关于Weyl积和模1的均匀分布。

Pub Date : 2018-01-01 Epub Date: 2017-09-26 DOI:10.1007/s00605-017-1100-8

Christoph Aistleitner, Gerhard Larcher, Friedrich Pillichshammer, Sumaia Saad Eddin, Robert F Tichy

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引用次数: 12

摘要

本文研究了πk=1N2sin（πxk）形式的三角乘积对N→ ∞ , 其中数ω=（xk）k=1N在单位区间[0，1]中均匀分布。主要结果是根据基础点ω的恒星差异匹配这些乘积的下限和上限，从而改进了Hlawka早期获得的结果（数论和分析（Papers in Honor of Edmund Landau，Plenum，New York），97-1181969）。此外，我们还考虑了当点ω是Kronecker或van der Corput序列的初始段时的特殊情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On Weyl products and uniform distribution modulo one.

In the present paper we study the asymptotic behavior of trigonometric products of the form $\prod_{k = 1}^{N} 2 sin (π x_{k})$ for $N \to \infty$ , where the numbers $ω = {(x_{k})}_{k = 1}^{N}$ are evenly distributed in the unit interval [0, 1]. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points $ω$ , thereby improving earlier results obtained by Hlawka (Number theory and analysis (Papers in Honor of Edmund Landau, Plenum, New York), 97-118, 1969). Furthermore, we consider the special cases when the points $ω$ are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.

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