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 obius function)
with various objects that arise from higher order Fourier analysis,
such as nilsequences.
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