Joint partial equidistribution of Farey rays in negatively curved manifolds and trees

TL;DR

This paper establishes a joint partial equidistribution result for Farey rays and related geometric structures in negatively curved manifolds and trees, extending previous results and providing new arithmetic applications.

Contribution

It introduces a novel joint partial equidistribution theorem for Farey rays and equidistributing hypersurfaces in negatively curved spaces and trees, with several arithmetic applications.

Findings

Proves joint partial equidistribution of Farey rays in hyperbolic and tree settings

Recovers and generalizes Marklof's result on Farey fractions

Provides new applications in Bruhat-Tits trees

Abstract

We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat-Tits trees.