Dynamical covering problems on the triadic Cantor set
type: Item
instance of: scholarly article
title: Dynamical covering problems on the triadic Cantor set
zbMATH Open document ID: 1392.11049
author: Jun Wu
author: Jian Xu
author: Baowei Wang
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2017-07-31T00:00:00Z
review text: Let \(Tx=3x \pmod 1\) be the natural map on the Cantor set \(\mathcal K\) and \(\mu\) the standard measure on \(\mathcal K\). The authors give a complete characterization of the size of the set \[ \mathcal D(x_0,\varphi):= \left\{y\in \mathcal K:|T^nx_0-y|<\varphi(n) \text{ for infinitely many }n \right\} \] for \(\mu\)-almost \(x_0\in \mathcal K\), being \(\varphi: \mathbb N\to \mathbb R^+\) a function with \(\varphi(n)\to 0\) as \(n\to \infty\). More precisely, in this paper they prove: Theorem 1.1 For \(\mu\)-almost \(x_0\in \mathcal K\), \[ \mathcal D(x_0,\varphi)= \begin{cases} 0, &\text{ if } \sum_{n\geq 1}\varphi^\gamma (n)<\infty, \\ 1, &\text{ if } \sum_{n\geq 1}\varphi^\gamma (n)=\infty, \end{cases} \] where \(\gamma:=\log 2/\log 3\) is the Hausdorff dimension of the Cantor set \(\mathcal K\). Theorem 1.2 Let \(f\) be a dimension function with \(f(x)/x^\gamma\) being increasing as \(x \to 0\). For \(\mu\)-almost \(x_0\in \mathcal K\), the \(f\)-Hausdorff measure of the set \(\mathcal D(x_0,\varphi)\) satisfies the following dichotomy law: \[ \mathcal D(x_0,\varphi)= \begin{cases} 0, &\text{ if } \sum_{n\geq 1}f(\varphi((n))<\infty, \\ \mathcal H^f(\mathcal K), &\text{ if } \sum_{n\geq 1}f(\varphi((n))=\infty, \end{cases} \] As a byproduct, they obtain an inhomogeneous counterpart of \textit{J. Levesley} et al.'s answer to a question of Mahler about the Diophantine approximation on \(\mathcal K\) [Math. Ann. 338, No. 1, 97--118 (2007; Zbl 1115.11040)]. The proofs are based on a dynamical Borel-Cantelli lemma, analogous to the one proved by \textit{W. Philipp} [Pac. J. Math. 20, 109--127 (1967; Zbl 0144.04201)], and on the ``mass transference principle'' by \textit{V. Beresnevich} and \textit{S. Velani} [Ann. Math. (2) 164, No. 3, 971--992 (2006; Zbl 1148.11033)].
reviewed by: Enrico Zoli
Mathematics Subject Classification ID: 11J83
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 11K60
zbMATH DE Number: 6754867
zbMATH Keywords: Borel-Cantelli lemma
zbMATH Keywords: mass transference principle
zbMATH Keywords: limsup sets
zbMATH Keywords: generalized Hausdorff measures
zbMATH Keywords: dynamical systems
zbMATH Keywords: dichotomy law
MaRDI profile type: Publication
full work available at URL: j.crma.2017.05.014
OpenAlex ID: W2732707852
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DOI: 10.1016/J.CRMA.2017.05.014
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description: scientific article; zbMATH DE number 6754867