NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

NEAR OPTIMAL BOUNDS IN FREIMAN’S THEOREM 3
2. Preliminaries
This part of the article contains basic definitions and notation we will use later on.
As usual, we set
A + B ={a + b : a ∈ A, b ∈ B},
and the k-fold sumset of A is denoted by kA.Byageneralized arithmetic progression
of dimension d, we mean every set of the form P = P1+···+Pd, where P1,...,Pd are
usual arithmetic progressions. The size of P is defined as the product |P1|···|Pd|.The
dimension and size of progression P are denoted by dim(P ) and size(P ), respectively.
If each x ∈ P has unique representation x = p1 +···+pd, pi ∈ Pi, thenwesay
that P is proper. Then the cardinality of P is equal to its size.
Let G, H be Abelian groups, and let A ⊆ G, B ⊆ H . We say that A is Fkisomorphic
(i.e., Freiman isomorphic of order k) toB if there exists a bijective map
ϕ : A → B such that
if and only if
x1 +···+xk = y1 +···+yk
ϕ(x1) +···+ϕ(xk) = ϕ(y1) +···+ϕ(yk)
for every x1,...,xk,y1,...,yk ∈ A.
We call (x,x ′ ,y,y ′ ) an additive quadruple if x + y = x ′ + y ′ . The number of
additive quadruples in X2 × Y 2 is denoted be E(X, Y ).
In this paper, by Zn we always mean Z/nZ. The Fourier coefficients of the
indicator function of a set X ⊆ Zn are defined by
ˆX(s) =
e −2πixs/n ,
x∈X
where s ∈ Zn. Parseval’s formula states that n−1
s=0 | ˆX(s)| 2 =|X|n. We observe also
that, for X, Y ⊆ Zn, wehave
E(X, Y ) = 1 n−1
| ˆX(s)|
n
2 | ˆY (s)| 2 . (1)
s=0
Our argument makes use of Plünnecke-Ruzsa’s inequality, which says, in particular
that if |A + A| ≤K|A|, then for all k, l ∈ N, wehave
|kA − lA| ≤K k+l |A|.