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arXiv · 2201.12974

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

Lulu Fang·Jihua Ma·Kunkun Song·Min Wu
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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 $n\to\infty$.

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Lulu Fang, Jihua Ma, Kunkun Song, Min Wu. 2022-01-31. Dimensions of certain sets of continued fractions with non-decreasing partial quotients. https://arxiv.org/abs/2201.12974

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