1.7. Linear equations in primes 117
To summarise, we have the following result, which essentially appears in
[GrTa2006]:
Theorem 1.7.5 (Criterion for Roth-pseudorandomness). Let ν be a mea-
sure obeying the Fourier-pseudorandomness assumption (1.55) and the re-
striction estimate (1.54) for some 2 q 3. Then ν is Roth-pseudorandom.
This turns out to be a fairly tractable criterion for establishing the Roth-
pseudorandomness of various measures, which in turn can be used to detect
progressions of length three (and related patterns) on various sparse sets,
such as the primes; see the next section.
The above arguments to establish Roth-pseudorandomness relied heavily
on linear Fourier analysis. Now we give an alternate approach that avoids
Fourier analysis entirely; it is less efficient and a bit messier, but will extend
in a fairly straightforward (but notationally intensive) manner to higher
order patterns. To do this, we replace the
u2
norm in Lemma 1.7.3 with
the U
2
norm, so now we have to verify a control by U
2
hypothesis and an
approximation by U
2
hypothesis.
We begin with the control by U
2
hypothesis. Instead of Fourier analysis,
we will rely solely on the Cauchy-Schwarz inequality, using a weighted ver-
sion of the arguments from Section 1.3 that first appeared in [GrTa2008b].
We wish to control the expression
Λ(f, g, h) = En,r∈Z/NZf(n)g(n + r)h(n + 2r)
where f, g, h are bounded in magnitude by ν + 1. For simplicity we will just
assume that f, g, h are bounded in magnitude by ν; the more general case
is similar but a little bit messier. For brevity we will also omit the domain
Z/NZ in the averages, and also abbreviate oN→∞(1) as o(1). We make the
change of variables (n, r) = (b + 2c, −a b c) to write this expression as
Ea,b,cf(b + 2c)g(a c)h(−2a b),
the point being that each term involves only two of the three variables a, b, c.
We can pointwise bound h by ν and estimate the above expression in
magnitude by
Ea,b|Ecf(b + 2c)g(a c)|ν(−2a b).
Since = 1 + o(1), we can use Cauchy-Schwarz and bound this by
(1 + o(1))(Ea,b|Ecf(b + 2c)g(a
c)|2ν(−2a

b))1/2
which we rewrite as
(1 + o(1))
(
Ea,b,c,c f(b + 2c)f(b + 2c )g(a c)g(a c )ν(−2a b)
)1/2
.
We now bound g by ν, to obtain
(1 + o(1))
(
Ea,c,c ν(a c)ν(a c )|Ebf(b + 2c)f(b + 2c )ν(−2a b)|
)1/2
.
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