and lengthy), but focuses instead on basic foundational and preparatory ma-
terial, and on the simplest illustrative examples of key results, and should
thus hopefully serve as a companion to the existing literature on the sub-
ject. In accordance with this complementary intention of this text, we also
present certain approaches to the material that is not explicitly present in
the literature, such as the abstract approach to Gowers-type norms (Section
2.2) or the ultrafilter approach to equidistribution (Section 1.1.3).
There is, however, one important omission in this text that should be
pointed out. In order to keep the material here focused, self-contained,
and of a reasonable length (in particular, of a length that can be mostly
covered in a single graduate course), I have focused on the combinatorial
aspects of higher order Fourier analysis, and only very briefly touched upon
the equally significant ergodic theory side of the subject. In particular, the
breakthrough work of Host and Kra [HoKr2005], establishing an ergodic-
theoretic precursor to the inverse conjecture for the Gowers norms, is not
discussed in detail here; nor is the very recent work of Szegedy [Sz2009],
[Sz2009b], [Sz2010], [Sz2010b] and Camarena-Szegedy [CaSz2010] in
which the Host-Kra machinery is adapted to the combinatorial setting.
However, some of the foundational material for these papers, such as the
ultralimit approach to equidistribution and structural decomposition, or the
analysis of parallelopipeds on nilmanifolds, is covered in this text.
This text presumes a graduate-level familiarity with basic real analysis
and measure theory, such as is covered in [Ta2011], [Ta2010], particularly
with regard to the “soft” or “qualitative” side of the subject.
The core of the text is Chapter 1, which comprises the main lecture
material. The material in Chapter 2 is optional to these lectures, except for
the ultrafilter material in Section 2.1 which would be needed to some extent
in order to facilitate the ultralimit analysis in Chapter 1. However, it is
possible to omit the portions of the text involving ultrafilters and still be able
to cover most of the material (though from a narrower set of perspectives).
I am greatly indebted to my students of the course on which this text was
based, as well as many further commenters on my blog, including Sungjin
Kim, William Meyerson, Joel Moreira, Thomas Sauvaget, Siming Tu, and
Mads Sørensen. These comments, as well as the original lecture notes for
this course, can be viewed online at
Thanks also to Ben Green for suggestions. The author is supported by
a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and
by the NSF Waterman award.