COURSE DESCRIPTION
"Additive combinatorics" is a very active subfield of mathematical research, which combines combinatorics, number theory, and harmonic analysis. Since it is difficult to define additive combinatorics in a clear short way, we instead present a few examples of main problems in this field:
 How large can a subset of \(\{1,2,3,...,n\}\) be without containing a 3term arithmetic progression in it; equivalently, showing that every subset that is sufficiently dense must contain such a progression (Roth's theorem)
 What subsets \(A \in \{1,2,3,...,n\}\) have a small sum set \(A+A=\{a+a' : a,a' \in A\}\) (the polynomial FreimanRuzsa conjecture).
 Are there subsets \(A \in \{1,2,3,...,n\} \) have both a small sum set \(A+A=\{a+a' : a,a' \in A\}\) and a small product set \(AA=\{aa' : a,a' \in A\}\) (the Erdős–Szemerédi conjecture).
Detailed lecture notes would be uploaded before each class (for example, see last year's polynomial method lecture notes). Other sources for additive combinatorics material are the book of Tao and Vu, lecture notes by Gowers, Ruzsa, Green, Soundararajan, and others, and surveys of Shachar Lovett (one and two).
Each course participant is expected to read one related paper and to present it in class. This webpage will contain a list of possible papers. If more than a few students enroll, some of the undergrad participants would instead submit 3 homework assignments. Students are also expected to attend most of the classes.
PREREQUISITES
The class requires a basic mathematical understanding, such as basic familiarity with combinatorics, probability, and groups. For other topics, such as the Fourier transform, we will go over the basics in class.
INSTRUCTORS
Adam Sheffer
HARRY BATEMAN INSTRUCTOR IN MATHEMATICS
276 Sloan, 6263954347,
adamsh@caltech.edu
TA's
 Pooya Vahidi, pvahidif@caltech.edu.LECTURE NOTES
Date  Description  Notes 

  Preliminaries  Chapter 0 
March 28th  Sets with small doubling  Chapter 1 
March 30th  Ruzsa's inequality and Plünnecke's inequality  Chapter 1 
April 1st  Freiman's theorem with bounded order and intro to the sumproduct problem  Chapter 2 
April 4th  Elekes' sumproduct proof and related results  Chapter 2 
April 6th  Solymosi's sumproduct bound and related results  Chapter 2 
April 8th  One final sumproduct variant and introduction to BalogSzemerediGowers  Chapter 3 
April 11th  Schoen's BSG proof  Chapter 3 
April 13th  BSG variants and beginning the proof of Sudakov, Szemerédi, and Vu  Chapter 3 
April 15th  Introduction to progressions in dense sets, Behrend's construction  Chapter 4 
April 18th  Basics of the Fourier transform over finite fields  Chapter 4 
April 20th  Meshulam's theorem  Chapter 4 
April 22nd  Roth's theorem  Chapter 4 
April 25th  Completing Roth and beginning Sanders' quasipolynomial FreimanRuzsa  Chapter 5 
April 27th  The BogolyubovRuzsa lemma  Chapter 5 
April 29th  No class today   
May 2nd  Completing Sanders' quasipolynomial FreimanRuzsa (almost)  Chapter 5 
May 4th  Rich translations via convolutions  Chapter 5 
May 6th  Gaurav Sinha presents the paper "A new bound on partial sumsets and differencesets, and applications to the Kakeya conjecture" by Katz and Tao   
May 9th  No class today.   
May 11th  No class today.   
May 13th  Ryan Yoo presents the paper "Integer sets containing no arithmetic progressions" by HeathBrown.   
May 16th  Laura Shou presents the paper "A trilinear maximal inequality via arithmetic combinatorics" by Demeter, Tao, and Thiele.   
May 18th  Cosmin Pohoata presents the paper "Progressionfree sets in \(Z^n_4\) are exponentially small" by Croot, Lev, and Pach.   
May 19th 10:30am, SLN 151 
(Makeup class) Third moment energy.  Chapter 6 
May 20th  Pooya Vahidi presents the paper "Near optimal bounds in Freiman's theorem" by Schoen.   
May 23rd  Bella Guo presents the paper "Embeddability properties of difference sets" by Di Nasso   
May 25th  Karlming Chen presents the paper "A probabilistic technique for finding almostperiods of convolutions" by Croot and Sisask   
May 27th  Angad Singh presents the paper "New counterexamples for sumsdifferences" by Lemm.   
June 3rd  (Makeup class) Alex McDonald presents the paper "A lowenergy decomposition theorem" by Balog and Wooley.   
READING
The following is a list of papers for students who give a talk in class. You may also suggest papers that are not on this list.
 New sumproduct type estimates over finite fields by RocheNewton, Rudnev, and Shkredov. This is the current best sumproduct bound over finite fields. It has a short proof that is based on an incidence bound, so if you present it you should also discuss this incidence bound.
 Higher moments of convolutions by Schoen and Shkredov. This paper uses several techniques to study a generalization of the concept of energy. Since it is too long for one class, you can either present a part of it, or two students can do it together during two class.
Near optimal bounds in Freiman’s theorem by Schoen. The title is self explanatory. In class we will prove a stronger bound for the special case of \({\mathbb F}^n_2\), while this paper focuses on \({\mathbb Z}\).A new bound on partial sumsets and differencesets, and applications to the Kakeya conjecture by Katz and Tao. Obtaining a bound for the Kakeya problem by examining sums and difference of sets. If you talk about this paper, you should also describe the Kakeya problem and the connection to it. On additive doubling and energy by Katz and Koester. Studying a property of sets that is called smoothing, and connecting it to doubling.
 If you are interested in connections between additive combinatorics and theoretical computer science, you can find many relevant papers in Lovett's survey.

In class we will prove the original bound for Roth's theorem. If you would like a more analysisoriented paper, you are welcome to present any later bound for the problem. For a list of these bound, see for example here (the latest bound by Bloom is probably too much for one class). A lowenergy decomposition theorem by Balog and Wooley. Proving that "any finite set of real numbers can be split into two parts, one part being highly nonadditive and the other highly nonmultiplicative". Further improvements to incidence and Becktype bounds over prime finite fields by Jones. Using a sumproduct bound to obtain incidence bound in finite fields.
A probabilistic technique for finding almostperiods of convolutions by Croot and Sisask. Introduces a powerful tool concerning convolutions of sets. We will introduce a special case of this result in class, and then you will talk about the general case.