Effective equidistribution of translates of maximal horospherical measures in the space of lattices

TL;DR

This paper establishes effective equidistribution results for translates of maximal horospherical measures in the space of lattices, providing explicit error terms for counting problems and applications to the Manin conjecture.

Contribution

It proves effective equidistribution for maximal horospherical measures and probability measures with densities in rank two and three, with applications to counting and number theory.

Findings

Effective equidistribution results with explicit error terms.

Error bounds for counting lifts of horospheres in lattice spaces.

Logarithmic error term for the Manin conjecture on flag varieties.

Abstract

Recently Mohammadi and Salehi-Golsefidy gave necessary and sufficient conditions under which certain translates of homogeneous measures converge, and they determined the limiting measures in the cases of convergence. The class of measures they considered includes the maximal horospherical measures. In this paper we prove the corresponding effective equidistribution results in the space of unimodular lattices. We also prove the corresponding results for probability measures with absolutely continuous densities in rank two and three. Then we address the problem of determining the error terms in two counting problems also considered by Mohammadi and Salehi-Golsefidy. In the first problem, we determine an error term for counting the number of lifts of a closed horosphere from an irreducible, finite-volume quotient of the space of positive definite $n\times n$ matrices of determinant one that intersect a ball with large radius. In the second problem, we determine a logarithmic error term for the Manin conjecture of a flag variety over $\mathbb{Q}$.