Dynamical covering problems on the triadic Cantor set
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)].
scientific article; zbMATH DE number 6754867
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