An effective equidistribution result for SL(2,R)\ltimes (R^2)^{oplus k} and application to inhomogeneous quadratic forms.

Abstract

Let G = SL(2, R) (R2) ⊕k and let Γ be a congruence subgroup of SL(2, (Z2) ⊕k. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in Γ\G which project to pieces of closed horocycles in SL(2, Z) SL(2, R). As an application, we prove an effective quantitative Oppenheim-type result for the quadratic form (m1 − α) 2 + (m2 − β) 2 − (m3 − α) 2 − (m4 − β) 2, for (α, β) ∈ R2 of Diophantine type, following the approach by Marklof [Ann. of Math. 158 (2003) 419–471] using theta sums.