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

Abstract

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