Proposition 4.15.
Assuming that:
f
:
𝔽
5
n
→
ℂ
∥
f
∥
∞
≤
1
∥
f
∥
U
3
(
G
)
≥
δ
|
𝔽
5
n
|
=
Ω
δ
(
1
)
Then
there exists
S
⊆
𝔽
5
n
with
|
S
|
=
Ω
δ
(
|
𝔽
5
n
|
)
and a function
ϕ
:
S
→
𝔽
5
n
^
such that
(i)
|
Δ
h
f
^
(
ϕ
(
h
)
)
|
=
Ω
δ
(
1
)
;
(ii)
There are at least
Ω
δ
(
|
𝔽
5
n
|
3
)
quadruples
(
s
1
,
s
2
,
s
3
,
s
4
)
∈
S
4
such that
s
1
+
s
2
=
s
3
+
s
4
and
ϕ
(
s
1
)
+
ϕ
(
s
2
)
+
ϕ
(
s
4
)
.