Math
8440

Textbook:
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.

I'll post below some links/notes relevant to the topics of the course.

Description:
The aim of
the course is to give an introduction to recent developments in
combinatorial number theory related
to

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 the primes.

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 the primes.

Basic
Notes
Supplementary
Notes
Exercises/Problems

1.1
Colorings.
1.1
Van der Waerden's Theorem
1.1
Combinatorics

1.2 The combinatorial proof. (Chapter 3) 1.2 The "original" combinatorial proof

1.3 The Balog-Szemeredi Theorem. 1.4* Background: Finite Fourier series

1.4 The Fourier-analytic proof. 1.4 The "short" Fourier analytic proof

1.5 Ergodic proof. 1.5 Ergodic approach (chapters 1-5)

1.2 The combinatorial proof. (Chapter 3) 1.2 The "original" combinatorial proof

1.3 The Balog-Szemeredi Theorem. 1.4* Background: Finite Fourier series

1.4 The Fourier-analytic proof. 1.4 The "short" Fourier analytic proof

1.5 Ergodic proof. 1.5 Ergodic approach (chapters 1-5)

II.
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
(AP's).

2.2 The Balog-Szemeredi Theorem.

2.2 The Balog-Szemeredi Theorem.

III.
Four term arithmetic progressions

If time permits we give a little bit of introduction to the methods to show the existence of AP's among the primes. Here are some extensive notes/papers.

4.1 Vinogradov's 3-primes Theorem.

4.2 Green's Theorem on 3-term AP's in subsets of primes.

4.3 Green-Tao Theorem on k-term AP's among the primes. (an introduction)

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).

3.1 4-term AP's

3.2 k-term AP's

3.1 4-term AP's

3.2 k-term AP's

IV.
Progressions among the primes

If time permits we give a little bit of introduction to the methods to show the existence of AP's among the primes. Here are some extensive notes/papers.

4.1 Vinogradov's 3-primes Theorem.

4.2 Green's Theorem on 3-term AP's in subsets of primes.

4.3 Green-Tao Theorem on k-term AP's among the primes. (an introduction)