The Local Hurwitz Constant and Diophantine Approximation on Hecke Groups

Define the Hecke group by «."((i r ".')> We call G (oo) the G -rationals, and M G (oo) the G -irrationals. The problem we treat here is the approximation of G -irrationals by Gq -rationals. Let M (a) be the upper bound of numbers c for which \a k/m\ < l/cm for all G -irrationals and infinitely many k/m e G (oo). Set h' = infQ M (a). We call h' the Hurwitz constant for by using A -continued fracti previously by D. Rosen. Write Gq . It is known that h'q = 2 , q even; h'q = 2(1 + (1 A?/2)2)l/2 , o odd. In this paper we prove this result ons, as developed pH-i (-irv2---«„ a — Q„-i mn_x(a)Ql i where £( = ±1 and P¡/Q¡ are the convergents of the A -continued fraction for q . Then M (a) = limn mn(a). We call mn(a) the local Hurwitz constant. In the final section we prove some results on the local Hurwitz constant. For example (Theorem 4), it is shown that if q is odd and en+x = en+2 = +1 , then rn¡ > ßq + 4) > h' for at least one of /' = n 1, n , n + 1 .