Convergence exponent of Dirichlet non-improvable numbers in the theory of continued fractions
Year: 2025
Type: preprint
Abstract: 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 $$ τ(x):= \inf\Big\{s \geq 0: \sum_{n \geq 1} a^{-s}_n(x)<\infty\Big\}. $$
Source: arXiv (Cornell University)
Authors: Tan, Xiaoyan, Zhang, Zhenliang
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7084166414
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