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

TL;DR

This paper investigates the Hausdorff dimension of sets of real numbers with specific growth rates of their largest partial quotient in continued fractions, extending previous results to the case where the growth rate is exponential with exponent 1/2.

Contribution

It proves that the Hausdorff dimension of the set where the maximum partial quotient grows like e^{n^{1/2}} is 1/2, completing the characterization for this growth rate.

Findings

Hausdorff dimension of F(1/2, α) is 1/2 for all α>0

Extends Liao and Rams' results to the case r=1/2

Supports the conjecture on dimension behavior at critical growth rate

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.