26 1. Higher order Fourier analysis
(ii) (Pequi is equidistributed) There exists a standard subtorus T of
Td,
such that Pequi takes values in T and is totally equidistributed on
[N]m
in this torus.
(iii) (Prat is rational) The coefficients αi,rat of Prat are standard rational
elements of Td. In particular, there is a standard positive integer
q such that qPrat = 0 and Prat is periodic with period q.
Proof. This is implicitly in [GrTa2011]; the result is phrased using the
language of single-scale equidistribution, but this easily implies the ultralimit
version.
1.2. Roth’s theorem
We now give a basic application of Fourier analysis to the problem of count-
ing additive patterns in sets, namely the following famous theorem of Roth
[Ro1953]:
Theorem 1.2.1 (Roth’s theorem). Let A be a subset of the integers Z whose
upper density
δ(A) := lim sup
N→∞
|A [−N, N]|
2N + 1
is positive. Then A contains infinitely many arithmetic progressions a, a +
r, a + 2r of length three, with a Z and r 0.
This is the first non-trivial case of Szemer´ edi’s theorem [Sz1975], which
is the same assertion but with length three arithmetic progressions replaced
by progressions of length k for any k.
As it turns out, one can prove Roth’s theorem by an application of linear
Fourier analysis by comparing the set A (or more precisely, the indicator
function 1A of that set, or of pieces of that set) against linear characters
n e(αn) for various frequencies α R/Z. There are two extreme cases to
consider (which are model examples of a more general dichotomy between
structure and randomness, as discussed in [Ta2008]). One is when A is
aligned almost completely with one of these linear characters, for instance,
by being a Bohr set of the form
{n Z : αn θ
R/Z
ε}
or, more generally, of the form
{n Z : αn U}
for some multi-dimensional frequency α
Td
and some open set U. In
this case, arithmetic progressions can be located using the equidistribution
theory from Section 1.1. At the other extreme, one has Fourier-uniform or
Fourier-pseudorandom sets, whose correlation with any linear character is
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