On Weyl products and uniform distribution modulo one

Christoph Aistleitner; Gerhard Larcher; Friedrich Pillichshammer; Sumaia Saad Eddin; Robert F. Tichy

doi:10.1007/s00605-017-1100-8

On Weyl products and uniform distribution modulo one

Christoph Aistleitner, Gerhard Larcher, Friedrich Pillichshammer^*, Sumaia Saad Eddin, Robert F. Tichy

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Pages (from-to)	365-391
Journal	Monatshefte fur Mathematik
Volume	185
Issue number	3
DOIs	https://doi.org/10.1007/s00605-017-1100-8
Publication status	Published - 2018

Keywords

Kronecker sequence
Star-discrepancy
Trigonometric product
van der Corput sequence

ASJC Scopus subject areas

General Mathematics

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10.1007/s00605-017-1100-8

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title = "On Weyl products and uniform distribution modulo one",

abstract = "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.",

keywords = "Kronecker sequence, Star-discrepancy, Trigonometric product, van der Corput sequence",

author = "Christoph Aistleitner and Gerhard Larcher and Friedrich Pillichshammer and Eddin, {Sumaia Saad} and Tichy, {Robert F.}",

year = "2018",

doi = "10.1007/s00605-017-1100-8",

language = "English",

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journal = "Monatshefte fur Mathematik",

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TY - JOUR

T1 - On Weyl products and uniform distribution modulo one

AU - Aistleitner, Christoph

AU - Larcher, Gerhard

AU - Pillichshammer, Friedrich

AU - Eddin, Sumaia Saad

AU - Tichy, Robert F.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Kronecker sequence

KW - Star-discrepancy

KW - Trigonometric product

KW - van der Corput sequence

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U2 - 10.1007/s00605-017-1100-8

DO - 10.1007/s00605-017-1100-8

M3 - Article

AN - SCOPUS:85029901955

SN - 0026-9255

VL - 185

SP - 365

EP - 391

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

IS - 3

ER -

On Weyl products and uniform distribution modulo one

Abstract

Keywords

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