Limiting Distributions of Translates of Divergent Diagonal Orbits

TL;DR

This paper establishes a topology on measures on homogeneous spaces and proves that certain divergent orbits equidistribute, leading to an asymptotic formula for counting integral points on varieties.

Contribution

It introduces a natural topology on measures and proves equidistribution of divergent orbits, with applications to counting integral points.

Findings

Pushforwards of infinite volume orbits equidistribute in the space

An asymptotic formula for counting integral points in large balls

New topology on measures on homogeneous spaces

Abstract

We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain infinite volume orbits equidistribute in the ambient space. As an application of our results we prove an asymptotic formula for the number of integral points in a ball on some varieties as the radius goes to infinity.