Baire category and the relative growth rate for partial quotients in continued fractions

Abstract

Let $[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]$ be the continued fraction expansion of an irrational number $x\in (0,1)$, and $q_n(x)$ be the denominator of its n-th convergent. In this note, the Baire category of the set

$$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}$$

for $\alpha ,\beta \in [0,\infty ]$ with $\alpha \le \beta$ is studied. We prove that the set $E(\alpha ,\beta )$ is residual if and only if $\alpha =0$ and $\beta =\infty$.