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Metrical Results on the Distribution of Fractional Parts of Powers of Real Numbers

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory Classical measure theory

Published online by Cambridge University Press:  29 November 2018

Yann Bugeaud ,
Lingmin Liao  and
Michał Rams
Yann Bugeaud
Affiliation:
IRMA UMR 7501, CNRS, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg, France ([email protected])
Lingmin Liao
Affiliation:
LAMA UMR 8050, CNRS, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France ([email protected])
Michał Rams
Affiliation:
Institute of Mathematics Polish Academy of Sciences ul. Śniadeckich 8, 00-656 Warszawa, Poland ([email protected])
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Abstract

We establish several new metrical results on the distribution properties of the sequence ({xn})n≥1, where {·} denotes the fractional part. Many of them are presented in a more general framework, in which the sequence of functions (x ↦ xn)n≥1 is replaced by a sequence (fn)n≥1, under some growth and regularity conditions on the functions fn.

Keywords

fractional parts of powersDiophantine approximationHausdorff dimension

MSC classification

Primary: 11K36: Well-distributed sequences and other variations
Secondary: 11J71: Distribution modulo one 28A80: Fractals
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 505 - 521
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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