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 Szemerédi's theorem, which says that arithmetic progressions of any finite length appear in any big-enough subset of the integers. Another is the Freiman-Ruzsa theorem, which says that if A is a set in some finite abelian group such that the set of pair-sums A+A has roughly the same size as A, then A is approximately a coset of a subgroup.
Major impetus was given to the field by a Fourier-analytic proof of Gowers. This proof 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. Building yet further this result and its 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.
See for instance CH16,17 of 'Abstract Algebra, Theory and Applications': http://abstract.pugetsound.edu/download.html
Aim of the course
To expose combinatorial, Fourier-analytic and algebraic techniques that together form the basis for much of the additive combinatorics tool box.
Lecturer
Jop Briët
Website
- Docent: Jop Briet