Course description

Additive combinatorics is relatively new area of mathematics that grew out of combinatorial number theory that brings together parts of other areas such as harmonic analysis, graph theory, extremal combinatorics, algebraic methods, analytic number theory, ergodic theory and probability theory. The area may be summarized as the theory of counting additive structures in subsets of integers or other abelian groups. A general question that drives much of the area asks what can be said about additive patterns in a set when the set is big.

One of the most fundamental results in the area is Szemeredi's theorem, which says that arithmetic progressions of any finite length appear in any subset of the integers with positive upper density. Another is the Freiman-Rusza theorem, which says that if the set of pair-sums A+A of elements from a set A has roughly the same size as A, then A must be contained in a generalized arithmetic progression. Major impetus was given to the field by Fourier-analytic proof by Gowers and the combinatorial proofs by Gowers, and Nagle, Rodl, Schacht and Skokan of Szemeredi's theorem. These proofs ingeniously expanded basic techniques used to prove the case of three-term arithmetic progressions and led to the development of a higher-order Fourier analysis and the notion of regularity for hypergraphs. Building yet further on these results and their proofs, Green and Tao later proved their now celebrated result that the prime numbers contain arbitrarily long arithmetic progressions.

One of the aims of the course is to expose key ideas behind the Fourier-analytic and graph-theoretic proofs of Szemeredi's theorem, in addition to the algebraic techniques used recently to tackle the problem for three-term arithmetic progressions in the finite-field setting (the so-called cap set problem).

Prerequisites

Basic knowledge of undergraduate graph theory (for instance Ch 1 of `Graph Theory' by Diestel: http://diestel-graph-theory.com/), linear algebra and algebra (familiarity with abelian groups, polynomial rings, fields).

Grading

30% homework assignments 70% final exam.
In case of a resit, the resit exam will count for 100% and the homework assignments will not be considered.

Lecture notes and literature

Lecture notes will be provided during the course. Additional optional reading: Tao, Terence, and Van H. Vu.

Additive combinatorics. Vol. 105. Cambridge University Press, 2006.

Lecturers

Jop Briët and Dion Gijswijt