Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequence

TL;DR

This paper investigates the Hausdorff dimension of sets of real numbers with continued fraction expansions where certain subsequences of partial quotients tend to infinity, revealing a dimension of 1/2 or 1 depending on the density of the subsequence.

Contribution

It characterizes the Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequences, depending on the subsequence's density.

Findings

Dimension of $E_{even}$ is 1/2.

Dimension of $E_{ ext{any subsequence}}$ is 1/2 or 1.

Dimension depends on the upper density of the subsequence.

Abstract

Let $x=[a_1(x),a_2(x),\ldots]$ be the continued fraction expansion of $x\in[0,1)$. We prove that the Hausdorff dimension of \begin{equation*}E_{even}=\{x\in[0,1)\colon a_{2n}(x)\to\infty\ (n\to\infty)\}.\end{equation*} is 1/2. In general, we study the set of continued fractions with unbounded partial quotients along subsequence \begin{equation*}E_{\{k_n\}}=\{x\in[0,1)\colon a_{k_n}(x)\to\infty\ (n\to\infty)\},\end{equation*} where $\{k_n\}\subset\mathbb{N}$ is a subsequence. We show that $E_{\{k_n\}}$ has Hausdorff dimension 1/2 or 1 according to whether the set of indices $\{k_n\}_{n\geq 1}$ has positive or zero upper density respectively.