Geodesic cusp excursions and metric diophantine approximation

TL;DR

This paper investigates how geodesics on hyperbolic surfaces approach cusps and connects these behaviors to metric Diophantine approximation, providing both new insights and proofs of classical theorems.

Contribution

It establishes a link between geodesic cusp excursions and Diophantine approximation, offering new results and proofs in the classical and Fuchsian group contexts.

Findings

Quantitative descriptions of geodesic cusp excursions

Connections between cusp excursions and Diophantine approximation

New proofs of classical Diophantine approximation theorems

Abstract

We derive several results that describe the rate at which a generic geodesic makes excursions into and out of a cusp on a finite area hyperbolic surface and relate them to approximation with respect to the orbit of infinity for an associated Fuchsian group. This provides proofs of some well known theorems from metric diophantine approximation in the context of Fuchsian groups. It also gives new results in the classical setting.