1.7. Linear equations in primes 123
Exercise 1.7.5. Suppose that ν obeys the hypotheses of Lemma 1.7.3 (with
the
u2
norm). Let f : Z/NZ R obey the pointwise bound 0 f 1
and has mean En∈Z/NZf(n) = δ; suppose also that one has the pseudo-
randomness bound supξ∈Z/NZ\0 |
ˆ(ξ)|
f = oN→∞(1). Show that Λ(f, f, f) =
δ3
+ oN→∞(1).
Exercise 1.7.6. Repeat the previous exercise, but with the
u2
norm re-
placed by the U
2
norm.
Informally, the above exercises show that if one wants to obtain asymp-
totics for three-term progressions in a set A which has positive relative
density with respect to a Roth-pseudorandom measure, then it suffices to
obtain a non-trivial bound on the exponential sums

n∈A
e(ξn) for non-zero
frequencies ξ.
For longer progressions, one uses higher-order Gowers norms, and a sim-
ilar argument (using the inverse conjecture for the Gowers norms) shows
(roughly speaking) that to obtain asymptotics for k-term progressions (or
more generally, linear patterns of complexity k−1) in a U
k−1-pseudorandom
measure (by which we mean that the analogue of Lemma 1.7.3 for the U
k−1
norm holds) then it suffices to obtain a non-trivial bound on sums of the
form

n∈A
F (g(n)Γ) for k−2-step nilsequences F (g(n)Γ). See [GrTa2010]
for further discussion.
1.7.2. A brief discussion of sieve theory. In order to apply the above
theory to find patterns in the primes, we need to build a measure ν with
respect to which the primes have a positive density, and for which one can
verify conditions such as the Fourier pseudorandomness condition (1.55),
the restriction estimate (1.54), linear forms conditions, and the correlation
condition (1.61).
There is an initial problem with this, namely that the primes themselves
are not uniformly distributed with respect to small moduli. For instance, all
primes are coprime to two (with one exception). In contrast, any measure ν
obeying the Fourier pseudorandomness condition (1.55) (which is implied by
the condition ν 1
U 2
= o(1), which would follow in turn from the linear
forms condition), must be evenly distributed in both odd and even residue
classes up to o(1) errors; this forces the density of the primes in ν to be at
most 1/2 + o(1). A similar argument using all the prime moduli less than
some parameter w shows in fact that the density of primes in ν is at most
pw
(1
1
p
) + oN→∞;w(1). Since

p
1
p
diverges to +∞,
p
(1
1
p
) diverges
to zero, and so we see that the primes cannot in fact have a positive density
with respect to any pseudorandom measure.
This difficulty can be overcome by a simple affine change of variables
known as the W -trick, where we replace the primes P = {2, 3, 5,... } by the
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