On the Markoff spectrum on the Hecke group of index six

TL;DR

This paper investigates the structure of the Markoff and Lagrange spectra associated with the Hecke group of index six, revealing their fractal nature, gaps, and isolated points near a critical value.

Contribution

It analyzes the spectra's behavior below a key accumulation point, showing they have positive Hausdorff dimension and identifying spectral gaps and isolated points.

Findings

Spectra have positive Hausdorff dimension below the critical point

Existence of maximal gaps in the spectra

Identification of an isolated spectral point

Abstract

The discrete part of the Markoff spectrum on the Hecke group of index 6 was determined by A.~Schmidt. In this paper, we study its Markoff and Lagrange spectra after the smallest accumulation point $4/\sqrt3$. We show that both the Markoff and Lagrange spectra below $4/\sqrt{3} + \epsilon$ have positive Hausdorff dimension for any positive $\epsilon$. We also find maximal gaps and an isolated point in the spectra.