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.
Comment: 10 pages. Some metric results on the extreme value theory for continued fractions are parallel to the classical results of i.i.d. random variables with Pareto-type distributions. This remark just collects some interesting results on the extreme value theory for continued fractions from the fractal points of view. The method in the proof of our main result is inspired by Xu