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 $G$ being one of $\operatorname{SO}(n, 1)^\circ$ for $n \geq 2$, $\operatorname{SL}_2(\mathbb R) \times \operatorname{SL}_2(\mathbb R)$, $\operatorname{SL}_3(\mathbb R)$, $\operatorname{SU}(2, 1)$, $\operatorname{Sp}_4(\mathbb R)$, 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.