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

TL;DR

The paper determines how the Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequences behaves. It proves a 1/2-dimensional threshold for the canonical even-index case and a density-driven dichotomy for general subsequences: dimension 1 when the index set has zero upper density and 1/2 when it has positive upper density. It further extends to Hirst-type sets E(D), showing dim_H equals the half of the convergence exponent tau(D) when the subsequence has positive density. The results illuminate a sharp dimension jump governed by the density of the chosen subsequence and connect to classical Diophantine-type fractal sets via robust covering and submersion techniques.

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.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: MR0004878
• Lemma 2.2: MR0004878
• Corollary 2.3: MR0004878
• Lemma 2.4: MR2215567
• Lemma 4.1
• proof
• Lemma 4.2
• proof