The Markoff and Lagrange spectra on the Hecke group H4

TL;DR

This paper studies the Markoff and Lagrange spectra on the Hecke group H4, showing their equivalence to classical spectra, analyzing their structure beyond the first accumulation point, and identifying gaps and bounds.

Contribution

It establishes the structure of spectra on H4 beyond the first accumulation point, including Hausdorff dimension and gaps, extending prior characterizations.

Findings

Spectra are identical to those of the unit circle.

Both spectra have positive Hausdorff dimension after the first accumulation point.

Identified gaps and provided bounds on Hall's ray.

Abstract

We consider the Markoff spectrum and the Lagrange spectrum on the Hecke group $\mathbf H_4$. They are identical to the Markoff and Lagrange spectra of the unit circle. The Markoff spectrum on $\mathbf H_4$ is also known as the Markoff spectrum of index 2 sublattices by Vulakh and the Markoff spectrum of 2-minimal forms or $C$-minimal forms by Schmidt. They characterized the spectrum up to the first accumulation point, independently. We show that, after the first accumulation point, both spectra have positive Hausdorff dimension. Then we find gaps in the spectra and give a bound on Hall's ray.