NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM
NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM
NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM
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136 MARGULIS and MOHAMMADI<br />
Indeed, an estimate like (43) would actually hold for f ˆ (by virtue of [Sch, Lemma<br />
2]) if A was defined by taking the union over all quasi-null subspaces. But we have<br />
replaced fˆ by f ˜,<br />
and this implies that we can take the union over Q instead. To see<br />
this, consider one of these subspaces, say L1. Assume that ξ ∈ L1 + w1ξ, where<br />
w1ξ ∈ Z4 . Now if there is k ∈ K such that (44) is satisfied with L = L1 and v ∈ Z4 ,<br />
then there is a constant cξ ≥ 1 depending on ξ such that d(atkH) 1/cξδ}. Hence there is c such that (43)<br />
holds.<br />
Now the proof of Theorem 4.1 will be completed if we can show that there is<br />
some η>0 depending on Qξ such that |AL t (δ)|
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