On Weyl products and uniform distribution modulo one

TL;DR

This paper investigates the asymptotic behavior of trigonometric products for uniformly distributed points, providing bounds related to discrepancy and analyzing special sequences like Kronecker and van der Corput.

Contribution

It offers improved bounds for these products based on discrepancy and extends analysis to specific sequences and probabilistic cases.

Findings

Established matching bounds in terms of star-discrepancy

Analyzed behavior for Kronecker and van der Corput sequences

Presented probabilistic analogues of the results

Abstract

In the present paper we study the asymptotic behavior of trigonometric products of the form $\prod_{k=1}^N 2 \sin(\pi x_k)$ for $N \to \infty$, where the numbers $\omega=(x_k)_{k=1}^N$ 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 $\omega$, thereby improving earlier results obtained by Hlawka in 1969. Furthermore, we consider the special cases when the points $\omega$ are the initial segment of a Kronecker or van der Corput sequence. The paper concludes with some probabilistic analogues.