We state Breuillard, Green and Tao’s rough classification of the finite approximate subgroups of an arbitrary group. This states that a finite approximate subgroup of an arbitrary group is contained in a union of a few cosets of a finite-by-nilpotent group, the nilpotent quotient of which has bounded step. We define coset nilprogressions, and show how to deduce a more detailed version of the Breuillard–Green–Tao theorem in which the approximate subgroup is contained in a union of a few translates of a coset nilprogression of bounded rank and step.

We motivate the definitions of sets of small doubling and approximate groups, and introduce their basic properties. We show that random sets of integers (suitably defined) have large expected doubling. We prove Freiman’s theorem that a subset of a group of doubling less than 2/3 is close to a finite subgroup. We prove the Plünnecke–Ruzsa inequalities, Ruzsa’s triangle inequality and Ruzsa’s covering lemma. We motivate in detail the definition of an approximate group, and reduce the study of sets of small doubling to the study of finite approximate groups. We show that the notions of small tripling and approximate group are stable under intersections and group homomorphisms. We introduce Freiman homomorphisms and present their basic properties.

We define sets of small doubling and approximate groups, and give brief motivation for aspects of the definitions. We give a brief overview of the history of approximate groups.