1.7. Linear equations in primes 113
A model example to keep in mind of a candidate for a Rothpseudoran
dom measure is a random sparse measure of some small density 0 p 1,
in which each ν(n) is an independent random variable that equals 1/p with
probability p and 0 otherwise. The case p = 1/ log N can be thought of as
a crude model for the primes (cf. Cram´ er’s random model for the primes).
Recall that the form Λ(f, g, h) is controlled by the U
2
norm in the sense
that one has the inequality
Λ(f, g, h) ≤ f
U 2(Z/NZ)
whenever f, g, h: Z/NZ → C are bounded in magnitude by 1, and similarly
for permutations. Actually one has the slightly more precise inequality
Λ(f, g, h) ≤ f
u2(Z/NZ)
where
f
u2(Z/NZ)
:= sup
ξ∈Z/NZ

ˆ(ξ)
f
as can easily be seen from the identity
(1.53) Λ(f, g, h) =
ξ∈Z/NZ
ˆ(ξ)ˆ(−2ξ)ˆ(ξ),
f g h
H¨ older’s inequality, and the Plancherel identity.
This suggests a strategy to establish the Rothpseudorandomness of a
measure by showing that functions f dominated by that measure can be
approximated in
u2
norm by functions dominated instead by the uniform
measure 1. Indeed, we have
Lemma 1.7.3 (Criterion for Rothpseudorandomness). Suppose we have a
measure ν with the following properties:
(i) (Control by
u2)
For any f, g, h: Z/NZ → R with the pointwise
bound f, g, h ≤ ν + 1, one has Λ(f, g, h) ≤ α( f u2(Z/NZ)) +
oN→∞(1), where α:
R+
→
R+
is a function with α(x) → 0 as
x → 0, and similarly for permutations.
(ii) (Approximation in
u2)
For any f : Z/NZ → R with the pointwise
bound 0 ≤ f ≤ ν, and any ε 0, there exists g : Z/NZ → R
with the pointwise bound 0 ≤ g ≤ 1 + on→∞;ε(1) such that f −
g
u2(Z/NZ)
≤ ε + on→∞;ε(1).
Then ν is Rothpseudorandom.
Proof. Let f : Z/NZ → C be such that 0 ≤ f ≤ ν and En∈Z/NZf ≥ δ. Let
ε 0 be a small number to be chosen later. We then use the decomposition
to split f = g + (f − g) with the above stated properties. Since
En∈Z/NZf(n) − g(n) ≤ f − g
u2(Z/NZ)
≤ ε + on→∞;ε(1)