In the present paper we study the asymptotic behavior of trigonometric products of the form $$\prod _{k=1}^N 2 \sin (\pi x_k)$$∏k=1N2sin(πxk) for $$N \rightarrow \infty $$N→∞, where the numbers $$\omega =(x_k)_{k=1}^N$$ω=(xk)k=1N 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 $$\omega $$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 $$\omega $$ω are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.

中文翻译：

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关于韦尔积和均匀分布模一

在本文中，我们研究 $$\prod _{k=1}^N 2 \sin (\pi x_k)$$∏k=1N2sin(πxk) 形式的三角积的渐近行为，对于 $$N \rightarrow \infty $$N→∞，其中数字 $$\omega =(x_k)_{k=1}^N$$ω=(xk)k=1N 均匀分布在单位区间 [0, 1] 内。主要结果是根据基础点 $$\omega $$ω 的星差匹配此类乘积的下限和上限，从而改进了 Hlawka 早期获得的结果（数论与分析（纪念 Edmund Landau 的论文） ，全会，纽约），97-118，1969）。此外，我们考虑当点 $$\omega $$ω 是 Kronecker 或 van der Corput 序列的初始段时的特殊情况。本文以一些概率类似物作为结论。