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We study Lagrange spectra at cusps of finite area Riemann surfaces. Thesespectra are penetration spectra that describe the asymptotic depths ofpenetration of geodesics in the cusps. Their study is in particular motivatedby Diophantine approximation on Fuchsian groups. In the classical case of themodular surface and classical Diophantine approximation, Hall proved in 1947that the classical Lagrange spectrum contains a half-line, known as a Hall ray.We generalize this result to the context of Riemann surfaces with cusps andDiophantine approximation on Fuchsian groups. One can measure excursion into acusp both with respect to a natural height function or, more generally, withrespect to any proper function. We prove the existence of a Hall ray for theLagrange spectrum of any non co-compact, finite covolume Fuchsian group withrespect to any given cusp, both when the penetration is measured by a heightfunction induced by the imaginary part as well as by any proper function closeto it with respect to the Lipschitz norm. This shows that Hall rays are stableunder (Lipschitz) perturbations. As a main tool, we use the boundary expansiondeveloped by Bowen and Series to code geodesics and produce a geometriccontinued fraction-like expansion and mimic the key ideas in Hall's originalargument. A key element in the proof of the results for proper functions is ageneralization of Hall's theorem on the sum of Cantor sets, where we considerfunctions which are small perturbations in the Lipschitz norm of the sum.

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces