Approximate (Abelian) groups

Abstract

Our aim is to discuss the structure of subsets of Abelian groups which behave 'a bit like' cosets (of subgroups). One version of 'a bit like' can be arrived at by relaxing the usual characterisation of cosets: a subset $S$ of an Abelian group is a coset if for every three elements $x,y,z\in S$ we have $x+y–z\in S$. What happens if this is not true 100% of the time but is true, say, 1% of the time? It turns out that this is a situation which comes up quite a lot, and one possible answer is called Freiman's theorem. We shall discuss it and some recent related quantitative advances.