Geodesic
Diophantine approximation on Hecke groups
Glasgow mathematical journal, Tome 26 (1985), pp. 117-127

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If α is a real irrational number, there exist infinitely many reduced rational fractions p/q for whichand √5 is the best constant possible. This result is due to A. Hurwitz. The following generalization was proposed in [2]. Let Г be a finitely generated fuchsian group acting on H+, the upper half of the complex plane. Let L be the limit set of Г P and the set of cusps (parabolic vertices). Assume ∞∊P. Then if α∊L–P, we havefor infinitely many p/q∊Г(∞), where k depends only on Г. Attention centers onk running over the set for which (1.2) holds. We call hthe Hurwitz constant for Г. When Г=Г(1), the modular group, (1.2) reduces to (1.1) and h(Г(l))=√5. A proof of (1.2) when Г is horocyclic (i.e., L=R, the real axis) was furnished by Rankin [4]; he also found upper and lower bounds for h. See also [3, pp. 334–5], where the theorem is proved for arbitrary finitely generated Г. 
Lehner, J. Diophantine approximation on Hecke groups. Glasgow mathematical journal, Tome 26 (1985), pp. 117-127. doi: 10.1017/S0017089500006121
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