Additive combinatorics is a relatively new area of mathematics that brings together areas such as combinatorial and analytic number theory, Ramsey theory, harmonic analysis, graph theory, extremal combinatorics, algebraic methods, 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 big-enough subset of the integers. Another is the Freiman-Rusza theorem, which says that if the set of pair-sums A+A of elements from a set of integers A has roughly the same size as A, then A must be 'close' to something called a 'generalized' arithmetic progression. Major impetus was given to the field by a Fourier-analytic proof of Gowers and combinatorial proofs of 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.

Prerequisites

  • Basic knowledge of graph theory.
    See for instance CH1 of 'Graph Theory' by Diestel: http://diestel-graph-theory.com/
  • Elementary group theory (groups, homomorphisms, structure of finite abelian groups).
    See for instance CH3,11,13 of 'Abstract Algebra, Theory and Applications': http://abstract.pugetsound.edu/download.html
  • Basic linear algebra (linear system, vector space, matrix rank).
  • Some familiarity with (finite) fields, rings and polynomials can be helpful.
    See for instance CH16,17 of 'Abstract Algebra, Theory and Applications': http://abstract.pugetsound.edu/download.html

Aim of the course
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).

Lecturers
Jop Briët and Dion Gijswijt