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 H¨ 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: