NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

142 MARGULIS and MOHAMMADI
(ii) There exists a split integral form Q ′ with coefficients bounded by a fixed power
of T τ2 and 1 ≤ λ ∈ R satisfying Q − (1/λ)Q ′ ≤T −ρ such that the number
of quasi-null subspaces of Q with norm between T/2 and T which are not null
subspaces of Q ′ is O(T 1−τ2 ).
We refer to [EMM2, Section 10] and also Section 7 of this paper for a more careful
analysis of this theorem. The main ingredient in the proof is the system of inequalities
from [EMM1]. The main difference is that these inequalities are applied to a certain
dilated lattice in 2 R 4 = R 6 .
Let us now outline the rest of the proof. Let Qξ be the inhomogeneous quadratic
form as in the statement of Theorem 1.9. We apply the above theorem with appropriate
parameters ρ,τ2 to be determined later. Let T ≥ 2 be given. Now if (i) above holds
(which is always the case if Q is not EWAS), then we already have a good control on
the number of quasi-null subspaces in question, and we will get the desired control
on the measure of the set
L AL t (δ, ε). Hence, we may assume that (ii) above holds.
Again, if L is not a null subspace of the appropriate approximation of Q, we proceed
as in case (i). So we need to consider the contribution from quasi-null subspaces which
are null subspaces of some rational approximation. In this case, using Lemma 5.3,we
are reduced to the case where only one translate of L has contribution. We then use
the Diophantine property of ξ, and we get a control on the number of such subspaces.
This completes the proof.
We need to fix some more notations before proceeding with the above outline.
If Q is a rational form, we may choose µ1 small enough such that all µ1-quasi-null
subspaces are null subspaces. Also in this case, by replacing Q with a scalar multiple,
we may and will assume that Q is a primitive integral form.
Let T ≥ 2 be a fixed number. Recall from Theorem 5.6 that there are two
possibilities for quasi-null subspaces. The case which requires more study is case
(ii), so let us assume that we are in this case. Let QT = Q ′ (resp., QT = Q) ifQ
is an irrational form (resp., if Q is split integral form), where Q ′ is given as in (ii)
of Theorem 5.6. LetQt(δ, ε, τ2) be the set of all such quasi-null subspaces (resp.,
null subspaces if Q is an rational) which are not exceptional subspaces and such that
A2 δ (•,ε) is nonempty for them. Let L ∈ Qt(δ, ε, τ2) be such a subspace. We further
assume that L is of the first type and that T/2 ≤vL≤T. Since QT is a split integral form, after possibly multiplying by a scalar bounded
by a fixed power of T τ2 , there exists a nonsingular integral matrix such that QT (v) =
B(pv). Furthermore, Theorem 5.6 guarantees that we may choose p such that its
entries are bounded by a fixed power of T τ2 . Recall that the null subspaces of B are
of two types based on the fact that the corresponding vector vL is in V1 or V2. From
now on by a null space of first kind for B, we mean the following.