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

Abstract

Let G = SL(2, R) n, let Γ = Γn 0 , where Γ0 is a co-compact lattice in SL(2, R), let F(x) be a non-singular quadratic form and let u(x1, ..., xn) := 1 x1 0 1 ×...× 1 xn 0 1 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 x0 ∈ G/Γ, the sparse subset {u(x)x0 : x ∈ Z n, F(x) = 0} equidistributes in G/Γ as long as n ≥ 481, independent of the spectral gap of Γ