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arXiv · 2512.06741

Hausdorff dimension of the Cartesian product of exact approximation set in $\beta$-expansions

Wanjin Cheng·Xinyun Zhang
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Abstract

In this paper, we study the metrical theory of Cartesian products of exact approximation sets in $\beta$-expansions. More precisely, for an integer $d \ge 2$ and real numbers $\beta_i > 1$ $(1 \le i \le d)$, we consider the set of points $x_i \in [0,1)$ is approximable by its convergents in the $\beta_i$-expansion to order $\psi_i$, but not to any better order. For any non-increasing functions $\psi_i$, we determine the Hausdorff dimension of the Cartesian product of these sets.

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Wanjin Cheng, Xinyun Zhang. 2025-12-07. Hausdorff dimension of the Cartesian product of exact approximation set in $\beta$-expansions. https://arxiv.org/abs/2512.06741

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