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 Eν = 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

.