Quantitative distortion and the Hausdorff dimension of continued fractions

TL;DR

This paper establishes quantitative distortion bounds for iterated function systems generating continued fractions, enabling precise computation of Hausdorff dimension bounds for these fractal sets in real and complex cases.

Contribution

It introduces a new quantitative distortion theorem and provides practical bounds on Hausdorff dimension via solvable Moran-type equations.

Findings

Derived upper and lower bounds for Hausdorff dimension

Implemented bounds using computer algebra systems

Applicable to both real and complex continued fractions

Abstract

We prove a quantitative distortion theorem for iterated function systems that generate sets of continued fractions. As a consequence, we obtain upper and lower bounds on the Hausdorff dimension of any set of real or complex continued fractions. These bounds are solutions to Moran-type equations in the convergents that can be easily implemented in a computer algebra system.