Theorem 1.19
(Balog–Szemeredi–Gowers, Schoen)
.
Assuming that:
A
⊆
G
is finite
E
(
A
)
≥
η
|
A
|
3
for some
η
>
0
Then
there exists
A
′
⊆
A
of size at least
c
1
(
η
)
|
A
|
such that
|
A
′
+
A
′
|
≤
|
A
′
|
c
2
(
η
)
, where
c
1
(
η
)
and
c
2
(
η
)
are polynomial in
η
.