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 $n \geq 1$, write $T_n(x) = \max_{1 \leq k \leq n}\{a_k(x)\}$. We are interested in the Hausdorff dimension of the fractal set $$E_\phi = \left\{x \in (0,1): \lim_{n \to \infty} \frac{T_n(x)}{\phi(n)} =1\right\},$$ where $\phi$ is a positive function defined on $\mathbb{N}$ with $\phi(n) \to \infty$ as $n \to \infty$. 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 $\phi(n)$ tends to infinity with a doubly exponential rate as $n$ goes to infinity.