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Bernoulli decomposition and arithmetical independence between sequences

Part of: Diophantine approximation, transcendental number theory Classical measure theory Measure-theoretic ergodic theory

Published online by Cambridge University Press:  14 January 2020

HAN YU Open the ORCID record for HAN YU [Opens in a new window]
HAN YU*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, CB3 0WB, UK email hy351@maths.cam.ac.uk
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Abstract

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In this paper, we study the set

$$\begin{eqnarray}A=\{p(n)+2^{n}d~\text{mod}~1:n\geq 1\}\subset [0,1],\end{eqnarray}$$
where $p$ is a polynomial with at least one irrational coefficient on non-constant terms, $d$ is any real number and, for $a\in [0,\infty )$, $a~\text{mod}~1$ is the fractional part of $a$. With the help of a method recently introduced by Wu, we show that the closure of $A$ must have full Hausdorff dimension.

Keywords

independence of sequencesBernoulli decompositiondisjointness between dynamical systems

MSC classification

Primary: 11J71: Distribution modulo one
Secondary: 28D20: Entropy and other invariants 28A80: Fractals
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1590 - 1600
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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