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

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.