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 \[\left\{x\in(0,1): a_1(x)\leq a_2(x)\leq \cdots,\ \limsup\limits_{n\to\infty}\frac{\log a_n(x)}{ψ(n)}=1\right\}\] for any $\psi :\mathbb{N}\to {\mathbb{R}}^{+}$ satisfying $\psi (n)\to \infty$ as $n\to \infty$.

14 pages