Zero-full law for well approximable sets in missing digit sets
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910880846839808 |
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| author | Li, Bing Li, Ruofan Wu, Yufeng |
| author_facet | Li, Bing Li, Ruofan Wu, Yufeng |
| contents | Let $b \geq 3$ be an integer and $C(b,D)$ be the set of real numbers in $[0,1]$ whose base $b$ expansion only consists of digits in a set $D \subseteq \{0,...,b-1\}$. We study how close can numbers in $C(b,D)$ be approximated by rational numbers with denominators being powers of some integer $t$ and obtain a zero-full law for its Hausdorff measure in several circumstances. When $b$ and $t$ are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann., 338:97-118, 2007) and generalize their theorem. When $b$ and $t$ are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_03936 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Zero-full law for well approximable sets in missing digit sets Li, Bing Li, Ruofan Wu, Yufeng Number Theory Dynamical Systems Let $b \geq 3$ be an integer and $C(b,D)$ be the set of real numbers in $[0,1]$ whose base $b$ expansion only consists of digits in a set $D \subseteq \{0,...,b-1\}$. We study how close can numbers in $C(b,D)$ be approximated by rational numbers with denominators being powers of some integer $t$ and obtain a zero-full law for its Hausdorff measure in several circumstances. When $b$ and $t$ are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann., 338:97-118, 2007) and generalize their theorem. When $b$ and $t$ are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed. |
| title | Zero-full law for well approximable sets in missing digit sets |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2302.03936 |