On the theorem of Davenport and generalized Dedekind sums

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• 作者：Bence Borda ^{bordabence85@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11K38 ; 11F20
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：172
• 期：Complete
• 页码：1-20
• 全文大小：352 K

文摘

A symmetrized lattice of 2n points in terms of an irrational real number α is considered in the unit square, as in the theorem of Davenport. If α   is a quadratic irrational, the square of the mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302244&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=574eddd640880fad2e077594a688a241" title="Click to view the MathML source">L²mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">L2math> discrepancy is found to be mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302244&_mathId=si2.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=59d9a11b02032eccd4fc3da1ebe7d7e0" title="Click to view the MathML source">c(α)log⁡n+O(log⁡log⁡n)mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">c(α)mathvariant="normal">log⁡n+O(mathvariant="normal">log⁡mathvariant="normal">log⁡n)math> for a computable positive constant mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302244&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=ff1a7a534d05961405d7be6fd8b901a2" title="Click to view the MathML source">c(α)mathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll">c(α)math>. For the golden ratio φ  , the value mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302244&_mathId=si4.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=e23f971a8fa8e36bf4ad18898a70cdd6">mage" height="20" width="84" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302244-si4.gif">mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll">c(φ)mathvariant="normal">log⁡nmath> yields the smallest mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302244&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=574eddd640880fad2e077594a688a241" title="Click to view the MathML source">L²mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">L2math> discrepancy of any sequence of explicitly constructed finite point sets in the unit square. If the partial quotients mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302244&_mathId=si5.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=e871ef59796745910b1b41215453eb2e" title="Click to view the MathML source">a_kmathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll">akmath> of α   grow at most polynomially fast, the mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302244&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=574eddd640880fad2e077594a688a241" title="Click to view the MathML source">L²mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">L2math> discrepancy is found in terms of mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302244&_mathId=si5.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=e871ef59796745910b1b41215453eb2e" title="Click to view the MathML source">a_kmathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll">akmath> up to an explicitly bounded error term. It is also shown that certain generalized Dedekind sums can be approximated using the same methods. For a special generalized Dedekind sum with arguments a, b   an asymptotic formula in terms of the partial quotients of mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302244&_mathId=si6.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=83de3cc77dfa3d613c13e8422b81e01c">mage" height="16" width="10" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302244-si6.gif">mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll">abmath> is proved.