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
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
and so it will suﬃce by the Cauchy-Schwarz-Gowers inequality, followed by
the H¨ older inequality, to show that
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
Fj(n + hj)Fj(n + hj + a)Fj(n + hj + b)Fj(n + hj + a + b)|
which we can upper bound by
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 +
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
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
τ(hi − hj), where τ : Z/NZ →
is a function
obeying the moment bounds
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: