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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.06741 |
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Table of Contents:
- In this paper, we study the metrical theory of Cartesian products of exact approximation sets in $β$-expansions. More precisely, for an integer $d \ge 2$ and real numbers $β_i > 1$ $(1 \le i \le d)$, we consider the set of points $x_i \in [0,1)$ is approximable by its convergents in the $β_i$-expansion to order $ψ_i$, but not to any better order. For any non-increasing functions $ψ_i$, we determine the Hausdorff dimension of the Cartesian product of these sets.