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Abstract We establish effective versions of Oppenheim’s conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed quadratic forms and generic shifts. Our results complement our previous paper [13] where we considered generic forms and fixed shifts. In this paper, we use ergodic theorems and in particular we establish a strong spectral gap with effective bounds for some representations of orthogonal groups, which do not possess Kazhdan’s property $(T)$.
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Effective Density for Inhomogeneous Quadratic Forms I: Generic Forms and Fixed Shifts
Abstract We establish effective versions of Oppenheim’s conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result, which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the 2nd moment of Siegel transforms on certain congruence quotients of $$SL_n(\mathbb{R}),$$ which we believe to be of independent interest. In a sequel, we use different techniques to treat the companion problem concerning generic shifts and fixed quadratic forms.
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- Award ID(s):
- 1651563
- PAR ID:
- 10186868
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnaa206
