We give a new proof of Moeckels result that for any finite index subgroup of the modular group, almost every real number has its regular continued fraction approximants equidistributed into the cusps of the subgroup according to the weighted cusp wid
ths. Our proof uses a skew product over a cross-section for the geodesic flow on the modular surface. Our techniques show that the same result holds true for approximants found by Nakadas alpha-continued fractions, and also that the analogous result holds for approximants that are algebraic numbers given by any of Rosens lambda-continued fractions, related to the infinite family of Hecke triangle Fuchsian groups.
