On Weyl products and uniform distribution modulo one

On Weyl products and uniform distribution modulo one In the present paper we study the asymptotic behavior of trigonometric products of the form $$\prod _{k=1}^N 2 \sin (\pi x_k)$$ ∏ k = 1 N 2 sin ( π x k ) for $$N \rightarrow \infty $$ N → ∞ , where the numbers $$\omega =(x_k)_{k=1}^N$$ ω = ( 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 $$\omega $$ ω , 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 $$\omega $$ ω are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals
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Publisher
Springer Vienna
Copyright
Copyright © 2017 by The Author(s)
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-017-1100-8
Publisher site
See Article on Publisher Site

Abstract

In the present paper we study the asymptotic behavior of trigonometric products of the form $$\prod _{k=1}^N 2 \sin (\pi x_k)$$ ∏ k = 1 N 2 sin ( π x k ) for $$N \rightarrow \infty $$ N → ∞ , where the numbers $$\omega =(x_k)_{k=1}^N$$ ω = ( 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 $$\omega $$ ω , 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 $$\omega $$ ω are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.

Journal

Monatshefte f�r MathematikSpringer Journals

Published: Sep 26, 2017

References

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