NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

INHOMOGENEOUS QUADRATIC FORMS 149
This finishes the proof of Theorem 4.1.
7. Proof of Theorem 1.10
The proof of Theorem 1.10 is very similar to that of Theorem 1.9. Indeed, our study
in this case is simpler as we are dealing with the case where the homogeneous part Q
is rational. Hence, we only need to consider the contribution of null subspaces to the
counting function.
As before, let q ∈ SL3(R) be such that Q(v) = B(qv) for all v ∈ R 3 . Since Q is
rational, we may assume that q is in PGL3(Q). Let p ∈ GL3(Q) be a representative
for q. Let = qZ 3 , and define qξ = q(Z 3 + ξ). As in Section 4, letX(qξ)
be the set of vectors in qξ not contained in qL, where L ⊂ Z 3 is an exceptional
(1-dimensional) subspace for Q. An argument like that in Lemma 6.1 shows that there
are at most three such subspaces when Q is rational and that ξ is an irrational vector.
For any continuous compactly supported function f on R 3 , define
f ˜(g
: qξ) =
v∈X(qξ )
f (gv). (74)
As in previous discussions, we reduce the proof of Theorem 1.10 to the following
theorem.
THEOREM 7.1
Let G, H, K, and {at} be as in Section 2 for the signature (2, 1) case. Let Qξ be a
quadratic form of signature (2, 1) as in the statement of Theorem 1.10.Letq∈ SL3(R)
and qξ be as above. Let ν be a continuous function on K. Then we have
lim sup
t→∞
K
˜f (atk : qξ)ν(k) dk ≤
G/ Ɣ
f ˆ(g)
dµ(g)
K
ν(k) dk, (75)
where µ is the H ⋉R 3 -invariant probability measure on the closed orbit H ⋉R 3 ·qξ.
The proof of this theorem is very similar to that of Theorem 4.1. We will use the
same notations as those in the previous sections for the sake of simplicity. The main
notational difference to bear in mind is that in previous sections L denotes a 2dimensional
subspace, where in this section L is a 1-dimensional (null) subspace.
As before, we need to study the subsets of K where the function ˜f is “large”.
We start by recalling the following. There is c>0 such that, for all large t and small