the)h (framew)m(ork)f(of)h(De\014nition)g(3.5)g(is)f(maximal)i(in)e(these)h (cases.)249 2675 y FA(Remark.)39 b Fz(Ev)m(en)25 b(for)f Fr(F)g Fz(=)h Fr(F)1267 2689 y Fw(st)1351 2675 y Fz(there)f(are)h(not)g (enough)f(constrain)m(ts)h(to)g(assure)f(uniform)f(con)m(tin)m(uit)m(y) -8 b(.)83 2788 y(The)27 b(function)g Fy(')p Fz(\()p Fy(x)p Fz(\))g(=)e(cos)16 b Fy(x)1113 2755 y Fw(2)1179 2788 y Fz(is)28 b(asymptotically)i(syndetic)e(without)f(b)s(eing)g (uniformly)g(con)m(tin)m(uous)h(on)83 2901 y Ft(R)p Fz(.)64 b(And)37 b(one)i(can)f(get)i(asymptotic)f(repleteness)g(just)f(b)s (ecause)g(of)h(the)f(b)s(eha)m(viour)g(at)h(+)p Fv(1)p Fz(,)h(so)e(that)83 3014 y(the)29 b(b)s(eha)m(viour)g(at)g Fv(\0001)g Fz(ma)m(y)g(lead)h(easily)g(to)f(the)g(failure)g(of)g (uniform)f(con)m(tin)m(uit)m(y)-8 b(.)43 b(T)-8 b(ak)m(e,)30 b(for)f(example,)83 3127 y Fy( )s Fz(\()p Fy(x)p Fz(\))d(=)f(cos)16 b Fy(x)578 3094 y Fw(1)p Fn(=)p Fw(2)719 3127 y Fz(on)30 b Ft(R)911 3141 y Fw(+)1000 3127 y Fz(and)g Fy( )s Fz(\()p Fy(x)p Fz(\))c(=)f(cos)16 b Fy(x)1672 3094 y Fw(2)1741 3127 y Fz(on)31 b Ft(R)1934 3141 y FF(\000)1992 3127 y Fz(.)249 3240 y(In)h(the)g(follo)m(wing)i(t)m(w)m(o)f(subsections)f (it)h(will)g(b)s(e)e(sho)m(wn)h(that)h(De\014nition)f(3.5)i(often)e (leads)h(to)g(large)83 3353 y(and)d(in)m(teresting)i Fy(C)780 3320 y FF(\003)818 3353 y Fz(-algebras,)g(so)f(it)g(is)f(w)m (orth)h(stating)83 3546 y FA(Corollary)k(3.6)46 b Fu(L)-5 b(et)28 b Fr(F)f Fu(b)-5 b(e)27 b(a)h(\014lter)g(on)g(the)g(lo)-5 b(c)g(al)5 b(ly)29 b(c)-5 b(omp)g(act,)31 b(non-c)-5 b(omp)g(act)29 b(gr)-5 b(oup)29 b Fy(X)7 b Fu(,)29 b(satisfying)f (\(H1\))83 3659 y(and)39 b(\(H2\).)57 b(L)-5 b(et)37 b Fv(A)h Fu(and)g Fv(B)i Fu(unital)e Fy(C)1428 3626 y FF(\003)1467 3659 y Fu(-sub)-5 b(algebr)g(as)39 b(of)f Fy(B)5 b(C)i Fu(.)55 b(Assume)38 b(that)g Fv(A)f Fu(is)h Fr(G)p Fu(-syndetic)g(and)h Fv(B)83 3772 y Fu(is)31 b Fr(G)250 3739 y FF(\000)p Fw(1)345 3772 y Fu(-r)-5 b(eplete)32 b(.)41 b(Then)32 b Fv(A)e Fu(and)j Fv(B)g Fu(ar)-5 b(e)33 b Fr(F)p Fu(-indep)-5 b(endent,)32 b(thus)f Fv(h)q(A)20 b(\001)g(B)s(i)2583 3799 y Fs(F)2660 3772 y Fu(is)32 b(c)-5 b(anonic)g(al)5 b(ly)33 b(isomorphic)g(to)83 3885 y Fv(A)156 3900 y Fs(F)222 3885 y Fv(\012)20 b(B)373 3900 y Fs(F)452 3885 y Fu(and)33 b Fy(X)726 3811 y Fq(\000)767 3885 y Fv(h)q(A)20 b(\001)g(B)s(i)1039 3912 y Fs(F)1085 3811 y Fq(\001)1159 3885 y Fu(is)33 b(c)-5 b(anonic)g(al)5 b(ly)34 b(home)-5 b(omorphic)37 b(to)c Fy(X)22 b Fz(\()q Fy(A)2644 3900 y Fs(F)2690 3885 y Fz(\))f Fv(\002)f Fy(X)i Fz(\()q Fy(B)3039 3900 y Fs(F)3085 3885 y Fz(\))p Fu(.)83 4125 y Fj(3.2)112 b(Syndeticit)m(y)38 b(in)g(the)f(sense)h(of)g (translations)h(and)f(minimalit)m(y)83 4297 y Fz(If)k(a)h(family)g Fr(M)g Fz(of)f(subsets)g(of)h Fy(X)50 b Fz(is)42 b(in)m(v)-5 b(arian)m(t)44 b(under)d(righ)m(t)i(translations,)j(the)d(fact)g(that)g Fy(')g Fz(is)g(of)83 4410 y(class)33 b Fr(M)f Fz(at)h(a)f(v)-5 b(alue)32 b Fy(\025)g Fz(can)g(b)s(e)f(expressed)h(in)f(translational)j (terms.)45 b(W)-8 b(e)33 b(presen)m(t)f(here)g(only)g(the)g(case)83 4523 y Fr(M)26 b Fz(=)f Fr(GS)p Fz(,)h(since)g(this)e(is)h(connected)h (\(at)f(least)h(in)f(the)g(case)h Fr(F)e Fz(=)h Fr(F)2409 4537 y Fw(st)2468 4523 y Fz(\))g(with)g(classical)i(t)m(yp)s(es)e(of)f (functions)83 4636 y(generalizing)32 b(the)f(usual)f(almost)h(p)s(erio) s(dic)f(functions.)83 4829 y FA(De\014nition)35 b(3.7)46 b Fu(L)-5 b(et)30 b Fy(')c Fv(2)f Fy(B)5 b(C)i Fu(,)29 b Fy(")c(>)g Fz(0)30 b Fu(and)h Fy(S)f Fv(\032)25 b Fy(X)7 b Fu(.)41 b(A)n(n)29 b(element)g Fy(y)k Fu(of)d Fy(X)36 b Fu(is)30 b(c)-5 b(al)5 b(le)-5 b(d)31 b Fz(an)e Fy(")p Fu(-)p Fz(translation)83 4942 y(elemen)m(t)38 b(for)h Fy(')g Fz(with)e(resp)s(ect)f(to)k Fy(S)j Fu(if)77 b Fv(j)p Fy(')p Fz(\()p Fy(y)s(x)p Fz(\))26 b Fv(\000)e Fy(')p Fz(\()p Fy(x)p Fz(\))p Fv(j)38 b Fy(<)e(")j Fu(for)g(al)5 b(l)39 b Fy(x)d Fv(2)g Fy(S)5 b Fu(.)60 b(The)39 b(set)g(of)f(al)5 b(l)40 b(these)83 5055 y(elements)33 b(wil)5 b(l)33 b(b)-5 b(e)32 b(denote)-5 b(d)35 b(by)d Fy(T)1244 5069 y Fn(")1281 5055 y Fz(\()p Fy(')p Fz(;)15 b Fy(S)5 b Fz(\))p Fu(.)249 5248 y Fz(The)21 b(set)g Fy(T)612 5262 y Fn(")649 5248 y Fz(\()p Fy(')p Fz(;)15 b Fy(S)5 b Fz(\))23 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5601 y Fn(")2621 5587 y Fz(\()p Fy(')p Fz(;)15 b Fy(S)5 b Fz(\))27 b(=)e Fv(\\)3035 5601 y Fn(x)p FF(2)p Fn(S)3172 5587 y Fy(T)3225 5601 y Fn(")3262 5587 y Fz(\()p Fy(')p Fz(;)15 b Fv(f)p Fy(x)p Fv(g)p Fz(\).)1880 5836 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 83 482 a FA(De\014nition)35 b(3.8)46 b Fu(L)-5 b(et)35 b Fy(')30 b Fv(2)e Fy(B)5 b(C)i Fu(.)47 b(We)34 b(shal)5 b(l)36 b(say)f(that)h Fy(')f Fz(is)g Fr(G)p Fu(-)p Fz(syndetic)e(in)f(the)h(sense)f(of)h(translations)83 594 y(in)f Fy(x)26 b Fv(2)e Fy(X)40 b Fu(if)33 b(the)g(set)f Fy(T)903 608 y Fn(")940 594 y Fz(\()p Fy(')p Fz(;)15 b Fv(f)p Fy(x)p Fv(g)p Fz(\))35 b Fu(is)e Fr(G)p Fu(-syndetic)g(for)g (any)g Fy(")26 b(>)f Fz(0)p Fu(.)83 795 y FA(Prop)s(osition)36 b(3.9)46 b Fu(Assume)23 b(that)i(the)f(\014lter)g Fr(F)e Fu(is)i(invariant)g(under)g(left)f(tr)-5 b(anslations.)42 b(F)-7 b(or)24 b(the)g(b)-5 b(ounde)g(d,)83 908 y(c)g(ontinuous)28 b(function)e Fy(')h Fu(on)g Fy(X)34 b Fu(and)28 b(the)f(c)-5 b(omplex)28 b(numb)-5 b(er)27 b Fy(\025)g Fu(the)g(fol)5 b(lowing)27 b(assertions)h(ar)-5 b(e)28 b(e)-5 b(quivalent:)249 1021 y(\(i\))33 b Fy(')g Fu(is)f Fr(G)p Fu(-syndetic)h(at)h Fy(\025)p Fu(.)249 1134 y(\(ii\))f Fy(')g Fu(is)f Fr(G)p Fu(-syndetic)h(in)g(the)g(sense)f(of)h(tr)-5 b(anslations)36 b(in)d(some)g(p)-5 b(oint)34 b Fy(x)e Fu(such)h(that)h Fy(')p Fz(\()p Fy(x)p Fz(\))26 b(=)f Fy(\025)p Fu(.)249 1247 y(\(iii\))33 b Fy(')g Fu(is)f Fr(G)p Fu(-syndetic)h(in)g(the)g (sense)f(of)h(tr)-5 b(anslations)36 b(in)c(any)i(p)-5 b(oint)34 b Fy(x)e Fu(such)h(that)h Fy(')p Fz(\()p Fy(x)p Fz(\))26 b(=)f Fy(\025)p Fu(.)249 1360 y(As)37 b(a)g(p)-5 b(articular)40 b(c)-5 b(ase,)38 b Fy(')g Fu(is)f Fr(G)p Fu(-syndetic)g(i\013)g(it)g(is)g Fr(G)p Fu(-syndetic)g(in)g(the)h (sense)f(of)g(tr)-5 b(anslations)40 b(in)83 1473 y(every)32 b(p)-5 b(oint)34 b(of)f Fy(')719 1440 y FF(\000)p Fw(1)829 1473 y Fz([)q Fy(R)924 1488 y Fs(F)970 1473 y Fz(\()p Fy(')p Fz(\)])q Fu(.)249 1673 y(Pr)-5 b(o)g(of:)40 b Fz(It)25 b(is)h(easily)h(sho)m(wn)e(that)h Fy(M)1504 1687 y Fn(")1541 1673 y Fz(\()p Fy(')p Fz(;)15 b Fy(')p Fz(\()p Fy(x)p Fz(\)\))28 b(=)d Fy(T)2068 1687 y Fn(")2105 1673 y Fz(\()p Fy(';)15 b(x)p Fz(\))p Fy(x)27 b Fz(for)f(an)m(y)g Fy(x)f Fv(2)g Fy(X)7 b Fz(.)39 b(Since)26 b Fr(F)f Fz(is)h(in)m(v)-5 b(arian)m(t)83 1786 y(under)23 b(left)i(translations,)i(the)e(family)g (of)g Fr(G)p Fz(-syndetic)h(sets)f(will)g(b)s(e)f(in)m(v)-5 b(arian)m(t)26 b(under)d(righ)m(t)i(translations,)83 1899 y(so)31 b Fy(M)283 1913 y Fn(")320 1899 y Fz(\()p Fy(')p Fz(;)15 b Fy(')p Fz(\()p Fy(x)p Fz(\)\))33 b(and)c Fy(T)932 1913 y Fn(")969 1899 y Fz(\()p Fy(';)15 b(x)p Fz(\))32 b(will)f(b)s(e)f(sim)m(ultaneously)h Fr(G)p Fz(-syndetic)g(.)1079 b Fg(\004)249 2012 y Fz(Our)24 b(rephrasing)g(of)i Fr(G)p Fz(-syndeticit)m(y)h(in)d(translational)j (terms)e(leads)h(to)g(connections)g(with)f(standard)83 2125 y(classes)31 b(of)g(functions,)f(and)g(some)h(of)f(these)h(do)f (form)g Fy(C)2047 2092 y FF(\003)2086 2125 y Fz(-algebras.)42 b(Some)30 b(preparation)h(is)f(needed.)249 2238 y(F)-8 b(or)35 b(eac)m(h)g Fy(y)f Fv(2)d Fy(X)42 b Fz(one)34 b(set)h Fy(r)1269 2252 y Fn(y)1310 2238 y Fy(;)15 b(l)1377 2252 y Fn(y)1451 2238 y Fz(:)31 b Fy(X)39 b Fv(!)32 b Fy(X)7 b Fz(,)35 b Fy(r)1927 2252 y Fn(y)1969 2238 y Fz(\()p Fy(x)p Fz(\))d(:=)f Fy(xy)37 b Fz(and)c Fy(l)2591 2252 y Fn(y)2633 2238 y Fz(\()p Fy(x)p Fz(\))f(:=)f Fy(y)s(x)j Fz(for)g(all)h Fy(x)c Fv(2)g Fy(X)7 b Fz(.)53 b(By)83 2351 y(dualit)m(y)-8 b(,)32 b(one)f(gets)g(automorphisms)f(of)g Fy(B)5 b(C)i Fz(:)40 b Fy(R)1784 2365 y Fn(y)1825 2351 y Fz(\()p Fy(')p Fz(\))27 b(:=)e Fy(')c Fv(\016)g Fy(r)2289 2365 y Fn(y)2360 2351 y Fz(and)30 b Fy(L)2599 2365 y Fn(y)2640 2351 y Fz(\()p Fy(')p Fz(\))d(:=)e Fy(')c Fv(\016)g Fy(l)3090 2365 y Fn(y)3162 2351 y Fz(for)30 b(all)h Fy(')26 b Fv(2)f Fy(B)5 b(C)i Fz(.)83 2464 y(Let)28 b(us)e(denote)h(b)m(y)g Fy(R)q(U)10 b(C)33 b Fz(the)27 b Fy(C)1233 2431 y FF(\003)1272 2464 y Fz(-algebra)i(of)e(all)h(b)s(ounded,)e(con)m(tin)m(uous)h (functions)g Fy(')g Fz(on)g Fy(X)34 b Fz(satisfying)83 2577 y(lim)209 2591 y Fn(y)r FF(!)p Fn(e)395 2577 y Fv(k)41 b Fy(R)550 2591 y Fn(y)592 2577 y Fy(')27 b Fv(\000)f Fy(')41 b Fv(k)p Fz(=)g(0.)69 b(Equiv)-5 b(alen)m(tly)d(,)44 b Fy(')d Fz(is)e(uniformly)g(con)m(tin)m(uous)i(with)e(resp)s(ect)h(to) g(the)g(left)83 2689 y(uniform)29 b(structure)h(on)g Fy(X)7 b Fz(.)249 2802 y(A)29 b Fy(C)418 2769 y FF(\003)457 2802 y Fz(-subalgebra)h Fv(C)k Fz(of)29 b Fy(R)q(U)10 b(C)35 b Fz(will)30 b(b)s(e)e(called)j Fu(an)e Fy(X)7 b Fz(-)p Fu(algebr)-5 b(a)31 b Fz(if)e(it)h(is)f(unital)h(and)e(in)m(v) -5 b(arian)m(t)31 b(under)83 2915 y(the)38 b(left)g(translations:)56 b Fy(L)1021 2929 y Fn(y)1062 2915 y Fv(C)43 b(\032)37 b(C)43 b Fz(for)37 b(all)i Fy(y)h Fv(2)d Fy(X)7 b Fz(.)62 b Fy(R)q(U)10 b(C)44 b Fz(itself)39 b(is)e(an)h Fy(X)7 b Fz(-algebra.)64 b(Once)37 b(again)i(b)m(y)83 3028 y(dualit)m(y)-8 b(,)45 b(the)c(sp)s(ectrum)f Fy(X)7 b Fz(\()p Fv(C)e Fz(\))42 b(of)f Fv(C)k Fz(b)s(ecomes)d(the)f(compact)h(space)f(of)g(a)g (\(top)s(ological\))j(dynamical)83 3141 y(system)34 b(with)g(group)g Fy(X)7 b Fz(.)53 b(If)34 b Fy(')h Fz(is)f(an)g(elemen)m(t)i(of)f Fy(R)q(U)10 b(C)d Fz(,)34 b(w)m(e)h(denote)g(b)m(y)f Fv(C)2769 3155 y Fn(')2853 3141 y Fz(the)h(smallest)g Fy(X)7 b Fz(-algebra)83 3254 y(con)m(taining)32 b Fy(')p Fz(.)249 3367 y(W)-8 b(e)42 b(sa)m(y)f(that)g(the)g Fy(X)7 b Fz(-algebra)42 b Fv(C)j Fz(is)c Fy(X)7 b Fz(-)p Fu(simple)42 b Fz(if)e(it)h(con)m(tains)h(no)e(prop)s(er)f(left)i(in)m(v)-5 b(arian)m(t)42 b(ideal.)83 3480 y(Plainly)-8 b(,)44 b(this)d(is)f (equiv)-5 b(alen)m(t)42 b(with)e(the)h(fact)g(that)g Fy(X)7 b Fz(\()p Fv(C)e Fz(\))42 b(is)f(minimal,)i(i.e.)72 b(it)41 b(con)m(taines)h(no)e(prop)s(er)83 3593 y(closed,)31 b(in)m(v)-5 b(arian)m(t)32 b(subset)e(\(equiv)-5 b(alen)m(tly)d(,)33 b(all)e(the)g(orbits)f(are)h(dense\).)249 3706 y(No)m(w)h(w)m(e)f(are)g 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Fy(C)3373 496 y Fw(0)3412 482 y Fz(-function)83 594 y(is)c(asymptotically)j (syndetic,)e(without)f(b)s(eing)g(minimal.)46 b(F)-8 b(urthermore,)33 b(for)f(a)h(minimal)f(function)g(the)83 707 y(range)42 b(and)f(the)g(asymptotic)i(range)f(are)g(the)g(same,)i (while)e(this)f(can)h(b)s(e)f(eviden)m(tly)h(false)h(for)e(man)m(y)83 820 y(asymptotically)46 b(syndetic)d(functions.)79 b(By)43 b(one)h(of)f(the)h(examples)g(of)f FA([3.1])h Fz(one)f(constructs)h (easily)83 933 y(functions)30 b(that)h(are)g(asymptotically)h(syndetic) f(without)f(b)s(eing)g Fr(G)2446 947 y Fw(F)-6 b(r)2518 933 y Fz(-syndetic.)249 1046 y(It)37 b(is)f(kno)m(wn)h(that)g(minimal)g (functions)f(form)g(a)h(closed,)i(left)e(in)m(v)-5 b(arian)m(t)38 b(subset)e(of)h Fy(B)5 b(C)43 b Fz(whic)m(h)36 b(is)83 1159 y(not)k(a)h(v)m(ector)g(space)g(in)f(general.)70 b(But,)43 b(fortunately)-8 b(,)44 b(there)c(is)g(a)h(w)m(a)m(y)g(to)g (put)e(in)m(to)i(evidence)g(large)83 1272 y Fy(C)155 1239 y FF(\003)194 1272 y Fz(-algebras)c(comp)s(osed)f(of)g(minimal)h (functions;)h(they)f(will)f(b)s(e)g Fr(G)2490 1286 y Fw(F)-6 b(r)2562 1272 y Fz(-syndetic,)38 b(th)m(us)e(asymptotically)83 1385 y(syndetic)d Fy(C)514 1352 y FF(\003)553 1385 y Fz(-algebras.)48 b(First,)34 b(a)f Fy(C)1379 1352 y FF(\003)1418 1385 y Fz(-algebra)h(con)m(tains)g(only)f(minimal)f(functions)g(i\013)h (it)g(is)g Fy(X)7 b Fz(-simple,)83 1498 y(i.e.)41 b(its)29 b(sp)s(ectrum)e(is)h(a)g(minimal)h(dynamical)g(system.)40 b(It)28 b(is)g(in)m(teresting)i(to)f(consider)f(maximal)h(\(under)83 1611 y(inclusion\))i Fv(C)551 1578 y FF(\003)591 1611 y Fz(-algebras)h(of)g(this)f(kind.)42 b(They)31 b(are)g(in)g (one-to-one)i(corresp)s(ondence)e(with)g(minimal)g(left)83 1724 y(in)m(v)-5 b(arian)m(t)44 b(ideals)g(of)f Fy(R)q(U)10 b(C)d Fz(.)77 b(Ev)m(en)44 b(the)f(in)m(tersection)h(of)f(all)h(the)f (maximal)h Fy(X)7 b Fz(-simple)44 b Fy(C)3375 1691 y FF(\003)3414 1724 y Fz(-algebras)83 1836 y(con)m(tains)c(man)m(y)f 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Fz(=)25 b Fy(W)13 b(AP)g Fz(,)28 b(but)g(in)g(most)h(cases)h(\(ex:)40 b Fy(X)c Fz(ab)s(elian\))29 b(the)g(inclusion)83 2966 y(is)34 b(strict.)53 b(Unfortunately)-8 b(,)36 b(w)m(eakly)f(almost)g (p)s(erio)s(dic)f(functions)f(ma)m(y)i(fail)g(to)f(ha)m(v)m(e)i(go)s(o) s(d)e(syndeticit)m(y)83 3078 y(prop)s(erties.)53 b(Let,)37 b(for)d(example,)j Fy(S)g Fz(=)32 b Fv(f)p Fy(x)1576 3092 y Fn(n)1623 3078 y Fv(g)1668 3092 y Fn(n)p FF(2)p Fm(N)1843 3078 y Fv(\032)g Fy(X)40 b Fv(\021)32 b Ft(Z)j Fz(suc)m(h)f(that)h Fy(x)2722 3092 y Fn(n)p Fw(+1)2882 3078 y Fv(\000)23 b Fy(x)3028 3092 y Fn(n)3107 3078 y Fv(!)33 b(1)p Fz(.)53 b(Then)34 b(the)83 3191 y(c)m(haracteristic)28 b(function)d Fy(\037)1045 3205 y Fn(S)1122 3191 y Fz(of)g Fy(S)31 b Fz(is)25 b(w)m(eakly)i(almost)g(p)s(erio)s(dic,)f(b)m(y)f(a)h (result)g(of)f([3)q(],)i(but)e Fy(S)30 b Fz(=)25 b Fy(M)3467 3205 y Fn(")3504 3191 y Fz(\()p Fy(\037)3596 3205 y Fn(S)3647 3191 y Fz(;)15 b(1\))83 3304 y(\()p Fy(")31 b Fz(small\))g(is)g(not)f 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FF(\003)305 942 y Fz(-algebras)h(of)e Fr(G)864 909 y FF(\000)p Fw(1)959 942 y Fz(-replete)h(functions.)83 1143 y FA(De\014nition)k(3.15)47 b Fu(\(a\))27 b(L)-5 b(et)28 b Fy( )g Fv(2)d Fy(B)5 b(C)33 b Fu(and)28 b Fy(M)36 b Fv(\032)24 b Fy(X)7 b Fu(.)41 b(We)26 b(set)i Fz(osc)2405 1157 y Fn(M)2484 1143 y Fz(\()p Fy( )s Fz(\))f(:)e Fy(X)33 b Fv(!)25 b Ft(R)2983 1157 y Fw(+)3042 1143 y Fu(,)55 b Fz([osc)3272 1157 y Fn(M)3351 1143 y Fz(\()p Fy( )s Fz(\)])16 b(\()p Fy(x)p Fz(\))26 b(:=)83 1256 y(sup)220 1278 y Fn(y)r FF(2)p Fn(M)398 1256 y Fv(j)p Fy( )s Fz(\()p Fy(xy)s Fz(\))c Fv(\000)e Fy( )s Fz(\()p Fy(x)p Fz(\))p Fv(j)p Fu(;)33 b(we)g(c)-5 b(al)5 b(l)34 b(it)e Fz(the)f(oscillation)h (of)f(the)g(function)h Fy( )k Fz(on)30 b(the)h(set)i Fy(M)10 b Fu(.)249 1369 y(\(b\))33 b(L)-5 b(et)32 b Fr(F)g Fu(and)i Fr(F)875 1336 y FF(0)930 1369 y Fu(two)f(\014lters)h(on)f Fy(X)7 b Fu(.)42 b(We)32 b(say)h(that)h(the)e(function)h Fy( )c Fv(2)c Fy(B)5 b(C)38 b Fz(has)30 b(v)-5 b(anishing)32 b Fr(G)3643 1336 y FF(\000)p Fw(1)3738 1369 y Fz(-)83 1482 y(oscillation)f(along)f(the)f(\014lter)j Fr(F)1182 1449 y FF(0)1236 1482 y Fu(if)f(for)h(al)5 b(l)31 b Fy(G)26 b Fv(2)f Fr(G)31 b Fu(one)h(has)g Fz(lim)2339 1497 y Fs(F)2381 1479 y Fp(0)2422 1482 y Fz([)q(osc)2569 1502 y Fn(G)2624 1483 y Fp(\000)p Ff(1)2711 1482 y Fz(\()p Fy( )s Fz(\)])27 b(=)e(0)p Fu(.)41 b(The)32 b(family)g(of)f(al)5 b(l)83 1595 y(these)33 b(functions)g(wil)5 b(l)33 b(b)-5 b(e)32 b(denote)-5 b(d)35 b(by)d Fy(V)20 b(O)1584 1615 y Fs(G)1638 1596 y Fp(\000)p Ff(1)1726 1595 y Fz(\()p Fr(F)1817 1562 y FF(0)1840 1595 y Fz(\))p Fu(.)249 1796 y Fz(The)32 b(b)s(eha)m(viour)h(required)f(b)m(y)h(the)g(prop)s(ert)m (y)f(of)h(b)s(elonging)g(to)h Fy(V)20 b(O)2660 1816 y Fs(G)2714 1797 y Fp(\000)p Ff(1)2801 1796 y Fz(\()p Fr(F)2892 1763 y FF(0)2916 1796 y Fz(\))33 b(is)g(quite)g(transparen)m(t)83 1909 y(in)c(certain)h(particular)g(cases.)41 b(Let)30 b(us)f(supp)s(ose,)f(for)h(instance,)i(that)f Fy(X)j Fz(=)25 b Ft(R)2799 1876 y 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Fz(\))f(en)m(tirely)-8 b(,)29 b(one)d(should)f(consider)g(the)i(case)f Fr(F)3407 900 y FF(0)3456 933 y Fz(=)f Fr(F)p Fz(:)37 b Fu(al)5 b(l)83 1046 y(the)37 b(functions)f(in)h Fy(V)20 b(O)894 1066 y Fs(G)948 1047 y Fp(\000)p Ff(1)1035 1046 y Fz(\()p Fr(F)p Fz(\))36 b Fu(ar)-5 b(e)38 b Fr(G)1427 1013 y FF(\000)p Fw(1)1521 1046 y Fu(-r)-5 b(eplete)37 b Fz(.)53 b(Ho)m(w)m(ev)m(er,)38 b(it)c(migh)m(t)i(happ)s(en)c(that)j(the)g(imp)s (ortan)m(t)83 1159 y(v)-5 b(alue)31 b Fy(\025)25 b Fz(=)p Fv(k)h Fy( )i Fv(k)670 1174 y Fs(F)747 1159 y Fz(still)k(b)s(elongs)e (to)h Fy(R)1431 1174 y Fs(F)1473 1155 y Fp(0)1500 1159 y Fz(\()p Fy( )s Fz(\))g(ev)m(en)g(if)g Fr(F)24 b Ft($)h Fr(F)2186 1126 y FF(0)2209 1159 y Fz(.)249 1272 y FA(Example.)51 b Fz(Prop)s(osition)34 b(3.16)h(\(b\))f(giv)m(es)h(us)e(a)h(hin)m(t)g (ab)s(out)g(the)g(w)m(a)m(y)h(the)f(class)g(of)g(all)h(the)f Fr(G)3643 1239 y FF(\000)p Fw(1)3738 1272 y Fz(-)83 1385 y(replete)c(functions)f(fails)h(to)g(form)f(an)g(algebra)h(or)g(at)g 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1860 y Ff(+)2067 1836 y Fz(\()p Fy( )s Fz(\))h(=)f([)p Fv(\000)p Fz(1)p Fy(;)15 b Fz(1])36 b(\(and)e(it)h(w)m(ould)f(b)s(e)g(imp)s (ossible,)83 1949 y(of)j(course,)i(to)e(b)s(e)g Fr(G)824 1916 y FF(\000)p Fw(1)918 1949 y Fz(-replete)h(outside)f(this)g(set\).) 61 b(The)37 b(same)g(is)g(true)f(for)h Fy( )2921 1916 y FF(0)2945 1949 y Fz(,)h(just)f(b)m(y)f(taking)i(in)m(to)83 2062 y(accoun)m(t)e(the)e(\014lter)g Fr(F)860 2076 y FF(\000)950 2062 y Fz(:=)e Fv(\000)p Fr(F)1205 2076 y Fw(+)1263 2062 y Fz(.)52 b(But)34 b(the)h(sum)e(and)g(the)i(pro)s(duct) d(of)j(the)f(t)m(w)m(o)i(functions)d Fy( )38 b Fz(and)33 b Fy( )3744 2029 y FF(0)83 2175 y Fz(are)38 b(no)f(longer)i (asymptotically)h(replete.)63 b(In)37 b Fy(X)44 b Fz(=)37 b Ft(R)2044 2142 y Fn(n)2091 2175 y Fz(,)j Fy(n)c Fv(\025)h Fz(2)h(one)g(has)g(ev)m(en)g(a)g(wider)f(freedom)g(to)83 2288 y(construct)31 b(coun)m(terexamples.)249 2401 y(Let)38 b(us)f(close)i(this)f(section)h(b)m(y)f(a)g(discussion)f(of)h(the)g(sp) s(ectrum)e(of)i(the)g Fy(C)2940 2368 y FF(\003)2979 2401 y Fz(-algebra)h Fy(V)20 b(O)3477 2421 y Fs(G)3531 2402 y Fp(\000)p Ff(1)3618 2401 y Fz(\()p Fr(F)3709 2368 y FF(0)3733 2401 y Fz(\))83 2514 y(that)36 b(is)f(useful)g(for)g (applications)i(to)f(sp)s(ectral)g(analysis,)h(as)f(hin)m(ted)f(in)g (the)h(In)m(tro)s(duction.)56 b(W)-8 b(e)36 b(pro)m(v)m(e)83 2627 y(\014rst)83 2839 y FA(Prop)s(osition)g(3.17)47 b Fu(Supp)-5 b(ose)40 b(that)g(for)g(al)5 b(l)39 b Fy(G)1783 2806 y FF(0)1843 2839 y Fv(2)d Fr(G)2011 2806 y FF(0)2074 2839 y Fu(and)k Fy(G)d Fv(2)f Fr(G)p Fu(,)k(we)f(have)h Fy(G)3029 2806 y FF(0)3053 2839 y Fy(G)d Fv(2)f Fr(G)3329 2806 y FF(0)3391 2839 y Fu(\(we)j(write)83 2986 y(brie\015y)31 b Fr(G)432 2953 y FF(0)455 2986 y Fr(G)26 b Fv(\032)f Fr(G)719 2953 y FF(0)743 2986 y Fu(\).)41 b(Then)1091 2963 y Fq(g)1084 2986 y Fy(B)5 b(C)1229 2921 y Fs(F)1271 2898 y Fp(0)1323 2986 y Fv(\032)25 b Fy(V)20 b(O)1561 3006 y Fs(G)1615 2987 y Fp(\000)p Ff(1)1702 2986 y Fz(\()p Fr(F)1793 2953 y FF(0)1816 2986 y Fz(\))p Fu(.)42 b(In)30 b(p)-5 b(articular,)33 b(if)d Fr(F)2621 2953 y FF(0)2674 2986 y Fu(is)h(in)f(addition)i(\014ner)f(than)h(the)83 3099 y(standar)-5 b(d)36 b(\014lter,)d Fy(V)20 b(O)841 3119 y Fs(G)895 3100 y Fp(\000)p Ff(1)982 3099 y Fz(\()p Fr(F)1073 3066 y FF(0)1096 3099 y Fz(\))33 b Fu(is)g(a)g Fy(C)1413 3066 y FF(\003)1452 3099 y Fu(-algebr)-5 b(a)33 b(of)g(the)g(\014rst)g(kind.)249 3346 y(Pr)-5 b(o)g(of:)40 b Fz(Let)27 b Fy( )i Fv(2)867 3323 y Fq(g)860 3346 y Fy(B)5 b(C)1005 3281 y Fs(F)1047 3258 y Fp(0)1100 3346 y Fz(and)26 b Fy(c)g Fz(=)f(lim)1560 3361 y Fs(F)1602 3342 y Fp(0)1644 3346 y Fy( )s Fz(.)40 b(Then)25 b(for)i(all)g Fy(")f(>)f Fz(0,)j Fy(M)2612 3364 y Fn("=)p Fw(2)2719 3346 y Fz(\()p Fy( )s Fz(;)15 b Fy(c)p Fz(\))28 b Fv(2)c Fr(F)3099 3313 y FF(0)3122 3346 y Fz(.)40 b(F)-8 b(or)27 b(an)m(y)g Fy(G)f Fv(2)f Fr(G)83 3459 y Fz(one)31 b(has)970 3572 y([osc)1117 3591 y Fn(G)1172 3573 y Fp(\000)p Ff(1)1259 3572 y Fz(\()p Fy( )s Fz(\)])16 b(\()p Fy(x)p Fz(\))26 b Fv(\024)g Fz(sup)1676 3650 y Fn(y)r FF(2)p Fn(G)1830 3572 y Fv(j)p Fy( )s Fz(\()p Fy(xy)2052 3534 y FF(\000)p Fw(1)2148 3572 y Fz(\))20 b Fv(\000)g Fy(c)p Fv(j)h Fz(+)f Fv(j)p Fy( )s Fz(\()p Fy(x)p Fz(\))h Fv(\000)f Fy(c)p Fv(j)p Fy(:)83 3810 y Fz(But)34 b(sup)404 3832 y Fn(y)r FF(2)p Fn(G)562 3810 y Fv(j)p Fy( )s Fz(\()p Fy(xy)784 3777 y FF(\000)p Fw(1)879 3810 y Fz(\))23 b Fv(\000)f Fy(c)p Fv(j)31 b Fy(<)g("=)p Fz(2)j(if)g Fy(xy)1580 3777 y FF(\000)p Fw(1)1707 3810 y Fz(is)g(in)f Fy(M)1999 3829 y Fn("=)p Fw(2)2107 3810 y Fz(\()p Fy( )s Fz(;)15 b Fy(c)p Fz(\))35 b(for)e(all)i Fy(y)e Fv(2)d Fy(G)2865 3777 y FF(\000)p Fw(1)2960 3810 y Fz(,)35 b(i.e.)51 b(if)33 b Fy(x)h Fz(b)s(elongs)f(to)83 3923 y Fv(\\)144 3937 y Fn(y)r FF(2)p Fn(G)287 3923 y Fy(M)375 3941 y Fn("=)p Fw(2)483 3923 y Fz(\()p Fy( )s Fz(;)15 b Fy(c)p Fz(\))p Fy(y)s Fz(.)42 b(If)30 b Fy(F)971 3890 y FF(0)1020 3923 y Fv(2)24 b Fr(F)1161 3890 y FF(0)1215 3923 y Fz(then)30 b Fv(\\)1483 3937 y Fn(z)s FF(2)p Fn(G)1624 3923 y Fy(F)1695 3890 y FF(0)1718 3923 y Fy(z)g Fv(2)25 b Fr(F)1932 3890 y FF(0)1955 3923 y Fz(,)30 b(b)m(y)g(using)g(the)h(h)m(yp)s(othesis)f (of)g(the)h(Prop)s(osition:)1075 4149 y Fg({)1135 4075 y Fq(\002)1174 4149 y Fv(\\)1235 4163 y Fn(y)r FF(2)p Fn(G)1378 4149 y Fy(F)1449 4111 y FF(0)1472 4149 y Fy(y)1520 4075 y Fq(\003)1583 4149 y Fz(=)25 b Fv([)1740 4163 y Fn(y)r FF(2)p Fn(G)1883 4149 y Fz(\()p Fg({)p Fy(F)2034 4111 y FF(0)2058 4149 y Fz(\))p Fy(y)k Fz(=)c(\()p Fg({)p Fy(F)2414 4111 y FF(0)2438 4149 y Fz(\))p Fy(G)h Fv(2)f Fr(G)2727 4111 y FF(0)2751 4149 y Fy(:)83 4316 y Fz(Th)m(us,)30 b(w)m(e)g(ha)m(v)m(e)i(sho)m(wn)e(that)h(for)f(ev)m(ery)h Fy(")26 b(>)f Fz(0)788 4541 y Fy(M)876 4555 y Fn(")928 4541 y Fz(\(osc)1085 4561 y Fn(G)1140 4542 y Fp(\000)p Ff(1)1227 4541 y Fz(\()p Fy( )s Fz(\);)15 b(0\))27 b Fv(\033)e Fy(M)1690 4560 y Fn("=)p Fw(2)1798 4541 y Fz(\()p Fy( )s Fz(;)15 b Fy(c)p Fz(\))22 b Fv(\\)2112 4468 y Fq(\002)2150 4541 y Fv(\\)2211 4555 y Fn(y)r FF(2)p Fn(G)2354 4541 y Fy(M)2442 4560 y Fn("=)p Fw(2)2549 4541 y Fz(\()p Fy( )s Fz(;)15 b Fy(c)p Fz(\))p Fy(y)2808 4468 y Fq(\003)2873 4541 y Fv(2)25 b Fr(F)3015 4504 y FF(0)3038 4541 y Fy(;)83 4708 y Fz(i.e.)42 b(lim)366 4724 y Fs(F)408 4705 y Fp(0)450 4708 y Fz([osc)597 4728 y Fn(G)652 4709 y Fp(\000)p Ff(1)738 4708 y Fz(\()p Fy( )s Fz(\)])27 b(=)e(0;)31 b(the)g(pro)s(of)e(of)i (the)f(\014rst)g(assertion)h(is)f(\014nished.)249 4856 y(Assume)g(no)m(w)g(that)h Fr(F)1028 4870 y Fw(st)1113 4856 y Fv(\032)25 b Fr(F)p Fz(.)40 b(Then)29 b Fy(C)1632 4870 y Fw(0)1697 4856 y Fv(\032)c Fy(C)7 b Fz(\()1936 4833 y(_)1900 4856 y Fy(X)g Fz(\))26 b(=)2145 4833 y Fq(g)2139 4856 y Fy(B)5 b(C)2284 4791 y Fs(F)2326 4799 y Ff(st)2408 4856 y Fv(\032)2510 4833 y Fq(g)2504 4856 y Fy(B)g(C)2648 4791 y Fs(F)2690 4768 y Fp(0)2742 4856 y Fv(\032)25 b Fy(V)c(O)2981 4876 y Fs(G)3035 4857 y Fp(\000)p Ff(1)3122 4856 y Fz(\()p Fr(F)3213 4823 y FF(0)3236 4856 y Fz(\).)401 b Fg(\004)249 4998 y FA(Example.)64 b Fz(Cho)s(ose)38 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Fp(\000)p Ff(1)3077 1846 y Fz(\()p Fr(F)3168 1813 y FF(0)3191 1846 y Fz(\))p Fy(=B)5 b(C)3417 1813 y Fs(F)3459 1789 y Fp(0)3485 1846 y Fz(\),)39 b(b)s(oth)83 1983 y(these)h(spaces)g(b)s(eing)f(in)m (v)-5 b(arian)m(t.)69 b(The)39 b(space)h Fy(X)22 b Fz([)q Fy(V)e(O)2024 2003 y Fs(G)2078 1984 y Fp(\000)p Ff(1)2165 1983 y Fz(\()p Fr(F)2256 1950 y FF(0)2279 1983 y Fz(\))2314 1998 y Fs(F)2356 1979 y Fp(0)2383 1983 y Fz(])40 b(can)g(b)s(e)f(iden)m (ti\014ed)g(with)g(the)h(set)g(of)83 2096 y(c)m(haracters)32 b(of)e Fy(V)21 b(O)762 2116 y Fs(G)816 2097 y Fp(\000)p Ff(1)903 2096 y Fz(\()p Fr(F)994 2063 y FF(0)1017 2096 y Fz(\))31 b(that)g(are)f(n)m(ull)h(on)f Fy(B)5 b(C)1883 2063 y Fs(F)1925 2040 y Fp(0)1951 2096 y Fz(.)249 2209 y(Let)41 b Fy( )46 b Fv(2)c Fy(V)20 b(O)772 2229 y Fs(G)826 2210 y Fp(\000)p Ff(1)913 2209 y Fz(\()p Fr(F)1004 2176 y FF(0)1028 2209 y Fz(\),)43 b Fy(x)g Fv(2)e Fy(X)48 b Fz(and)40 b Fy(G)j Fv(2)f Fr(G)f Fz(suc)m(h)f(that)i Fy(x)g Fv(2)g Fy(G)p Fz(.)72 b(Then)39 b(0)k 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Fv(\032)f Fy(R)q(U)10 b(C)d Fz(.)81 b(Th)m(us,)47 b(in)d(this)h (case,)k Fy(V)20 b(O)1966 3330 y Fs(G)2020 3311 y Fp(\000)p Ff(1)9 b Fz(\()p Fr(F)2198 3278 y FF(0)2221 3311 y Fz(\))45 b(is)f(indeed)g(an)g Fy(X)7 b Fz(-algebra)47 b(and)c Fy(X)52 b Fz(acts)83 3424 y Fu(c)-5 b(ontinuously)34 b Fz(on)f(its)h(sp)s(ectrum.)47 b(Note)34 b(that)g(in)e(the)i(case)g Fr(F)2248 3391 y FF(0)2300 3424 y Fz(=)29 b Fr(F)2456 3438 y Fw(st)2545 3424 y Fz(=)g Fr(F)k Fz(all)g(the)h(assumptions)e(ab) s(o)m(v)m(e)83 3537 y(are)39 b(v)m(eri\014ed.)67 b Fy(V)20 b(O)s Fz(\()p Fy(X)7 b Fz(\))40 b Fv(\021)f Fy(V)21 b(O)1213 3567 y Fs(G)1267 3537 y Fp(\000)p Ff(1)1267 3587 y(st)1354 3537 y Fz(\()p Fr(F)1445 3551 y Fw(st)1504 3537 y Fz(\))39 b(is)g(an)g Fy(X)7 b Fz(-algebra)41 b(of)e(the)g(\014rst)f(kind)g(and)g (the)h(sp)s(ectrum)f(of)83 3663 y Fy(V)20 b(O)s Fz(\()p Fy(X)7 b Fz(\))p Fy(=C)490 3677 y Fw(0)561 3663 y Fz(is)31 b(comp)s(osed)f(of)h(p)s(oin)m(ts)f(that)h(are)g(\014xed)e(b)m(y)h(the) h(action)h(of)e Fy(X)7 b Fz(.)83 3907 y Fj(3.4)112 b(Some)39 b(extensions)83 4078 y Fz(Let)31 b(us)f(men)m(tion)h(brie\015y)e(an)h (enlargemen)m(t)i(of)f(the)f(concepts)i(of)e Fr(G)p Fz(-syndeticit)m(y) i(and)e Fr(G)p Fz(-repleteness.)83 4291 y FA(De\014nition)35 b(3.19)47 b Fu(L)-5 b(et)33 b Fr(F)g Fu(and)h Fr(F)1282 4258 y FF(0)1338 4291 y Fu(two)h(\014lters)f(on)f Fy(X)7 b Fu(.)45 b(One)33 b(says)h(that)g(the)g(subset)f Fy(A)h Fu(of)f Fy(X)41 b Fu(is)33 b Fz(\()p Fr(G)p Fy(;)15 b Fr(F)3676 4258 y FF(0)3700 4291 y Fz(\))p Fu(-)83 4404 y Fz(syndetic)35 b Fu(if)f(ther)-5 b(e)35 b(exists)g(some)g Fy(G)29 b Fv(2)f Fr(G)35 b Fu(such)f(that)i Fy(AG)29 b Fv(2)f Fr(F)2251 4371 y FF(0)2274 4404 y Fu(.)47 b(A)n(nd)34 b Fy(B)f Fv(2)28 b Fy(X)42 b Fu(is)35 b(c)-5 b(al)5 b(le)-5 b(d)35 b Fz(\()p Fr(G)p Fy(;)15 b Fr(F)3408 4371 y FF(0)3432 4404 y Fz(\))p Fu(-)p Fz(replete)83 4517 y Fu(if)36 b(for)g(any)h Fy(G)32 b Fv(2)f Fr(G)36 b Fu(and)h Fy(F)1054 4484 y FF(0)1109 4517 y Fv(2)31 b Fr(F)1257 4484 y FF(0)1316 4517 y Fu(ther)-5 b(e)37 b(exists)f Fy(x)c Fv(2)f Fy(F)2045 4484 y FF(0)2104 4517 y Fu(such)36 b(that)i Fy(xG)31 b Fv(\032)h Fy(B)5 b Fu(.)51 b(\(A)n(nother)37 b(terminolo)-5 b(gy)83 4630 y(c)g(ould)34 b(b)-5 b(e)32 b(\\)g Fr(G)p Fu(-syndetic)i(\(r)-5 b(esp.)43 b Fr(G)p Fu(-r)-5 b(eplete\))33 b(in)g(the)g(dir)-5 b(e)g(ction)34 b(of)f Fr(F)2465 4597 y FF(0)2520 4630 y Fu(".\))249 4842 y Fz(Similar)d(terminology)i(will)f (b)s(e)f(used)f(for)h(functions.)249 4955 y(Ob)m(viously)-8 b(,)30 b(\()p Fr(G)p Fy(;)15 b Fr(F)895 4969 y Fw(tr)954 4955 y Fz(\)-syndetic)30 b(means)e Fr(G)p Fz(-syndetic)i(and)e(\()p Fr(G)p Fy(;)15 b Fr(F)2485 4969 y Fw(tr)2545 4955 y Fz(\)-replete)30 b(means)e Fr(G)p Fz(-replete.)42 b(But)83 5068 y(b)m(y)26 b(c)m(ho)s(osing)h(\014ner)e(\014lters)h Fr(F)1078 5035 y FF(0)1101 5068 y Fz(,)h(one)f(gets)h(more)g(\()p Fr(G)p Fy(;)15 b Fr(F)1924 5035 y FF(0)1948 5068 y Fz(\)-syndetic)26 b(and)g(less)g(\()p Fr(G)p Fy(;)15 b Fr(F)2903 5035 y FF(0)2927 5068 y Fz(\)-replete)28 b(subsets.)38 b(Let)83 5181 y(us)25 b(tak)m(e,)k(for)d(example,)h Fy(X)33 b Fz(=)25 b Ft(R)p Fz(,)i Fr(F)d Fz(=)h Fr(F)1480 5195 y Fw(st)1565 5181 y Fz(and)h Fr(F)1794 5148 y FF(0)1842 5181 y Fz(=)f Fr(F)1994 5195 y Fw(+)2052 5181 y Fz(,)i(the)f(\014lter)g (app)s(earing)g(in)f(a)i(previous)e(example.)83 5294 y(Then)33 b Ft(N)h Fz(is)g(\()p Fr(G)625 5308 y Fw(st)685 5294 y Fy(;)15 b Fr(F)781 5308 y Fw(+)840 5294 y Fz(\)-syndetic)35 b(without)g(b)s(eing)e Fr(G)1928 5308 y Fw(st)1988 5294 y Fz(-syndetic)i(and)e(\()p Fv(\0001)p Fy(;)15 b Fz(0])36 b(is)e Fr(G)3068 5308 y Fw(st)3128 5294 y Fz(-replete)h(without)83 5407 y(b)s(eing)30 b(\()p Fr(G)434 5421 y Fw(st)494 5407 y Fy(;)15 b Fr(F)590 5421 y Fw(+)649 5407 y Fz(\)-replete.)42 b(But)31 b(note)g(that)g(the)g(set)f([0)p Fy(;)15 b Fv(1)p Fz(\))32 b(is)f(\()p Fr(G)2392 5421 y Fw(st)2452 5407 y Fy(;)15 b Fr(F)2548 5421 y Fw(+)2607 5407 y Fz(\)-replete.)249 5520 y(Ho)m(w)m(ev)m(er,)33 b(there)d(are)h(situations)g(in)f(whic)m(h) g(nothing)h(new)f(app)s(ears:)1880 5836 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 83 482 a FA(Prop)s(osition)36 b(3.20)47 b Fu(If)75 b Fr(F)1074 449 y FF(0)1131 482 y Fv(\032)34 b Fr(F)j Fu(and)i Fr(F)e Fu(is)h(invariant)h(under)f(left) g(tr)-5 b(anslations,)42 b(then)c Fy(A)g Fu(is)f Fz(\()p Fr(G)p Fy(;)15 b Fr(F)3676 449 y FF(0)3700 482 y Fz(\))p Fu(-)83 594 y(syndetic)33 b(i\013)f Fy(A)h Fu(is)g Fr(G)p Fu(-syndetic.)249 799 y(Pr)-5 b(o)g(of:)38 b Fz(It)23 b(is)g(enough)g(to)g(pro)m(v)m(e)h(that,)h(under)d(the)h(assumptions)f (of)h(the)g(Prop)s(osition,)i(if)d Fy(A)h Fz(is)g(\()p Fr(G)p Fy(;)15 b Fr(F)3678 766 y FF(0)3702 799 y Fz(\)-)83 912 y(syndetic)33 b(then)f(it)i(is)f Fr(G)p Fz(-syndetic.)48 b(So)33 b(supp)s(ose)e(that)j(there)f(exists)g Fy(G)d Fv(2)f Fr(G)k Fz(suc)m(h)f(that)h Fy(AG)d Fv(\021)f Fy(F)3528 926 y Fw(0)3597 912 y Fv(2)g Fr(F)p Fz(.)83 1025 y(Set)43 b Fy(G)323 1039 y Fw(0)408 1025 y Fz(=)i Fg({)p Fy(F)627 1039 y Fw(0)713 1025 y Fv(2)g Fr(G)p Fz(;)k(w)m(e)43 b(ha)m(v)m(e)h(\()p Fy(AG)p Fz(\))29 b Fv([)f Fy(G)1730 1039 y Fw(0)1816 1025 y Fz(=)45 b Fy(X)7 b Fz(.)77 b(But)43 b Fy(A)f Fz(is)h(non-v)m(oid.)77 b(F)-8 b(or)43 b Fy(x)j Fv(2)f Fy(A)d Fz(one)h(has)83 1138 y Fy(G)154 1152 y Fw(0)244 1138 y Fz(=)50 b Fy(x)31 b Fv(\001)503 1064 y Fq(\002)541 1138 y Fy(x)593 1105 y FF(\000)p Fw(1)687 1138 y Fy(G)758 1152 y Fw(0)798 1064 y Fq(\003)886 1138 y Fv(\032)51 b Fy(A)1091 1064 y Fq(\002)1129 1138 y Fy(x)1181 1105 y FF(\000)p Fw(1)1275 1138 y Fy(G)1346 1152 y Fw(0)1386 1064 y Fq(\003)1424 1138 y Fz(,)e(th)m(us)c Fy(X)58 b Fz(=)50 b(\()p Fy(AG)p Fz(\))32 b Fv([)2299 1064 y Fq(\000)2340 1138 y Fy(A)p Fz([)p Fy(x)2485 1105 y FF(\000)p Fw(1)2580 1138 y Fy(G)2651 1152 y Fw(0)2691 1138 y Fz(])2716 1064 y Fq(\001)2808 1138 y Fz(=)50 b Fy(A)3012 1064 y Fq(\000)3054 1138 y Fy(G)21 b Fv([)f Fz([)p Fy(x)3304 1105 y FF(\000)p Fw(1)3398 1138 y Fy(G)3469 1152 y Fw(0)3509 1138 y Fz(])3534 1064 y Fq(\001)3622 1138 y Fz(and)83 1251 y Fy(G)h Fv([)f Fz([)p Fy(x)333 1218 y FF(\000)p Fw(1)427 1251 y Fy(G)498 1265 y Fw(0)538 1251 y Fz(])25 b Fv(2)g Fr(G)p Fz(.)2927 b Fg(\004)249 1363 y Fz(It)39 b(is)h(easy)g(to)g(c)m(hec)m(k)g(that)g (these)g(t)m(w)m(o)h(classes)f(in)m(tro)s(duced)e(ab)s(o)m(v)m(e)j (satify)f(a)f(result)h(analogue)g(to)83 1476 y(Lemma)31 b(3.2,)g(b)m(y)f(a)h(similar)g(pro)s(of.)83 1681 y FA(Lemma)k(3.21)46 b Fu(\(a\))33 b Fy(A)g Fu(is)g Fz(\()p Fr(G)p Fy(;)15 b Fr(F)1239 1648 y FF(0)1263 1681 y Fz(\))p Fu(-syndetic)33 b(i\013)64 b Fg({)p Fy(A)34 b Fu(is)e(not)h Fz(\()p Fr(G)2342 1648 y FF(\000)p Fw(1)2437 1681 y Fy(;)15 b Fr(F)2533 1648 y FF(0)2557 1681 y Fz(\))p Fu(-r)-5 b(eplete.)249 1794 y(\(b\))33 b Fy(B)k Fu(is)32 b Fz(\()p Fr(G)707 1761 y FF(\000)p Fw(1)802 1794 y Fy(;)15 b Fr(F)898 1761 y FF(0)922 1794 y Fz(\))p Fu(-r)-5 b(eplete)33 b(i\013)f Fg({)p Fy(B)38 b Fu(is)32 b(not)i Fz(\()p Fr(G)p Fy(;)15 b Fr(F)2009 1761 y FF(0)2033 1794 y Fz(\))p Fu(-syndetic.)249 1998 y Fz(Since)41 b(in)g(man)m(y)g(particular)g(cases)h(the)f(class)g Fr(M)h Fz(of)f(all)h(the)f(\()p Fr(G)p Fy(;)15 b Fr(F)2688 1965 y FF(0)2712 1998 y Fz(\)-syndetic)41 b(subsets)g(of)g Fy(X)48 b Fz(is)83 2111 y(indeed)27 b Fr(F)p Fz(-admissible,)g(one)h (can)f(apply)g(the)g(mac)m(hine)h(dev)m(elop)s(ed)f(in)g(Section)h(2)f (in)g(order)g(to)h(extend)f(the)83 2224 y(results)j(of)h(this)f (Section.)42 b(T)-8 b(o)30 b(b)s(e)g(sp)s(eci\014c,)g(b)m(y)h(a)g(pro)s (of)e(similar)i(to)g(that)g(of)f(Theorem)h(3.3)g(w)m(e)g(get)83 2429 y FA(Prop)s(osition)36 b(3.22)47 b Fu(Assume)27 b(that)i(the)f(\014lter)g Fr(F)f Fu(veri\014es)h(\(H1\))g(and)g(\(H2\)) g(and)h(that)f Fr(G)9 b Fv(\\)g Fr(F)3305 2396 y FF(0)3354 2429 y Fz(=)25 b Fv(;)p Fu(.)41 b(Then)83 2542 y(the)33 b(family)g(of)g Fz(\()p Fr(G)p Fy(;)15 b Fr(F)819 2509 y FF(0)843 2542 y Fz(\))p Fu(-syndetic)33 b(sets)g(is)g Fr(F)p Fu(-admissible.)249 2746 y Fz(Note)f(that)f(the)f(condition)h Fr(G)21 b Fv(\\)f Fr(F)1450 2713 y FF(0)1498 2746 y Fz(=)25 b Fv(;)31 b Fz(is)f(v)m(eri\014ed)g(as)h(so)s(on)f(as)h Fr(F)f Fz(and)f Fr(F)2831 2713 y FF(0)2884 2746 y Fz(are)i(comparable.) 249 2859 y(W)-8 b(e)35 b(lea)m(v)m(e)h(to)e(the)g(reader)g(the)g(task)g (of)g(reform)m(ulating)g(Theorem)g(3.3)g(and)f(Corollary)i(3.6)f(for)g (the)83 2972 y(presen)m(t,)45 b(more)c(general)i(case,)i(as)d(w)m(ell)h (as)e(\014nding)f(relev)-5 b(an)m(t)43 b(examples.)75 b(W)-8 b(e)43 b(end)d(this)i(Section)g(b)m(y)83 3085 y(indicating)34 b(a)g(rather)f(large)h(class)g(of)g(\()p Fr(G)p Fy(;)15 b Fr(F)1615 3052 y FF(0)1639 3085 y Fz(\)-syndetic)34 b(functions)f(that)h(can)f(b)s(e)g(constructed)h(from)e Fr(G)p Fz(-)83 3198 y(syndetic)f(functions.)83 3402 y FA(Prop)s(osition)36 b(3.23)47 b Fu(L)-5 b(et)38 b Fy(')c Fv(2)g Fy(B)5 b(C)44 b Fu(and)38 b Fy(\025)d Fv(2)f Ft(C)p Fu(.)56 b(Assume)38 b(that)h(ther)-5 b(e)38 b(exists)g(a)g(function)g Fy(')3454 3416 y Fw(0)3528 3402 y Fv(2)c Fy(B)5 b(C)83 3515 y Fu(which)34 b(is)e Fr(G)p Fu(-syndetic)h(at)g Fy(\025)g Fu(such)g(that)h Fz(lim)1607 3530 y Fs(F)1649 3511 y Fp(0)1675 3515 y Fz(\()p Fy(')22 b Fv(\000)d Fy(')1940 3529 y Fw(0)1980 3515 y Fz(\))26 b(=)f(0)p Fu(.)42 b(Then)33 b Fy(')g Fu(is)g Fz(\()p Fr(G)p Fy(;)15 b Fr(F)2881 3482 y FF(0)2905 3515 y Fz(\))p Fu(-syndetic)33 b(at)g Fy(\025)p Fu(.)249 3720 y(Pr)-5 b(o)g(of:)42 b Fz(F)-8 b(or)31 b(an)m(y)g Fy(x)25 b Fv(2)g Fy(X)38 b Fz(w)m(e)30 b(ha)m(v)m(e)1098 3916 y Fv(j)p Fy(')p Fz(\()p Fy(x)p Fz(\))21 b Fv(\000)f Fy(\025)p Fv(j)26 b(\024)f(j)p Fy(')p Fz(\()p Fy(x)p Fz(\))c Fv(\000)f Fy(')1993 3930 y Fw(0)2033 3916 y Fz(\()p Fy(x)p Fz(\))p Fv(j)h Fz(+)f Fv(j)p Fy(')2376 3930 y Fw(0)2416 3916 y Fz(\()p Fy(x)p Fz(\))h Fv(\000)f Fy(\025)p Fv(j)p Fy(;)83 4112 y Fz(whic)m(h)39 b(implies)g(that)g Fy(M)962 4126 y Fn(")995 4135 y Ff(1)1034 4112 y Fz(\()p Fy(')27 b Fv(\000)e Fy(')1310 4126 y Fw(0)1350 4112 y Fz(;)15 b(0\))27 b Fv(\\)f Fy(M)1672 4126 y Fn(")1705 4135 y Ff(2)1743 4112 y Fz(\()p Fy(')1837 4126 y Fw(0)1877 4112 y Fz(;)15 b Fy(\025)p Fz(\))41 b Fv(\032)e Fy(M)2244 4126 y Fn(")2277 4135 y Ff(1)2311 4126 y Fw(+)p Fn(")2399 4135 y Ff(2)2437 4112 y Fz(\()p Fy(')p Fz(;)15 b Fy(\025)p Fz(\))41 b(for)e(all)g Fy(")3024 4126 y Fw(1)3064 4112 y Fy(;)15 b(")3146 4126 y Fw(2)3226 4112 y Fy(>)39 b Fz(0.)67 b(So)39 b(it)g(is)83 4225 y(enough)30 b(to)h(sho)m(w)g(that)f (if)h Fy(M)1101 4239 y Fw(1)1166 4225 y Fv(2)25 b Fr(F)1308 4192 y FF(0)1361 4225 y Fz(and)k Fy(M)1625 4239 y Fw(2)1695 4225 y Fz(is)i Fr(G)p Fz(-syndetic,)g(then)f Fy(M)2565 4239 y Fw(1)2625 4225 y Fv(\\)20 b Fy(M)2794 4239 y Fw(2)2864 4225 y Fz(is)30 b(\()p Fr(G)p Fy(;)15 b Fr(F)3157 4192 y FF(0)3181 4225 y Fz(\)-syndetic.)249 4338 y(Let)38 b Fy(G)f Fv(2)f Fr(G)h Fz(suc)m(h)g(that)h Fy(M)1236 4352 y Fw(2)1275 4338 y Fy(G)f Fz(=)g Fy(X)7 b Fz(;)41 b(there)c(is)g(no)g(loss)h(of)f(generalit)m(y)i(to)f(supp)s(ose)e(that) h Fy(e)g Fv(2)f Fy(G)p Fz(.)83 4451 y(Then)903 4564 y(\()p Fy(M)1026 4578 y Fw(1)1086 4564 y Fv(\\)20 b Fy(M)1255 4578 y Fw(2)1295 4564 y Fz(\))p Fy(G)26 b Fz(=)f(\()p Fy(M)1646 4578 y Fw(1)1686 4564 y Fy(G)p Fz(\))c Fv(\\)f Fz(\()p Fy(M)2017 4578 y Fw(2)2057 4564 y Fy(G)p Fz(\))26 b(=)f Fy(M)2373 4578 y Fw(1)2412 4564 y Fy(G)h Fv(\033)f Fy(M)2693 4578 y Fw(1)2758 4564 y Fv(2)g Fr(F)2900 4526 y FF(0)2923 4564 y Fy(;)83 4726 y Fz(th)m(us)30 b(\()p Fy(M)406 4740 y Fw(1)466 4726 y Fv(\\)20 b Fy(M)635 4740 y Fw(2)675 4726 y Fz(\))p Fy(G)26 b Fv(2)f Fr(F)949 4693 y FF(0)1002 4726 y Fz(to)s(o.)2542 b Fg(\004)83 4931 y FA(Corollary)35 b(3.24)46 b Fu(If)37 b Fr(F)935 4898 y FF(0)991 4931 y Fv(\032)c Fr(F)p Fu(,)k Fy(')1275 4945 y Fw(0)1352 4931 y Fu(is)g Fr(G)p Fu(-syndetic)g(and)h Fy(')24 b Fv(\000)f Fy(')2332 4945 y Fw(0)2405 4931 y Fv(2)32 b Fy(B)5 b(C)2644 4898 y Fs(F)2686 4874 y Fp(0)2749 4931 y Fu(then)37 b Fy(')h Fu(is)f Fz(\()p Fr(G)p Fy(;)15 b Fr(F)3356 4898 y FF(0)3380 4931 y Fz(\))p Fu(-syndetic)83 5044 y(\(at)33 b(any)h(value)e(of)h(its)g Fr(F)p Fu(-r)-5 b(ange\).)249 5248 y(Pr)g(o)g(of:)37 b Fz(This)21 b(follo)m(ws)i 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y(the)k(Conference)f(\\Op)s(erator)h(Algebras)g(and)e (Mathematical)36 b(Ph)m(ysics",)e(Constan)m(ta,)g(July)-8 b(,)33 b(2001,)270 4828 y(Ed.)g(J.-M.)h(Com)m(b)s(es,)g(J.)g(Cun)m(tz,) g(G.)g(E.)f(Elliott,)k(G.)d(Nenciu,)g(H.)g(Sieden)m(top,)h(S.)e (Str\025)-45 b(atil\025)g(a,)36 b(Theta)270 4941 y(F)-8 b(undation,)31 b(297-312.)83 5129 y([13])47 b(S.)e(C.)h(P)m(o)m(w)m (er,)k Fu(Commutator)g(Ide)-5 b(als)48 b(and)f(Pseudo)-5 b(di\013er)g(ential)49 b Fy(C)2676 5096 y FF(\003)2715 5129 y Fu(-A)n(lgebr)-5 b(as)p Fz(,)49 b(Qart.)d(J.)f(Math.)270 5242 y(Oxford)29 b(Ser.)h(\(2\))i FA(31)f Fz(\(1980\),)i(467-489.)83 5429 y([14])47 b(S.)26 b(C.)f(P)m(o)m(w)m(er,)k Fu(F)-7 b(r)i(e)g(dholm)31 b(T)-7 b(o)i(eplitz)30 b(Op)-5 b(er)g(ators)32 b(and)d(Slow)h(Oscil)5 b(lation)p Fz(,)28 b(Can.)d(J.)h(Math.)h FA(32)f Fz(\(1980\),)270 5542 y(1058-1071.)1880 5836 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 83 482 a Fz([15])47 b(S.)28 b(C.)h(P)m(o)m(w)m (er,)i Fu(Char)-5 b(acters)34 b(of)d Fy(C)1426 449 y FF(\003)1465 482 y Fu(-A)n(lgebr)-5 b(as,)31 b(the)h(Joint)f(Normal)i (Sp)-5 b(e)g(ctrum)33 b(and)f(a)g(Pseudo)-5 b(di\013er-)270 594 y(ential)33 b Fy(C)595 561 y FF(\003)634 594 y Fu(-A)n(lgebr)-5 b(a)p Fz(,)30 b(Pro)s(c.)h(Edin)m(burgh)d(Math.)j(So)s(c.)g(\(2\).)g FA(24)g Fz(\(1981\),)i(47-53.)1880 5836 y(19)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0306030547403--