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

Strombergsson, Andreas; Vishe, Pankaj

Abstract

Let G = SL(2, R) (R2) ⊕k and let Γ be a congruence subgroup of SL(2, Z) (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.

Citation

Strombergsson, A., & Vishe, P. (2020). An effective equidistribution result for SL(2,R)\ltimes (R^2)^{oplus k} and application to inhomogeneous quadratic forms. Journal of the London Mathematical Society, 102(1), 143-204. https://doi.org/10.1112/jlms.12316

• Journal Article Type: Article
• Acceptance Date: Mar 2, 2020
• Online Publication Date: Apr 7, 2020
• Publication Date: 2020-08
• Deposit Date: Mar 24, 2020
• Publicly Available Date: Sep 18, 2020
• Journal: Journal of the London Mathematical Society
• Print ISSN: 0024-6107
• Electronic ISSN: 1469-7750
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 102
• Issue: 1
• Pages: 143-204
• DOI: https://doi.org/10.1112/jlms.12316
• Public URL: https://durham-repository.worktribe.com/output/1267601