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 L²$L^2$ discrepancy is found to be c(α)log⁡n+O(log⁡log⁡n)$c(\alpha )\log n+O(\log \log n)$ for a computable positive constant c(α)$c(\alpha )$. For the golden ratio φ  , the value $\sqrt{c(\phi )\log n}$ yields the smallest L²$L^2$ discrepancy of any sequence of explicitly constructed finite point sets in the unit square. If the partial quotients a_k$a_k$ of α   grow at most polynomially fast, the L²$L^2$ discrepancy is found in terms of 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 $\frac{a}{b}$ is proved.