1.7. Linear equations in primes 121

for all k ≥ 1 and all F1,...,Fk bounded in magnitude by ν + 1.

We summarise this discussion as follows:

Theorem 1.7.7 (Dense model theorem). If (1.58), (1.59) hold, then the

approximation in U

2

hypothesis in Lemma 1.7.3 holds.

There is nothing too special about the U

2

norm here; one could work

with higher Gowers norms, or indeed with any other norm for which one has

a reasonably explicit description of the dual.

The abstract version of the theorem was first (implicitly) proven in

[GrTa2008b], and made more explicit in [TaZi2008]. The methods there

were different (and somewhat more complicated). To prove approximation,

the basic idea was to write g = E(f|B) for some carefully chosen σ-algebra

B (built out of dual functions that correlated with things like the residual

f − E(f|B)). This automatically gave the non-negativity of g; the upper

bound on g came from the bound E(f|B) ≤ E(ν|B), with the latter expres-

sion then being bounded by the Weierstrass approximation theorem and

(1.59).

To summarise, in order to establish the Roth-pseudorandomness of a

measure μ, we have at least two options. The first (which relies on Fourier

analysis, and is thus largely restricted to complexity 1 problems) is to estab-

lish the Fourier pseudorandomness bound (1.55) and the restriction estimate

(1.54). The other (which does not require Fourier analysis) is to establish a

finite number of linear forms conditions, as well as the estimate (1.59).

Next, we informally sketch how one can deduce (1.59) from a finite

number of linear forms conditions, as well as a crude estimate

(1.60) ν = O(N

o(1))

and a condition known as the correlation condition. At the cost of oversim-

plifying slightly, we express this condition as the assertion that

(1.61) En∈Z/NZν(n + h1) . . . ν(n + hk)

k

1

whenever h1,...,hk ∈ Z/NZ are distinct, thus the k-point correlation func-

tion of ν is bounded for each k. For the number-theoretic applications, one

needs to replace the 1 on the right-hand side by a more complicated expres-

sion, but we will defer this technicality to the exercises. We remark that for

each fixed k, the correlation condition would be implied by the linear forms

condition, but it is important that we can make k arbitrarily large.

For simplicity of notation we assume that the Fj are bounded in mag-

nitude by ν rather than by ν + 1. We begin by expanding out (1.59) as

|En,h1,1,...,h2,k (ν(n) − 1)

k

j=1

Fj(n + h1,j)Fj(n + h2,j)Fj(n + h1,j + h2,j)|.