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On Weyl products and uniform distribution modulo one
Christoph Aistleitner1, Gerhard Larcher2, Friedrich Pillichshammer2
11Institute for Analysis and Number Theory, Graz University of Technology, Graz, Austria.
Abstract:
In the present paper we study the asymptotic behavior of trigonometric products of the form for , where the numbers 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 , 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 are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.
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