A note on dyadic approximation in Cantor's set
Let \(K\) denote the middle third Cantor set, i.e., the set of numbers \(x \in [0,1]\) whose ternary expansion uses only the digits \(0\) and \(2\), and let \(\gamma = \log 2/ \log3\), the Hausdorff dimension of \(K\). Denote by \(\mu\) the natural measure on \(K\), i.e., the \(\gamma\)-dimensional Hausdorff measure restricted to \(K\). The study of Diophantine approximation of numbers in \(K\) is an active area of research, and the approximation of elements in \(K\) by rationals with denominators \(3^n\) is very well understood. By contrast, approximation by rationals with denominators \(2^n\) is not nearly as well understood. For a function \(\psi:\mathbb{N} \rightarrow [0,\infty)\) and a real number \(y \in\mathbb{R}\), let \[ W_2(\psi, y) = \{x \in \mathbb{R} : \Vert 2^n x - y \Vert < \psi(n) \text{ for infinitely many }n \in\mathbb{N}\}, \] where \(\Vert \cdot \Vert\) denotes the distance to the nearest integer. In earlier work, \textit{A. Baker} [``Approximating elements of the middle third Cantor set with dyadic rationals'', Preprint, \url{arXiv:2203.12477}] conjectured that the set \(W_2(\psi, y)\) is \(\mu\)-null (resp. \(\mu\)-full) according to the convergence (resp. divergence) of the series \(\sum_{n=1}^\infty \psi(n)\). However, not much is known unless \(\psi(n)\) is of the form \(\psi_\tau(n) = n^{-\tau}\). For the latter type of approximation function, it was shown by \textit{D. Allen} et al. [Sel. Math., New Ser. 29, No. 1, Paper No. 11, 49 p. (2023; Zbl 07626251)] that \(\mu(W_2(\psi_\tau, 0)) = 0\) for \(\tau > 1/\gamma\); and that conditional on a certain unproven assertion, the conjecture holds for this type of function and \(y=0\). In the opposite direction, Baker [loc. cit.] proved that \(\mu(W_2(\psi_\tau, y)) = 1\) for \(\tau \le 0.01\). In the present paper, the authors prove that unconditionally, \(\mu(W_2(\psi_\tau, 0)) = 0\) for \(\tau > (0.922(1-\gamma) + 1)/(\gamma(2-\gamma))\).
scientific article; zbMATH DE number 7637414