Lemma 2.22.
Assuming that:
A
⊆
[
N
]
of density
α
>
0
N
>
5
0
α
−
2
A
contains no non-trivial 3 term arithmetic progressions
p
a prime in
[
N
3
,
2
N
3
]
let
A
′
=
A
∩
[
p
]
⊆
ℤ
∕
p
ℤ
Then
one of the following holds:
(i)
𝟙
sup
t
≠
0
|
𝟙
A
′
^
(
t
)
|
≥
α
2
1
0
(where the Fourier coefficient is computed in
ℤ
∕
p
ℤ
)
(ii)
There exists an interval
J
⊆
[
N
]
of length
≥
N
3
such that
|
A
∩
J
|
≥
α
(
1
+
α
4
0
0
)
|
J
|