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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; Vishe, Pankaj

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


Authors

Andreas Strombergsson



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

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Copyright 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.




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