A sparse equidistribution result for $(\mathrm{SL}(2,\mathbb{R})/\Gamma_0)^n$

Abstract

Let $G=\mathrm{SL}(2,\mathbb{R})^n$, let $\Gamma=\Gamma_0^n$, where $\Gamma_0$ is a co-compact lattice in $\mathrm{SL}(2,\mathbb{R})$, let $F(\mathbf{x})$ be a non-singular quadratic form and let $u(x_1,...,x_n)$ denote the unipotent elements in $G$ which generate the standard $n$ dimensional horospherical subgroup, consisting of $2\times 2$ upper triangular unipotent matrices in each co-ordinate. We prove that in absence of any local obstructions for $F$, given any $x_0\in G/\Gamma$, the sparse subset $\{u(\mathbf{x})x_0:\in\mathbb{Z}^n, F(\mathbf{x})=0\}$ equidistributes in $G/\Gamma$ as long as $n\geq 481$, independent of the spectral gap of $\Gamma_0$.