Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism

TL;DR

This paper develops a formula using thermodynamic formalism to precisely or asymptotically determine the Hausdorff dimension of limit sets resulting from small perturbations of conformal iterated function systems, with applications to number theory.

Contribution

It introduces a series-form formula for Hausdorff dimensions under perturbations of CIFS, extending classical results in Diophantine approximation.

Findings

Derived an exact and asymptotic series formula for Hausdorff dimensions

Strengthened Hensley's 1992 asymptotic formula for continued fractions

Provided new bounds for sets with bounded or large partial quotients

Abstract

We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism \`a la Sinai--Ruelle--Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarn\'ik and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all $\leq N$; as well as its counterpart for reals whose partial quotients are all $\geq N$ due to Good from 1941.