NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

126 MARGULIS and MOHAMMADI
planes” interfere with each other. This fact prevents us from successfully using the
ideas in [EMM2] when the signature is (2, 1). The absence of such an asymptotic
in the homogeneous case with explicit Diophantine conditions is responsible for the
“extra” assumption (i) in Theorem 1.10. Indeed, using this assumption, we are able to
reduce the study to an investigation of the contribution from null vectors. We are then
able to impose and utilize an explicit Diophantine condition on ξ (see Section 7 for
details).
Let us also mention that the Diophantine conditions in Theorems 1.10 and 1.9
are necessary. Similar to [EMM1, Theorem 2.2], [Mark, Theorem 1.3], and [Sar,
Theorem 2], it is possible to construct sets of second Baire category of quadratic
forms for which the above fail. To be more explicit, it is shown in [EMM1], using
category arguments, that there exists a dense set of second Baire category γ such
that NQ,(1/8, 2,Ti) >Ti(log Ti) 1−ε for an infinite sequence Ti, where Qγ (x) =
x2 1 + x2 2 − γ 2 (x2 3 + x2 4 ). Nowifwetakeanyξ∈ Z4 , then the same holds for Q γ
ξ .
This already gives the examples we wanted. However, these examples are in a sense
“degenerate" since ξ ∈ Z4 . We will reproduce the argument in [EMM1] to get more
“generic" counterexamples. First, note that an argument like that in [Mark, Appendix
10] shows that, for any (α, ς) ∈ Q × Q4 ,wehave
# v ∈ Z 4 : Q α
ς (v) = 0 and v ≤T ∼ Cς,αT 2 log T. (15)
Now the same argument as in [EMM1, Lemma 3.15] (see also [Mark, Section 9])
gives: for every ε>0 and any S>0, the set (β,ζ) ∈ R × R 4 for which there exists
T>Ssuch that NQ β ,ζ,(1/4, 1,T) ≥ T 2 (log T ) 1−ε is dense.
Given that S>0, letUS be the set of ordered pairs (γ,ξ) ∈ R × R 4 for which
there exists (β,ζ) ∈ R × R 4 and T>Ssuch that
NQ β ,ζ,(1/4, 1,T) ≥ T 2 (log T ) 1−ε , |β − γ |