Traditionally, Fourier analysis has been focused on the analysis of functions
in terms of linear phase functions such as the sequence n e(αn) :=
In recent years, though, applications have arisen—particularly in connection
with problems involving linear patterns such as arithmetic progressions—in
which it has been necessary to go beyond the linear phases, replacing them
to higher order functions such as quadratic phases n e(αn2). This has
given rise to the subject of quadratic Fourier analysis and, more generally,
to higher order Fourier analysis.
The classical results of Weyl on the equidistribution of polynomials (and
their generalisations to other orbits on homogeneous spaces) can be inter-
preted through this perspective as foundational results in this subject. How-
ever, the modern theory of higher order Fourier analysis is very recent in-
deed (and still incomplete to some extent), beginning with the breakthrough
work of Gowers [Go1998], [Go2001] and also heavily influenced by paral-
lel work in ergodic theory, in particular, the seminal work of Host and Kra
[HoKr2005]. This area was also quickly seen to have much in common with
areas of theoretical computer science related to polynomiality testing, and in
joint work with Ben Green and Tamar Ziegler [GrTa2010], [GrTa2008c],
[GrTaZi2010b], applications of this theory were given to asymptotics for
various linear patterns in the prime numbers.
There are already several surveys or texts in the literature (e.g.
[Gr2007], [Kr2006], [Kr2007], [Ho2006], [Ta2007], [TaVu2006]) that
seek to cover some aspects of these developments. In this text (based on a
topics graduate course I taught in the spring of 2010), I attempt to give a
broad tour of this nascent field. This text is not intended to directly substi-
tute for the core papers on the subject (many of which are quite technical
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