Counting, Mixing and Equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds

TL;DR

This paper investigates how expanding horospheres distribute evenly in certain infinite volume geometric spaces and applies these findings to count spheres in Apollonian packings.

Contribution

It introduces new results on horosphere equidistribution in infinite volume manifolds and applies them to the Apollonian sphere packing counting problem.

Findings

Proved equidistribution of expanding horospheres in geometrically finite rank one manifolds.

Applied equidistribution results to derive orbital counting formulas in Apollonian packings.

Enhanced understanding of geometric and dynamical properties of infinite volume manifolds.

Abstract

In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in apollonian sphere packing.