The Lagrange spectrum of a Veech surface has a Hall ray

TL;DR

This paper proves that the Lagrange spectrum of any Veech surface contains a Hall ray, using boundary expansion and Cantor set sums to analyze geodesic coding in Teichmüller disks.

Contribution

It establishes the existence of a Hall ray in the Lagrange spectrum of Veech surfaces, extending classical results to a broader geometric context.

Findings

Lagrange spectra of Veech surfaces contain a Hall ray

Boundary expansion effectively codes geodesics in Teichmüller disks

Large values in the spectrum are expressed as sums of Cantor sets

Abstract

We study Lagrange spectra of Veech translation surfaces, which are a generalization of the classical Lagrange spectrum. We show that any such Lagrange spectrum contains a Hall ray. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics in the corresponding Teichm\"uller disk and prove a formula which allows to express large values in the Lagrange spectrum as sums of Cantor sets.