Convergence exponent of Dirichlet non-improvable numbers in the theory of continued fractions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909812456947712 |
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| author | Tan, Xiaoyan Zhang, Zhenliang |
| author_facet | Tan, Xiaoyan Zhang, Zhenliang |
| contents | Let $x \in [0,1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x), \cdots,a_n(x),\cdots]$ and $q_n(x)$ be the denominator of its $n$-th convergent.
We establish, for any $α,β$ in $[0,+\infty]$, the Hausdorff dimension formula of the intersections of the sets of Dirichlet non-improvable numbers and the level set of convergent exponent, i.e. $$ G(α,β): =\left\{x\in[0,1)\colon τ(x)=α,\,\,\text{and} \,\, \limsup_{n\to\infty}\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}\geqβ\right\}, $$ and $$ E(α,β): =\left\{x\in[0,1)\colon τ(x)=α,\,\,\text{and} \,\, \limsup_{n\to\infty}\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}=β\right\}, $$ where $$ τ(x):= \inf\Big\{s \geq 0: \sum_{n \geq 1} a^{-s}_n(x)<\infty\Big\}. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23059 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence exponent of Dirichlet non-improvable numbers in the theory of continued fractions Tan, Xiaoyan Zhang, Zhenliang Number Theory 11K55, 28A80 Let $x \in [0,1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x), \cdots,a_n(x),\cdots]$ and $q_n(x)$ be the denominator of its $n$-th convergent. We establish, for any $α,β$ in $[0,+\infty]$, the Hausdorff dimension formula of the intersections of the sets of Dirichlet non-improvable numbers and the level set of convergent exponent, i.e. $$ G(α,β): =\left\{x\in[0,1)\colon τ(x)=α,\,\,\text{and} \,\, \limsup_{n\to\infty}\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}\geqβ\right\}, $$ and $$ E(α,β): =\left\{x\in[0,1)\colon τ(x)=α,\,\,\text{and} \,\, \limsup_{n\to\infty}\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}=β\right\}, $$ where $$ τ(x):= \inf\Big\{s \geq 0: \sum_{n \geq 1} a^{-s}_n(x)<\infty\Big\}. $$ |
| title | Convergence exponent of Dirichlet non-improvable numbers in the theory of continued fractions |
| topic | Number Theory 11K55, 28A80 |
| url | https://arxiv.org/abs/2509.23059 |