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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.17177 |
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| _version_ | 1866911531170529280 |
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| author | Bugeaud, Yann Kaneko, Hajime Kim, Dong Han |
| author_facet | Bugeaud, Yann Kaneko, Hajime Kim, Dong Han |
| contents | Let $ξ$ be a real number and $b \ge 2$ an integer. We study the relationship between the irrationality exponent of $ξ$ and the subword complexity $p(n, \mathbf{x})$ of the $b$-ary expansion $\mathbf{x}$ of $ξ$, where $p(n, \mathbf{x})$ counts the number of distinct blocks of length $n$ in $\mathbf{x}$, for $n \ge 1$. If the irrationality exponent of $ξ$ is equal to $2$, which is the case for almost all real numbers $ξ$, we show that the limit superior of the sequence $(p(n, \mathbf{x}) / n)_{n \ge 1}$ is at least equal to 4/3. The proof is based on a careful study of the evolution of the Rauzy graphs of infinite words of low complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17177 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the irrationality exponent of real numbers with low complexity expansion Bugeaud, Yann Kaneko, Hajime Kim, Dong Han Number Theory Combinatorics Dynamical Systems 11A63, 11J82 (primary), 68R15 (secondary) Let $ξ$ be a real number and $b \ge 2$ an integer. We study the relationship between the irrationality exponent of $ξ$ and the subword complexity $p(n, \mathbf{x})$ of the $b$-ary expansion $\mathbf{x}$ of $ξ$, where $p(n, \mathbf{x})$ counts the number of distinct blocks of length $n$ in $\mathbf{x}$, for $n \ge 1$. If the irrationality exponent of $ξ$ is equal to $2$, which is the case for almost all real numbers $ξ$, we show that the limit superior of the sequence $(p(n, \mathbf{x}) / n)_{n \ge 1}$ is at least equal to 4/3. The proof is based on a careful study of the evolution of the Rauzy graphs of infinite words of low complexity. |
| title | On the irrationality exponent of real numbers with low complexity expansion |
| topic | Number Theory Combinatorics Dynamical Systems 11A63, 11J82 (primary), 68R15 (secondary) |
| url | https://arxiv.org/abs/2510.17177 |