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.
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