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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/2501.09582 |
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| _version_ | 1866915105737801728 |
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| author | Baker, Simon Bender, George |
| author_facet | Baker, Simon Bender, George |
| contents | Given some integer $m \geq 3$, we find the first explicit collection of countably many intervals in $(1,2)$ such that for any $q$ in one of these intervals, the set of points with exactly $m$ base $q$ expansions is nonempty and moreover has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli, which guarantees that the intersection of a family of compact subsets of $\mathbb{R}^d$ has positive Hausdorff dimension under certain conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09582 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On finitely many base $q$ expansions Baker, Simon Bender, George Dynamical Systems Number Theory Given some integer $m \geq 3$, we find the first explicit collection of countably many intervals in $(1,2)$ such that for any $q$ in one of these intervals, the set of points with exactly $m$ base $q$ expansions is nonempty and moreover has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli, which guarantees that the intersection of a family of compact subsets of $\mathbb{R}^d$ has positive Hausdorff dimension under certain conditions. |
| title | On finitely many base $q$ expansions |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2501.09582 |