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 $n\ge 481$, independent of the spectral gap of ${\Gamma }_0$.

24 Pages, 0 figures. Revision: improved introduction, minor edits and added Remark 4.2