Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications

TL;DR

This paper proves effective equidistribution of large torus translates in certain arithmetic homogeneous spaces, leading to precise counting formulas for matrices with given characteristic polynomials, extending Eskin-Mozes-Shah results.

Contribution

It establishes effective equidistribution results for translates of tori in rank at most 2 groups, including SL(3,R), with applications to matrix counting problems.

Findings

Effective equidistribution for large torus translates in SL(3,R)/Γ.

Asymptotic counting formula with power saving error term.

Extension of Eskin-Mozes-Shah theorems to new settings.

Abstract

Let $\Gamma < G$ be an arithmetic lattice in a noncompact connected semisimple real algebraic group. For many such $G$ of rank at most $2$, in particular $G = \operatorname{SL}_3(\mathbb R)$, we prove effective equidistribution of large translates of tori in $G/\Gamma$. As an application, we obtain an asymptotic counting formula with a power saving error term for integral $3 \times 3$ matrices with a specified characteristic polynomial. These effectivize celebrated theorems of Eskin-Mozes-Shah.