NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

152 MARGULIS and MOHAMMADI
Some important properties of these intervals were proved in [EMM2] and recalled in
Lemma 5.1. What is important for us in Section 7 is property (ii) in Lemma 5.1.This
property, tailored to our current assumptions, gives
|A L
t
−t
L e δ
1/2 (δ)| ≤|It (δ)| ≈ . (81)
T
Proof of Theorem 7.1
Let M be a large number which is fixed for now. Let t>0 be a large number. Assume
also that j is a large number. We assume throughout that η = η1, where η1 is as in
Lemma 7.3. LetL0,L1 be nonexceptional null subspaces such that {〈v Lk , pξ〉B} <
1/2 j for k = 0, 1 and such that v L0 ≤v L1 are minimal with these properties.
Indeed, we fix any two such subspaces if there are more than two subspaces satisfying
these conditions. Assume that e t /2 j+i0+1 ≤v L 2 jη2 , where η2 is as in Lemma 7.3. Now using this
lemma, we see that, for all i ≤ i0, the number of null subspaces L with e t /2 j+i+1 ≤
v L i0, the subspaces of norm at most e t /2 i+j have
no contribution to the set At(1/2 j ), that is, A L t (1/2j ) =∅for all such subspaces L.
Hence if we use this fact and (81), we have: if j is such that Case 1 holds, then
we have
Lnull
AL
1
t
2j
≤
i
≤ 1
2 j(1+η)
et /(2i+j+1 )≤vL