On Weyl products and uniform distribution modulo one

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.