1.7. Linear equations in primes 119

precisely, the Hahn-Banach theorem) that yields the same result, due inde-

pendently to Gowers [Go2010] and to Reingold-Trevisan-Tulsiani-Vadhan

[ReTrTuVa2008]. This approach also has the benefit of giving somewhat

sharper quantitative refinements.

The first task is to represent the U

2

norm in a dual formulation. The

starting point is that the expression

f

4

U

2(Z/NZ)

= En,a,bf(n)f(n + a)f(n + b)f(n + a + b)

whenever f : Z/NZ → R, can be rewritten as

f

4

U

2(Z/NZ)

= f, Df

L2(Z/NZ)

where the dual function Df = D2f : Z/NZ → R is defined by

Df(n) := Ea,bf(n + a)f(n + b)f(n + a + b).

Define a basic anti-uniform function to be any function of the form DF ,

where F : Z/NZ → R obeys the pointwise bound |F | ≤ ν + 1. To obtain

the approximation property, it thus suﬃces to show that for every ε 0, for

N suﬃciently large depending on ε, and any f : Z/NZ → R with 0 ≤ f ≤ ν,

one can decompose f = f1 + f2 where 0 ≤ f1 ≤ 1 and |f2, DF | ≤

ε4

for

all basic anti-uniform functions DF . Indeed, if one sets F := f2, the latter

bound gives f2

4

U

2(Z/NZ)

≤

ε4,

and the desired decomposition follows.

In order to apply the Hahn-Banach theorem properly, it is convenient

to symmetrise and convexify the space of basic anti-uniform functions. De-

fine an averaged anti-uniform function to be any convex combination of

basic anti-uniform functions and their negations, and denote the space of

all such averaged anti-uniform functions as B. Thus B is a compact convex

symmetric subset of the finite-dimensional real vector space

L2(Z/NZ)

that

contains a neighbourhood of the origin; equivalently, it defines a norm on

L2(Z/NZ).

Our task is then to show (for fixed ε and large N) that for any

f ∈ Z/NZ → R with 0 ≤ f ≤ ν + 1, the sets

U := {(f1,f2) ∈

L2(Z/NZ)

∩

L2(Z/NZ)

: f1 + f2 = f}

and

V := {(f1,f2) ∈

L2(Z/NZ)

∩

L2(Z/NZ)

: 0 ≤ f1 ≤ 1; f2,φ ≤

ε4

for all φ ∈ B}

have non-empty intersection.

The point of phrasing things this way is that U and V are both closed

convex subsets of the finite-dimensional vector space

L2(Z/NZ)∩L2(Z/NZ),

and so the Hahn-Banach theorem is applicable23. Indeed, suppose that there

23One could also use closely related results, such as the Farkas lemma; see [Ta2008, §1.16]

for more discussion.