Combinatorial Number Theory
Additive Combinatorics by Terence Tao and Van Vu.
It is not at all necessary to by the book.
I'll post below some
links/notes relevant to the topics of the course.
The aim of
the course is to give an introduction to recent developments in
combinatorial number theory related
arithmetic progressions in sets of positive density of the integers,
and among the primes.
The course will consist of roughly three
parts, and if time permits go a little bit into similar results among
I. Roth' Theorem.
basic approaches to proving Roth' theorem and some variations of it,
which shows the existence of 3-term arithmetic progressions in dense
sets of the integers.
Freiman's theorem and the circle method
We discuss the tools from combinatorics
and number theory needed for the Fourier analytic proof (due to
Gowers) of Szemeredi's theorem for 4-term arithmetic progressions
The Balog-Szemeredi Theorem.
Four term arithmetic progressions
Finally we discuss Gowers
proof for 4-term AP's. Also the link for his proof of the general
case for k-term AP's is posted here (though we will not discuss the
general case in detail).
Progressions among the primes