Distribution of approximants and geodesic flows

TL;DR

This paper provides a new proof that continued fraction approximants are equidistributed into cusps for various types of continued fractions, using geodesic flow techniques on the modular surface.

Contribution

It introduces a novel proof method for equidistribution of approximants, extending results to Nakada's ontinued fractions and Rosen's ontinued fractions associated with Hecke groups.

Findings

Equidistribution of continued fraction approximants into cusps for modular groups.

Extension of equidistribution results to Nakada's ontinued fractions.

Extension of results to Rosen's ontinued fractions and Hecke triangle groups.

Abstract

We give a new proof of Moeckel's 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 widths. 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 Nakada's \alpha-continued fractions, and also that the analogous result holds for approximants that are algebraic numbers given by any of Rosen's \lambda-continued fractions, related to the infinite family of Hecke triangle Fuchsian groups.