Metric Theory for Continued Fractions with Multiple Large Partial Quotients

TL;DR

This paper develops a metric theory for real numbers with multiple large partial quotients in their continued fraction expansions, analyzing measure and dimension properties when at least r partial quotients are large among the first n terms.

Contribution

It extends classical metric results by considering multiple large partial quotients simultaneously, providing measure and dimension results for these sets.

Findings

Determines Lebesgue measure of sets with r large partial quotients.

Calculates Hausdorff dimension of these sets.

Generalizes previous results to multiple large partial quotients.

Abstract

The presence of large partial quotients can invalidate many classical limit theorems in the metric theory of continued fractions. A commonly employed strategy to overcome this problem is to discard the largest partial quotient when formulating variant forms of such theorems. However, this method will fail when dealing with at least two large partial quotients. Motivated by recent work of Tan, Tian, and Wang [Sci. China Math., 2023], we investigate the metric theory of real numbers that contain at least $r$ large partial quotients among the first $n$ terms of their continued fraction expansions. Specifically, let $[a_1(x), a_2(x), \ldots ]$ be the continued fraction expansion of a real number $x \in [0, 1)$. We determine the Lebesgue measure and Hausdorff dimension of the following set: \[ F(r, \psi)=\Big\{ x \in [0,1): \exists 1 \leq k_1< \cdots < k_r \leq n, a_{k_i} (x) \geq \psi(n)~(i=1, \ldots, r)\ \text{for i.m.}~n \in \mathbb{N} \Big\}, \] where `i.m.' stands for `infinitely many', $r \geq 1$ and $\psi$ is a positive function defined on $\mathbb{N}$.