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