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 2244&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=574eddd640880fad2e077594a688a241" title="Click to view the MathML source">L²$L^2$ discrepancy is found to be 2244&_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)$c(\alpha )\log n+O(\log \log n)$ for a computable positive constant 2244&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=ff1a7a534d05961405d7be6fd8b901a2" title="Click to view the MathML source">c(α)$c(\alpha )$. For the golden ratio φ  , the value 2244&_mathId=si4.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=e23f971a8fa8e36bf4ad18898a70cdd6">2244-si4.gif">$\sqrt{c(\phi )\log n}$ yields the smallest 2244&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=574eddd640880fad2e077594a688a241" title="Click to view the MathML source">L²$L^2$ discrepancy of any sequence of explicitly constructed finite point sets in the unit square. If the partial quotients 2244&_mathId=si5.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=e871ef59796745910b1b41215453eb2e" title="Click to view the MathML source">a_k$a_k$ of α   grow at most polynomially fast, the 2244&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=574eddd640880fad2e077594a688a241" title="Click to view the MathML source">L²$L^2$ discrepancy is found in terms of 2244&_mathId=si5.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=e871ef59796745910b1b41215453eb2e" title="Click to view the MathML source">a_k$a_k$ 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 2244&_mathId=si6.gif&_user=111111111&_pii=S0022314X16302244&_rdoc=1&_issn=0022314X&md5=83de3cc77dfa3d613c13e8422b81e01c">2244-si6.gif">$\frac{a}{b}$ is proved.