NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

INHOMOGENEOUS QUADRATIC FORMS 139
(i) there exists λL ∈ such that, if for some λ ∈ there is kφ ∈ EL t (δ, kθ,ε) for
which the plane (L + λ + ζ )φ intersects B(r), then L + λ = L + λL;
(ii) for all λ ∈ , we have maxkφ (at(kθ,kφ)[λ + ζ ]) Lφ ⊥ >c/(δ(1−ε)/2 ).
Furthermore, if (ii) holds, then there exists a computable constant η1 > 0 such that,
for 0 0 such that the function hλ is (C1,β1) good. It follows from the
definition of (C, α)-good functions (see [KM]) that fλ(φ) =(at(kθ,kφ)(v L ∧ (λ +
ζ )) is (C2,β2)-good for some C2,β2 > 0.
Recall now that
c
maxkφ∈E (at(kθ,kφ)[λ + ζ ]) ⊥ Lφ > δ (1−ε)/2 ,
δ1+ε