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METRIC RESULTS ON THE DISCREPANCY OF SEQUENCES $(a_{n}{\it\alpha})_{n\geqslant 1}$ MODULO ONE FOR INTEGER SEQUENCES $(a_{n})_{n\geqslant 1}$ OF POLYNOMIAL GROWTH

Part of: Exponential sums and character sums Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  05 February 2016

Christoph Aistleitner  and
Gerhard Larcher
Christoph Aistleitner
Affiliation:
Institute of Financial Mathematics and Applied Number Theory, University Linz, Altenbergerstrasse 69, 4040 Linz, Austria email aistleitner@math.tugraz.at
Gerhard Larcher
Affiliation:
Institute of Financial Mathematics and Applied Number Theory, University Linz, Altenbergerstrasse 69, 4040 Linz, Austria email gerhard.larcher@jku.at
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Abstract

An important result of Weyl states that for every sequence $(a_{n})_{n\geqslant 1}$ of distinct positive integers the sequence of fractional parts of $(a_{n}{\it\alpha})_{n\geqslant 1}$ is uniformly distributed modulo one for almost all ${\it\alpha}$ . However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $(\{a_{n}{\it\alpha}\})_{n\geqslant 1}$ for almost all ${\it\alpha}$ . In particular, it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences $(a_{n})_{n\geqslant 1}$ and for some special cases such as the Kronecker sequence $(\{n{\it\alpha}\})_{n\geqslant 1}$ or the sequence $(\{n^{2}{\it\alpha}\})_{n\geqslant 1}$ . In the present paper we answer the question for a large class of sequences $(a_{n})_{n\geqslant 1}$ including as a special case all polynomials $a_{n}=P(n)$ with $P\in \mathbb{Z}[x]$ of degree at least 2.

MSC classification

Primary: 11K38: Irregularities of distribution, discrepancy 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension 11L07: Estimates on exponential sums

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 478 - 491
Copyright
Copyright © University College London 2016 

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