A sparse equidistribution result for $(\operatorname {SL}(2,\mathbb {R})/\Gamma _0)^{n}$

Abstract:

Let $G=\operatorname {SL}(2,\mathbb {R})^n$, let $\Gamma =\Gamma _0^n$, where $\Gamma _0$ is a co-compact lattice in $\operatorname {SL}(2,\mathbb {R})$, let $F(\mathbf {x})$ be a non-singular quadratic form and let $u(x_1,\dots ,x_n)≔{\bigl (\begin {smallmatrix} 1 & x_1 \\ 0 & 1 \end {smallmatrix}\bigr )} \times \dots \times {\bigl (\begin {smallmatrix} 1 & x_n \\ 0 & 1 \end {smallmatrix}\bigr )}$ denote unipotent elements in $G$ which generate an $n$ dimensional horospherical subgroup. We prove that in the absence of any local obstructions for $F$, given any $x_0\in G/\Gamma$, the sparse subset $\{u(\mathbf {x})x_0:\mathbf {x}\in \mathbb {Z}^n, F(\mathbf {x})=0\}$ equidistributes in $G/\Gamma$ as long as $n\geq 481$, independent of the spectral gap of $\Gamma _0$.