1.7. Linear equations in primes 113
A model example to keep in mind of a candidate for a Roth-pseudoran-
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
older’s inequality, and the Plancherel identity.
This suggests a strategy to establish the Roth-pseudorandomness 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 Roth-pseudorandomness). 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 Roth-pseudorandom.
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)
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