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

TL;DR

This paper determines the Hausdorff dimension of specific sets of continued fractions with non-decreasing partial quotients, characterized by a growth condition involving a function \\psi(n), expanding understanding of their fractal structure.

Contribution

It provides the Hausdorff dimension for sets of continued fractions with non-decreasing partial quotients under a general growth condition, a novel extension in the field.

Findings

Hausdorff dimension of the set with non-decreasing partial quotients is obtained

Dimension formula applies to any growth function \\psi(n) \\to \\infty

Results extend previous work on continued fractions and fractal dimensions

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