Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

TL;DR

This paper introduces a simple, efficient method to rigorously estimate the Hausdorff dimension of limit sets in various mathematical contexts, with applications to Diophantine spectra, Zaremba's conjecture, and spectral bounds of infinite area surfaces.

Contribution

We develop a practical approach for obtaining rigorous bounds on Hausdorff dimensions of limit sets, applicable to multiple areas including number theory and geometric analysis.

Findings

Confirmed and strengthened conjectures on Lagrange and Markov spectra differences.

Improved estimates related to Zaremba's conjecture in number theory.

Bounded the bottom of the Laplacian spectrum for infinite area surfaces.

Abstract

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas red of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira arXiv:1803.01230. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain-Kontorovich arXiv:1107.3776v2, Huang arXiv:1310.3772v4 and Kan arXiv:1604.04884. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen. In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in section 3 in a way that is straightforward to implement.