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. $$G(\alpha ,\beta ):=\left\{x\in [0,1):\tau (x)=\alpha ,\text{and}\underset{n\to \infty }{\mathrm{limsup}}\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}\ge \beta \right\},$$ and $$E(\alpha ,\beta ):=\left\{x\in [0,1):\tau (x)=\alpha ,\text{and}\underset{n\to \infty }{\mathrm{limsup}}\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}=\beta \right\},$$ where $$\tau (x):=\inf \{s\ge 0:\sum _{n\ge 1}{a}_n^{-s}(x)<\infty \}.$$