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

Strombergsson, Andreas and Vishe, Pankaj (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). pp. 143-204.

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.

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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1112/jlms.12316
Publisher statement:© 2020 The Authors. The Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Date accepted:02 March 2020
Date deposited:24 March 2020
Date of first online publication:07 April 2020
Date first made open access:11 April 2020

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