A remark on Liao and Rams' result on distribution of the leading partial quotient with growing speed $e^{n^{1/2}}$ in continued fractions

Abstract

For a real $x\in(0,1)\setminus\mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots]$ be its continued fraction expansion. Denote by $T_n(x):= max \{a_k(x): 1\leq k\leq n\}$ the leading partial quotient up to $n$. For any real $\alpha\in(0,\infty), \gamma\in(0,\infty)$, let $F(\gamma,\alpha):=\{x\in(0,1)\setminus\mathbb{Q}: \lim_{n\rightarrow\infty}\frac{T_n(x)}{e^{n^\gamma}}=\alpha\}$. For a set $E\subset (0,1)\setminus\mathbb{Q}$, let $dim_H E$ be its Hausdorff dimension. Recently Lingmin Liao and Michal Rams [LR, Theorem 1.3] show that $dim_H F(\gamma,\alpha)$ is $1$ if $r\in(0,1/2)$, it is $1/2$ if $r\in(1/2,\infty)$ for any $\alpha\in(0,\infty)$. In this paper we show that $dim_H F(1/2,\alpha)=1/2$ for any $\alpha\in(0,\infty)$ following Liao and Rams' method, which supplements their result.