NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

INHOMOGENEOUS QUADRATIC FORMS 147
Proof of Theorem 4.1
We may and will assume that ˜f is nonnegative. Now define
A(r) = ∈ X : α1() >r . (63)
Let gr be a continuous function on X such that gr() = 0 if /∈ A(r), gr() = 1
for all ∈ A(r + 1), and0 ≤ gr() ≤ 1 if r ≤ α1() ≤ r + 1. Let the constant c
(depending on f and ξ) beasin(43), and let s ∈ N be a large number. Define
( fgr) ˜ ≤
s = ( fgr)χ ˜
{ : fgr ˜ ()≤c2s } and ( fgr) ˜ >
s = ( fgr)χ ˜
{ : fgr ˜ ()>c2s }. (64)
Now let µ be as in the statement of Theorem 4.1.Sincef˜−fgr ˜ is bounded and
continuous, Theorems A.3 and A.4 imply that
lim sup ( f˜ − fgr)(atkqξ)ν(k) ˜
dk ≤ fdµ ˆ νdk. (65)
t→∞ K
G/ Ɣ K
Hence Theorem 4.1 will be proved if we show that: given ɛ>0, we can choose r0
such that, if r>r0, then lim supt K ˜ fgr dk < ɛ.
Now let ɛ>0 be an arbitrary small number. Let us first control the contribution
of ( fgr) ˜ > s when s is large enough. Let M>0 beanumberfixed(fornow)andlarge
enough such that 1/M < ɛ/6C,where C is a universal constant appearing in (70) and,
in particular, is independent of µ1 in the definition of quasi-null subspaces. Further
assume that M is large enough such that all exceptional subspaces have norm less
than M. Let µ1 in the definition of quasi-null subspace be small enough such that,
for all L ∈ Qt(δ, ε) with vL 0
be large enough such that the conclusions of Theorems 4.2 and 4.5, for this µ1 which
we chose, as well as of Corollaries 5.5 and 5.11, hold for δ = 1/2j whenever j>s.
Recall now that we have
k ∈ K : ˜f (atkξ) >c2 j ⊂ k ∈ K : α13(atkξ) > 2 j
1
∪ Bt
2j
1
∪ At
2j
,
where At(1/2 j ) and Bt(1/2 j ) are as in (43). Let Ct(1/2 j ) ={k ∈ K : α13(atkξ) >
2 j }.Wehave
K
h >
s (atKqξ) dk ≤
s