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 Teichmüller disk and prove a formula which allows to express large values in the Lagrange spectrum as sums of Cantor sets.

Main result extended to any closed and SL(2,R)-invariant set (closedness of Lagrange Spectrum and density of values corresponding to closed geodesics). Reorganized exposition. New material is in Appendix BInternational audienceWe introduce Lagrange Spectra of closed-invariant loci for the action of SL(2,R) on the moduli space of translation surfaces, generalizing the classical Lagrange Spectrum, and we analyze them with renormalization techniques. A formula for the values in such spectra is established in terms of the Rauzy-Veech induction and it is used to show that any invariant locus has closed Lagrange spectrum and values corresponding to pseudo-Anosov elements are dense. Moreover we show that Lagrange spectra of arithmetic Teichm\"uller discs contain an Hall's ray, giving an explicit bound for it via a second formula which uses the classical continued fraction algorithm. In addition, we show the equivalence of several definitions of bounded Teichm\"uller geodesics and bounded type interval exchange transformations and we prove quantitative estimates on excursions to the boundary of moduli space in terms of norms of positive matrices in the Rauzy-Veech induction