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.