On good approximations and Bowen-Series expansion

TL;DR

This paper studies the continued fraction expansion of real numbers influenced by non-uniform lattices in PSL(2,R), establishing metric relations between convergents and geometric approximations.

Contribution

It introduces new metric relations linking continued fraction convergents to geometric notions of approximation in the context of non-uniform lattices.

Findings

Established metric relations between convergents and geometric approximations.

Extended classical continued fraction theory to non-uniform lattice actions.

Provided insights into Diophantine approximation in hyperbolic geometry.

Abstract

We consider the continued fraction expansion of real numbers under the action of a non-uniform lattice in PSL(2,R) and prove metric relations between the convergents and a natural geometric notion of good approximations.