NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

124 MARGULIS and MOHAMMADI
As in [EMM1], we also have the following uniform version of Theorem 1.5. Let
I(p, q) denote the space of inhomogeneous quadratic forms whose homogeneous
parts are quadratic forms of signature (p, q) and discriminant ±1.
THEOREM 1.6
Let D be a compact subset of I(p, q) with p ≥ 3 and q ≥ 1, and let n = p + q.
Then for every interval (a,b) and every θ>0, there exists a finite subset P of D
such that each Qξ ∈ P is a rational form and, for every compact subset F ⊂ D \ P ,
there exists T0 such that, for all Qξ ∈ F and T ≥ T0, we have
(1 − θ)λQ,(b − a)T n−2 ≤ NQ,ξ,(a,b,T ) ≤ (1 + θ)λQ,(b − a)T n−2 , (10)
where λQ, is as in (3).
The proofs of the above theorems are straightforward inhomogeneous versions of the
arguments and ideas developed in [DM] and[EMM1].
As we mentioned before, Theorem 1.5 fails in signature (2, 2) and (2, 1). In this,
paper, we prove an inhomogeneous version of Theorem 1.3. Indeed,asin[EMM2],
one needs to assume certain Diophantine conditions on the quadratic form. Using
similar ideas, we also give a partial result in the (2, 1) case (see Theorem 1.10 below).
We start with the following definitions.
Definition 1.7
Let κ>0. A vector ξ = (ξ1,...,ξn) ∈ R n is called κ-Diophantine if there exist C =
C(ξ) > 0 such that, for all 0