Hausdorff dimension of the Cartesian product of exact approximation set in $β$-expansions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914185492824064 |
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| author | Cheng, Wanjin Zhang, Xinyun |
| author_facet | Cheng, Wanjin Zhang, Xinyun |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06741 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of the Cartesian product of exact approximation set in $β$-expansions Cheng, Wanjin Zhang, Xinyun Number Theory 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. |
| title | Hausdorff dimension of the Cartesian product of exact approximation set in $β$-expansions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.06741 |