122 1. Higher order Fourier analysis
Shifting hi,j by hi for some h1,h2 and reaveraging, we can rewrite this as
|Eh1,1,...,h2,k En,h1,h2 (ν(n) 1)Fh1 (n + h1)Fh2 (n + h2)Fh1+h2 (n + h1 + h2)|
where hi := (hi,1,...,hi,k) for i = 1, 2 and
F(v1,...,vk)(n) :=
k
j=1
Fj(n + vj).
The inner expectation is the Gowers inner product of ν 1, Fh1 , Fh2 , and
Fh1+h2 . Using the linear forms condition we may assume that
ν 1
U
2(Z/NZ)
= o(1)
and so it will suffice by the Cauchy-Schwarz-Gowers inequality, followed by
the older inequality, to show that
Eh1,1,...,h2,k Fh1
4
U 2(Z/NZ)
K
1
and similarly for h2 and h1 + h2.
We just prove the claim for h1, as the other two cases are similar. We
expand the left-hand side as
|En,a,b,h1,...,hk
k
j=1
Fj(n + hj)Fj(n + hj + a)Fj(n + hj + b)Fj(n + hj + a + b)|
which we can upper bound by
|En,a,b,h1,...,hk
k
j=1
ν(n + hj)ν(n + hj + a)ν(n + hj + b)ν(n + hj + a + b)|.
We can factorise this as
Ea,b|Enν(n)ν(n + a)ν(n + b)ν(n + a +
b)|k.
Using (1.61), we see that the inner expectation is Ok(1) as long as 0,a,b,a+b
are distinct; in all other cases they are O(N
o(1)),
by (1.60). Combining these
two cases we obtain the claim.
Exercise 1.7.4. Show that (1.59) also follows from a finite number of linear
forms conditions and (1.61), if the Fj are only assumed to be bounded in
magnitude by ν + 1 rather than ν, and the right-hand side of (1.61) is
weakened to

1≤ij≤m
τ(hi hj), where τ : Z/NZ
R+
is a function
obeying the moment bounds
En∈Z/NZτ(n)q
q
1 for each q 1.
The above machinery was geared to getting Roth-type lower bounds on
Λ(f, f, f); but it also can be used to give more precise asymptotics:
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