The Markoff and Lagrange spectra on the Hecke group H4

2022/06/11 by Dong Han Kim, Kim, Dong Han, Deokwon Sim +1
Mathematics · #11J06 #11J70 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2206.05441

openalex publication_date 2022/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Markoff spectrum and the Lagrange spectrum on the Hecke group \mathbf H₄. They are identical to the Markoff and Lagrange spectra of the unit circle. The Markoff spectrum on \mathbf H₄ 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.

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