Hausdorff dimension of the set of real numbers of Lagrange value three

TL;DR

This paper proves that the set of real numbers with a Lagrange value of 3 has Hausdorff dimension zero, using a generalization involving Teichmüller space and modular groups.

Contribution

It introduces a novel approach connecting Lagrange values with Teichmüller space to determine Hausdorff dimension.

Findings

Hausdorff dimension of the set is zero

Generalization applies to subgroups of the modular group

Method links number theory with geometric structures

Abstract

We show that the set of real numbers of Lagrange value 3 has Hausdorff dimension zero by showing the appropriate generalization for each element of the Teichmueller space of the appropriate subgroup of the classical modular group.