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 $G={\mathrm{SL}}_3(\mathbb{R})$, 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.

Included Proposition 4.20; 61 pages, 1 figure