On Weyl products and uniform distribution modulo one

In the present paper we study the asymptotic behavior of trigonometric products of the form (Formula presented.) for (Formula presented.), where the numbers (Formula presented.) 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 (Formula presented.), 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 (Formula presented.) are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.