NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

QUANTITATIVE VERSION OF THE OPPENHEIM
CONJECTURE FOR INHOMOGENEOUS
QUADRATIC FORMS
GREGORY MARGULIS and AMIR MOHAMMADI
Abstract
We prove a quantitative version of the Oppenheim conjecture for inhomogeneous
quadratic forms. We also give an application to eigenvalue spacing on flat 2-tori with
Aharonov-Bohm flux.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2. Passage to space of inhomogeneous lattices . . . . . . . . . . . . . . . 128
3. The case where p ≥ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4. The case of signature (2, 2) . . . . . . . . . . . . . . . . . . . . . . . . 132
5. Contribution from quasi-null subspaces . . . . . . . . . . . . . . . . . 136
6. Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7. Proof of Theorem 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A. Equidistribution of spherical averages . . . . . . . . . . . . . . . . . . . 153
B. Number of quasi-null subspaces . . . . . . . . . . . . . . . . . . . . . . 157
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
1. Introduction
Let Q be a nondegenerate indefinite quadratic form on R n . Let ξ ∈ R n beavector,
and define the (inhomogeneous) quadratic form Qξ by
Qξ(x) = Q(x + ξ) for all x ∈ R n . (1)
We will refer to Q = Q0 as the homogeneous part of Qξ. We say that Qξ has signature
(p, q) if Q does. Recall that a quadratic form Qξ is called irrational if it is not a scalar
DUKE MATHEMATICAL JOURNAL
Vol. 158, No. 1, c○ 2011 DOI 10.1215/00127094-1276319
Received 4 March 2010. Revision received 22 July 2010.
2010 Mathematics Subject Classification. Primary 11E20; Secondary 58J50.
Margulis’s work partially supported by National Science Foundation grant DMS-0801195.
121