Diophantine approximation on Hecke groups

Extract

If α is a real irrational number, there exist infinitely many reduced rational fractions p/q for which

and √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 have

for infinitely many p/q∊Г(∞), where k depends only on Г. Attention centers on

k 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.