## 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:
• Gowers uniformity norms and arithmetic progressions in sets of positive density
• Freimann's theorem: structure of sets with small sumsets
• Randomness of almost primes: the linear forms condition of Green-Tao
• The Green-Tao theorem: arithmetic progressions in the primes
• Bounded gaps between the primes

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).

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

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