Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917554093555712 |
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| author | Tang, Yuefeng |
| author_facet | Tang, Yuefeng |
| contents | Let $x=[a_1(x),a_2(x),\ldots]$ be the continued fraction expansion of $x\in[0,1)$. We prove that the Hausdorff dimension of \begin{equation*}E_{even}=\{x\in[0,1)\colon a_{2n}(x)\to\infty\ (n\to\infty)\}.\end{equation*} is 1/2. In general, we study the set of continued fractions with unbounded partial quotients along subsequence \begin{equation*}E_{\{k_n\}}=\{x\in[0,1)\colon a_{k_n}(x)\to\infty\ (n\to\infty)\},\end{equation*} where $\{k_n\}\subset\mathbb{N}$ is a subsequence. We show that $E_{\{k_n\}}$ has Hausdorff dimension 1/2 or 1 according to whether the set of indices $\{k_n\}_{n\geq 1}$ has positive or zero upper density respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequence Tang, Yuefeng Number Theory Dynamical Systems Let $x=[a_1(x),a_2(x),\ldots]$ be the continued fraction expansion of $x\in[0,1)$. We prove that the Hausdorff dimension of \begin{equation*}E_{even}=\{x\in[0,1)\colon a_{2n}(x)\to\infty\ (n\to\infty)\}.\end{equation*} is 1/2. In general, we study the set of continued fractions with unbounded partial quotients along subsequence \begin{equation*}E_{\{k_n\}}=\{x\in[0,1)\colon a_{k_n}(x)\to\infty\ (n\to\infty)\},\end{equation*} where $\{k_n\}\subset\mathbb{N}$ is a subsequence. We show that $E_{\{k_n\}}$ has Hausdorff dimension 1/2 or 1 according to whether the set of indices $\{k_n\}_{n\geq 1}$ has positive or zero upper density respectively. |
| title | Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequence |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2510.24064 |