Additive Combinatorics and Number Theory


Math 8440 - Spring 2020




Textbook:      There is no official textbook for the course but I will use the Lecture notes and papers listed below.
                      

Description:  Description:  The aim is to give an introduction to some fundamental results of additive combinatorics and number theory.
More specifically I plan to discuss the following topics:




I. Freiman's theorem and Four term arithmetic progressions


We introduce the tools from additive number theory needed for the Fourier analytic proof  (due to Gowers) of Szemeredi's theorem for 4-term arithmetic progressions (AP's).
Notes: K. Soundararjan: Additive Combinatorics

Lecture 1      Lecture 2      Lecture 3      Lecture 4     Lecture 5       Lecture 6        Lecture 7       Lecture 8      Lecture 9   Lecture 10



II. Hypergraph Regularity and Arithmetic Progressions in the Primes

We introduce the (weak) hypergraph regularity lemma and use it to prove the Green-Tao theorem on long arithmetic progressions in the primes. The missing facts from analytic number theory i.e. the so-called linear forms conditions will be discussed afterwards.
Notes: Tao: A variant of the Hypergraph Removal Lemma
           Conlon-Fox-Zhao: A relative Szemredi Theorem

Lecture 9      Lecture 10      Lecture 11      Lecture 12     Lecture 13       Lecture 14    Lecture 15        Lecture 16


 
III. Sieve theory I. - The linear forms condition

We introduce a variant of the sieve methods due to Goldston-Yildirim to finish the proof the Green-Tao theorem.
Notes: Green-Tao: The primes contain arbitrary long arithmetic progressions

Lecture 17      Lecture 18      Lecture 19      Lecture 20    







                                                                                                         IV. Sieve theory II. - Bounded gaps between the primes

       
We discuss the Maynard-Tao theorem on bounded gaps between the primes.
         Notes: A. Granville: Primes in intervals of bounded length

         Lecture 21      Lecture 22      Lecture 23      Lecture 24     Lecture 25       Lecture 26