NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

INHOMOGENEOUS QUADRATIC FORMS 145
Proof
We may and will assume that L is of the first type. We continue to use the notations
as in Lemma 5.7. In particular, let M = pL, andlet{v1,v2} be the standard basis
for M. Then Lemma 5.7 implies that {〈vi, pξ〉B} ≤(cδ 1−η2 )/(T 1/2−τ3 ). Recall that
v1 = (m, 0,n,0) and v2 = (0,m,0,n), where gcd(m, n) = 1, and we have
T 1−τ3 ≤v M =m 2 + n 2 ≤ T 1+τ3 , (60)
where τ3 is a fixed multiple of the constant τ2 appearing in Theorem 5.6. We also note
that 〈v1 , pξ〉B = m(pξ)4 − n(pξ)2 and that 〈v2 , pξ〉B(pξ)1 − m(pξ)3.
Let τ ′ 1 ≪ 1/32κ be chosen. Then choose τ2 in Theorem 5.6 such that τ2 T τ ′ 1. Taking τ2 even smaller, we may and will assume that
τ3