Convergence and Counting in Infinite Measure

TL;DR

This paper studies the behavior of lattice points in negatively curved spaces, providing detailed asymptotic descriptions for their counting functions, including both convergent and divergent cases.

Contribution

It introduces a new framework for analyzing lattices in negative curvature, with precise asymptotic results for counting functions in both convergent and divergent scenarios.

Findings

Asymptotic formulas for lattice counting functions

Construction of specific convergent and divergent lattices

Detailed analysis of lattice behavior in negative curvature

Abstract

We construct convergent and divergent lattices in negative curvature and give a precise asymptotic description of the behavior of their counting function.