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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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Second moment of the light-cone Siegel transform and applications
We study the light-cone Siegel transform, transforming functions on the light cone of a rational indefinite quadratic form Q to a function on the homogenous space $$\SO_Q(\Z)\backslash \SO_Q(\R)$$. In particular, we prove a second moment formula for this transform for forms of signature (n+1,1), and show how it can be used for various applications involving counting integer points on the light cone. In particular, we prove some new results on intrinsic Diophantine approximations on ellipsoids as well as on the distribution of values of random linear and quadratic forms on the light cone.
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- Award ID(s):
- 1651563
- PAR ID:
- 10513724
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Advances in Mathematics
- Volume:
- 432
- Issue:
- C
- ISSN:
- 0001-8708
- Page Range / eLocation ID:
- 109270
- 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/10.1016/j.aim.2023.109270
