Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T12:04:24.277Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Check for updates
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

References

27th Brazilian Mathematical Olympiad, third round, problem 6, 2005.Google Scholar
Downarowicz, T.. Entropy in Dynamical Systems. Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
Einsiedler, M. and Ward, T.. Ergodic Theory: With a View towards Number Theory (Graduate Texts in Mathematics) . Springer, London, 2011.10.1007/978-0-85729-021-2CrossRefGoogle Scholar
Falconer, K.. Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. John Wiley & Sons, Chichester, 2005.Google Scholar
Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation. Math. Syst. Theory 1(1) (1967), 1–49.10.1007/BF01692494CrossRefGoogle Scholar
Feng, D.-J. and Xiong, Y.. Affine embeddings of Cantor sets and dimension of 𝛼𝛽-sets. Israel J. Math. 226(2) (2018), 805–826.10.1007/s11856-018-1713-1CrossRefGoogle Scholar
Katznelson, Y.. On 𝛼𝛽-sets. Israel J. Math. 33(1) (1979), 1–4.CrossRefGoogle Scholar
Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability (Cambridge Studies in Advanced Mathematics) . Cambridge University Press, Cambridge, 1999.Google Scholar
Ornstein, D. and Weiss, B.. Unilateral codings of Bernoulli systems. Israel J. Math. 21 (1975), 159–166.CrossRefGoogle Scholar
Pollicott, M. and Yuri, M.. Dynamical Systems and Ergodic Theory (London Mathematical Society Student Texts) . Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
Wu, M.. A proof of Furstenberg’s conjecture on the intersections of × p and × q-invariant sets. Ann. of Math. (2) 189(3) (2019), 707–751.CrossRefGoogle Scholar
Yu, H.. Multi-rotations on the unit circle. J. Number Theory 200 (2019), 316–328.CrossRefGoogle Scholar