110 1. Higher order Fourier analysis

To illustrate the main ideas, we will focus on the following result of

Green [Gr2005]:

Theorem 1.7.1 (Roth’s theorem in the primes [Gr2005]). Let A ⊂ P be

a subset of primes whose upper density lim supN→∞ |A ∩ [N]|/|P ∩ [N]| is

positive. Then A contains infinitely many arithmetic progressions of length

three.

This should be compared with Roth’s theorem in the integers (Section

1.2), which is the same statement but with the primes P replaced by the

integers Z (or natural numbers N). Indeed, Roth’s theorem for the primes is

proven by transferring Roth’s theorem for the integers to the prime setting;

the latter theorem is used as a “black box”. The key diﬃculty here in

performing this transference is that the primes have zero density inside the

integers; indeed, from the prime number theorem we have |P ∩ [N]| = (1 +

o(1))

N

log N

= o(N).

There are a number of generalisations of this transference technique. In

[GrTa2008b], the above theorem was extended to progressions of longer

length (thus transferring Szemer´ edi’s theorem to the primes). In a series

of papers [GrTa2010, GrTa2011, GrTa2008c, GrTaZi2010b], related

methods are also used to obtain an asymptotic for the number of solutions

in the primes to any system of linear equations of bounded complexity. This

latter result uses the full power of higher order Fourier analysis, in particular,

relying heavily on the inverse conjecture for the Gowers norms; in contrast,

Roth’s theorem and Szemer´ edi’s theorem in the primes are “softer” results

that do not need this conjecture.

To transfer results from the integers to the primes, there are three basic

steps:

(i) A general transference principle, that transfers certain types of ad-

ditive combinatorial results from dense subsets of the integers to

dense subsets of a suitably “pseudorandom set” of integers (or more

precisely, to the integers weighted by a suitably “pseudorandom

measure”).

(ii) An application of sieve theory to show that the primes (or more

precisely, an aﬃne modification of the primes) lie inside a suitably

pseudorandom set of integers (or more precisely, have significant

mass with respect to a suitably pseudorandom measure).

(iii) If one is seeking asymptotics for patterns in the primes, and not

simply lower bounds, one also needs to control correlations between

the primes (or proxies for the primes, such as the M¨ obius function)

with various objects that arise from higher order Fourier analysis,

such as nilsequences.