Convergence exponent of Dirichlet non-improvable numbers in the theory of continued fractions

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 $\alpha,\beta$ 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. $$ G(\alpha,\beta): =\left\{x\in[0,1)\colon \tau(x)=\alpha,\,\,\text{and} \,\, \limsup_{n\to\infty}\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}\geq\beta\right\}, $$ and $$ E(\alpha,\beta): =\left\{x\in[0,1)\colon \tau(x)=\alpha,\,\,\text{and} \,\, \limsup_{n\to\infty}\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}=\beta\right\}, $$ where $$ \tau(x):= \inf\Big\{s \geq 0: \sum_{n \geq 1} a^{-s}_n(x)<\infty\Big\}. $$