Summer School: Additive Combinatorics

The summer school is dedicated to graduate students and young researchers, and aims to give an introduction to recent techniques and topics of additive combinatorics. The lectures of the summer school will concentrate on recent developments of the polynomial method, some combinatorial methods of additive combinatorics, and the introduction of Fourier analytic techniques connected to them. The main topics will be presented by top researchers of the area.

Lecture sessions

Christian Elsholtz: Large structures without certain patterns

Abstract: We study point sets in (Z/mZ)^n, when m is fixed and n tends to infinity, especially with the constraint that they do not contain any arithmetic progression of length k, or that there are no three collinear points (so called caps). In these lectures the emphasis is on different methods to achieve lower bounds.

In the case of caps (over prime fields) we will similarly improve previous lower bound constructions and achieve in F_5 a "density" which supersedes the one of the most frequently studied case F_3. We also discuss results on progression-free sets in the integers.

In another part we intend to discuss improvements on the longest gaps between integers which are sums of two squares.

Péter Pál Pach: Polynomial method

Abstract: During the last few decades the polynomial method has developed into a powerful tool in combinatorics. It involves encoding combinatorial problems to the nonvanishing of some polynomials, and then investigating the resulting polynomial question. Usually a common step is to show that a low-degree (non-zero) polynomial can not vanish on a certain set (which is sufficiently large). We will start with exploring several applications of the Combinatorial Nullstellensatz, including classical problems like the Cauchy-Davenport theorem and other questions. The solution of the finite field Kakeya conjecture is also going to be discussed. Finally, we will study the CLP polynomial method which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like Z_4^n and Z_3^n are exponentially small (compared to the size of the group). We will also look at the slice rank formulation of the technique and explore some further applications, for instance, the solution of the Erdős-Szeméredi sunflower conjecture.

Sean Prendiville: Nonlinear Fourier analysis and nonlinear progressions

Abstract: We introduce some of the ideas behind higher order Fourier analysis, in particular the use of this theory to locate patterns in sets of arithmetic interest (e.g. dense sets or primes). As an example application, we detail how to count certain polynomial progressions in dense subsets of a finite field - an application which exhibits the essential features of the theory in an elementary manner.

Julia Wolf: Linear methods in additive combinatorics

Abstract: In Lecture 1, we introduce Fourier analysis on finite abelian groups. As an application, we shall prove Bogolyubov’s lemma, and then move on to illustrating the limitations of the technique. In Lecture 2, we shall cover a probabilistic technique due to Croot and Sisask, which allows us to detect ‘almost periodicity’ in certain subsets of finite abelian groups. As an application, we give a simple proof of a Roth-type theorem. In Lecture 3, we intend to give a sketch of the recent breakthrough result of Kelley and Meka on the size of 3-term progression-free sets in the integers, carried out in the finite-field setting.

Registration 

The registration for the summer school is open till October 15, 2023.

                                           Link to Registration Form

The registration is not automatic, after filling the form we have to approve it. We will send confirmation that the registration is successful.

If you are an MSc or PhD student, we ask in the registration form the name and email address of advisor / mentor for asking a brief reference email. 

If you are a postdoc, please send your CV by email to faap.semester@renyi.hu, with the subject "[Summer school: Additive Combinatorics] CV - APPLICANT'S FULL NAME"

Fund

We can cover the accommodation for a limited number of participants. Please, indicate on the registration form whether you need such financial support. In case you do, please, do not arrange your accommodation yet, as we may do it for you. We are going to make the decision of support before the end of October.