The Hausdorff dimension of the intersection of $ψ$-well approximable numbers and self-similar sets
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| Format: | Preprint |
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2025
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| _version_ | 1866912659657457664 |
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| author | Chen, Suxuan |
| author_facet | Chen, Suxuan |
| contents | Let $ψ:\mathbb{N}\rightarrow\mathbb{R}_+$ be a monotonically non-increasing function, and let $ψ_v:\mathbb{N}\rightarrow\mathbb{R}_+$ be defined by $ψ_v(q)=1/q^v$. In this article, we consider self-similar sets whose iterated function systems satisfy the open set condition. For functions $ψ$ that do not decrease too rapidly, we give a conjecturally sharp upper bound on the Hausdorff dimension of the intersection of $ψ$-well approximable numbers and such self-similar sets. When $ψ=ψ_v$ for some $v$ greater than 1 and sufficiently close to $1$, we give a lower bound for this Hausdorff dimension, which asymptotically matches the upper bound as $v\downarrow 1$. In particular, we show that the set of very well approximable numbers has full Hausdorff dimension within self-similar sets, thus confirming a conjecture of Levesley, Salp, and Velani. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17096 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Hausdorff dimension of the intersection of $ψ$-well approximable numbers and self-similar sets Chen, Suxuan Dynamical Systems Number Theory Let $ψ:\mathbb{N}\rightarrow\mathbb{R}_+$ be a monotonically non-increasing function, and let $ψ_v:\mathbb{N}\rightarrow\mathbb{R}_+$ be defined by $ψ_v(q)=1/q^v$. In this article, we consider self-similar sets whose iterated function systems satisfy the open set condition. For functions $ψ$ that do not decrease too rapidly, we give a conjecturally sharp upper bound on the Hausdorff dimension of the intersection of $ψ$-well approximable numbers and such self-similar sets. When $ψ=ψ_v$ for some $v$ greater than 1 and sufficiently close to $1$, we give a lower bound for this Hausdorff dimension, which asymptotically matches the upper bound as $v\downarrow 1$. In particular, we show that the set of very well approximable numbers has full Hausdorff dimension within self-similar sets, thus confirming a conjecture of Levesley, Salp, and Velani. |
| title | The Hausdorff dimension of the intersection of $ψ$-well approximable numbers and self-similar sets |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2510.17096 |