Combinatorial Number Theory


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.

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.


I. Roth' Theorem.

We discuss  three 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.

           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)
             


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.                                     
2.3 Freiman's Theorem.
2.4 Weyl's lemma and Sarkozy's Theorem          
2.5 Sums of squares and the "circle method".


III. 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).

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)