A remark on the extreme value theory for continued fractions

Let $x$ be a irrational number in the unit interval and denote by its continued fraction expansion $[a_1(x),a_2(x),\cdots ,a_n(x),\cdots ]$. For any $n\ge 1$, write $T_n(x)={\max }_{1\le k\le n}\{a_k(x)\}$. We are interested in the Hausdorff dimension of the fractal set $$E_{\varphi }=\{x\in (0,1):\underset{n\to \mathrm{\infty }}{\lim }\frac{T_n(x)}{\varphi (n)}=1\},$$ where $\varphi$ is a positive function defined on $\mathbb{N}$ with $\varphi (n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. Some partial results have been obtained by Wu and Xu, Liao and Rams, and Ma. In the present paper, we further study this topic when $\varphi (n)$ tends to infinity with a doubly exponential rate as $n$ goes to infinity.