Fractal geometry of continued fractions with large coefficients and dimension drop problems

TL;DR

This paper investigates how the Hausdorff dimension of sets of continued fractions decreases as their coefficients grow large, extending previous results and applying new methods to related dimension problems.

Contribution

It generalizes and extends existing studies on the dimension drop phenomenon for continued fractions with large coefficients, providing new results and alternative proofs.

Findings

Reproved a result on dimensions of Borel-Bernstein sets.

Fulfilled the dimension gap proposed by Liao and Rams.

Established new results on the dimension theory of liminf and limsup sets.

Abstract

In 1928, Jarn\'{\i}k \cite{Jar} obtained that the set of continued fractions with bounded coefficients has Hausdorff dimension one. Good \cite{Goo} observed a dimension drop phenomenon by proving that the Hausdorff dimension of the set of continued fractions whose coefficients tend to infinity is one-half. For the set of continued fractions whose coefficients tend to infinity rapidly, Luczak \cite{Luc} and Feng et al. \cite{FWLT} showed that its Hausdorff dimension decreases even further. Recently, Liao and Rams \cite{LR16} also observed an analogous dimension drop phenomenon when they studied the subexponential growth rate of the sum of coefficients. In this paper, we consolidate and considerably extend the studies of the abovementioned problem into a general dimension drop problem on the distribution of continued fractions with large coefficients. As applications, we use a different approach to reprove a result of Wang and Wu on the dimensions of the Borel-Bernstein sets \cite{WW}, fulfil the dimension gap proposed by Liao and Rams \cite{LR16}, and establish several new results concerning the dimension theory of liminf and limsup sets related to the maximum of coefficients.