An effective equidistribution result for SL(2,R)⋉(R2)⊕k and application to inhomogeneous quadratic forms

Abstract

Let G = SL(2, R) ⋉ (R 2 ) ⊕k and let Γ be a congruence subgroup of SL(2, Z) ⋉ (Z 2 ) ⊕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 quadraticfollowing the approach by Marklof [24] using theta sums.1 Apply [5, Thm. 3] with d = 2 and M = 12 and use the anti-automorphism (M, ( x x ′ )) → ( t M, t M x ′ −x ) of G to translate from the setting with G/Γ in [5] into our setting with X = Γ\G. As noted in [5, Remark 7.2], the proof of [5, Thm. 3] extends trivially to the case when Γ is an arbitrary subgroup of SL(2, Z) ⋉ (Z 2 ) ⊕k of finite index.

Some new results in quantitative Diophantine approximation

“…Then the inhomogeneous quadratic form is F (x + θ) for x ∈ R s . Values of inhomogeneous forms at integer points have been studied extensively [4,[25][26][27]33]. In [15,16], Ghosh, Kelmer and Yu proved effective results for inhomogeneous quadratic forms.…”

Effective equidistribution of expanding translates in the space of affine lattices

“…However, in special settings effective equidistribution results have been obtained e.g. [Str15,BV16] for Ĝ " SL 2 pRq ˙R2 , [SV20] for Ĝ " SL 2 pRq ˙pR 2 q 'k , and [Pri18] for Ĝ " SL 3 pRq ˙R3 . The purpose of the present paper is to establish an effective equidistribution result for Ĝ " SL d pRq ˙Rd . )…”