The Markov and Lagrange spectra on the Hecke group $\mathbf H_4$
Volume 224 / 2026
Acta Arithmetica 224 (2026), 47-70
MSC: Primary 11J06; Secondary 11J70
DOI: 10.4064/aa250827-16-2
Published online: 9 June 2026
Abstract
We consider the Markov spectrum and the Lagrange spectrum on the Hecke group $\mathbf H_4$. They are identical to the Markov and Lagrange spectra on the unit circle. The Markov spectrum on $\mathbf H_4$ is termed the Markov spectrum on index 2 sublattices by Vulakh and the Markov spectrum on 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.
