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