On Weyl products and uniform distribution modulo one.

IF 0.8 4区数学 Q2 MATHEMATICS

Monatshefte fur Mathematik 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

Abstract

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.

Abstract Image

查看原文本刊更多论文

关于Weyl积和模1的均匀分布。

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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来源期刊

Monatshefte fur Mathematik 数学-数学

CiteScore

1.60

自引率

11.10%

发文量

155

审稿时长

4-8 weeks

期刊介绍： The journal was founded in 1890 by G. v. Escherich and E. Weyr as "Monatshefte für Mathematik und Physik" and appeared with this title until 1944. Continued from 1948 on as "Monatshefte für Mathematik", its managing editors were L. Gegenbauer, F. Mertens, W. Wirtinger, H. Hahn, Ph. Furtwängler, J. Radon, K. Mayrhofer, N. Hofreiter, H. Reiter, K. Sigmund, J. Cigler. The journal is devoted to research in mathematics in its broadest sense. Over the years, it has attracted a remarkable cast of authors, ranging from G. Peano, and A. Tauber to P. Erdös and B. L. van der Waerden. The volumes of the Monatshefte contain historical achievements in analysis (L. Bieberbach, H. Hahn, E. Helly, R. Nevanlinna, J. Radon, F. Riesz, W. Wirtinger), topology (K. Menger, K. Kuratowski, L. Vietoris, K. Reidemeister), and number theory (F. Mertens, Ph. Furtwängler, E. Hlawka, E. Landau). It also published landmark contributions by physicists such as M. Planck and W. Heisenberg and by philosophers such as R. Carnap and F. Waismann. In particular, the journal played a seminal role in analyzing the foundations of mathematics (L. E. J. Brouwer, A. Tarski and K. Gödel). The journal publishes research papers of general interest in all areas of mathematics. Surveys of significant developments in the fields of pure and applied mathematics and mathematical physics may be occasionally included.