On the frequency of partial quotients of regular continued fractions

TL;DR

This paper investigates the Hausdorff dimensions of sets of real numbers in [0,1) with specified partial quotient frequencies in their continued fractions, revealing a lower bound of 1/2 and a variational principle for their dimensions.

Contribution

It introduces a modified variational principle to determine the Hausdorff dimensions of these sets, extending understanding of continued fraction expansions.

Findings

Hausdorff dimensions are always at least 1/2

Dimensions are characterized by a modified variational principle

Provides new insights into the structure of continued fractions

Abstract

We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are given by a modified variational principle.