The Markoff and Lagrange spectra on the Hecke group H4

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.