Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of $E_2$

TL;DR

This paper presents a rigorous computational method to accurately estimate the Hausdorff dimension of continued fraction Cantor sets, exemplified by the set of reals with digits 1 and 2, achieving over 100 decimal places of precision.

Contribution

It introduces a novel, rigorous algorithm based on transfer operators and holomorphic extensions to precisely bound the Hausdorff dimension of dynamically defined Cantor sets.

Findings

Achieved over 100 decimal place accuracy for the dimension of E_2

Developed methods for bounding transfer operator approximation numbers

Provided explicit bounds on Hausdorff dimension using complex analysis techniques

Abstract

We prove that the algorithm of [13] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set $E_2$ (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25. The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension.