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 $T_n(x):=max\{a_k(x):1\le k\le n\}$ the leading partial quotient up to $n$. For any real $\alpha \in (0,\infty ),\gamma \in (0,\infty )$, let $F(\gamma ,\alpha ):=\{x\in (0,1)\setminus \mathbb{Q}:{\lim }_{n\to \infty }\frac{T_n(x)}{e^{n^{\gamma }}}=\alpha \}$. For a set $E\subset (0,1)\setminus \mathbb{Q}$, let $di{m}_{H}E$ be its Hausdorff dimension. Recently Lingmin Liao and Michal Rams [LR, Theorem 1.3] show that $di{m}_{H}F(\gamma ,\alpha )$ 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 $di{m}_{H}F(1/2,\alpha )=1/2$ for any $\alpha \in (0,\infty )$ following Liao and Rams' method, which supplements their result.

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