Dimensions of certain sets of continued fractions with non-decreasing partial quotients

Abstract

Let $[a_1(x),a_2(x),a_3(x),\cdots ]$ be the continued fraction expansion of $x\in (0,1)$. This paper is concerned with certain sets of continued fractions with non-decreasing partial quotients. As a main result, we obtain the Hausdorff dimension of the set

$$\begin{aligned} \left\{ x\in (0,1): a_1(x)\le a_2(x)\le \cdots ,\ \limsup \limits _{n\rightarrow \infty }\frac{\log a_n(x)}{\psi (n)}=1\right\} \end{aligned}$$

for any $\psi :\mathbb {N}\rightarrow \mathbb {R}^+$ satisfying $\psi (n)\rightarrow \infty$ as $n\rightarrow \infty$.