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 prove Tointon’s theorem that a finite approximate subgroup of a residually nilpotent group is contained in a union of a few cosets of a finite-by-nilpotent group in which the nilpotent quotient is of bounded step. We first prove it in the special case in which G is nilpotent of unbounded step, and finish the chapter by showing how to extend this to the general residually nilpotent case. As part of the proof we show that if a nilpotent group G is a central extension of a finite approximate group A then the commutator subgroup of G is contained in a bounded power of A. We also show that if A is an approximate subgroup of a nilpotent group then a large piece of A can be written as a bounded series of some bounded extensions and some central extensions.

We present Green and Ruzsa’s proof of Freiman’s theorem in an arbitrary abelian group. More specifically, we show that a finite set A of small doubling inside an abelian group is contained in a relatively small coset progression of bounded rank. We introduce the basics of discrete Fourier analysis, and how it relates to sets of small doubling. We prove the Green–Ruzsa result that a set of small doubling in an arbitrary abelian group has a Freiman model in a relatively small finite abelian group. We then prove Bogolyubov’s lemma that a small iterated sum set of this model must contain a relatively large Bohr set of low rank. Combined with the material of the previous chapter, this shows that A contains a relatively large coset progression of low rank. We then deduce the main theorem of the chapter using Chang’s covering argument. In the exercises we guide the reader to a simpler version of the argument yielding the same result in the special case in which A is a set of integers.

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 prove Breuillard and Green’s theorem that a finite approximate subgroup of a soluble complex linear group G of bounded degree is contained in a union of a few cosets of a nilpotent group of bounded step. We first treat the special case in which G is an upper-triangular group. An important ingredient is Solymosi’s sum-product theorem over the complex numbers, which we state and prove. We introduce some basic representation theory and use it to prove that a soluble complex linear group of bounded degree has a subgroup of bounded index that is conjugate to an upper-triangular group; this is a special case of a result of Mal’cev. We then use this to extend from the upper-triangular case to the general soluble case.

We present Breuillard, Green and Tao’s theorem that a finite approximate subgroup of a complex linear group of bounded degree is contained in a union of a few cosets of a nilpotent subgroup of bounded step. We state two substantial ingredients without proof. The first is a result of Mal’cev and Platinov that a virtually soluble complex linear group of bounded degree has a soluble subgroup of bounded index. The second is Breuillard’s uniform Tits alternative, which states that if a finitely generated complex linear group of bounded degree is not virtually soluble then there exist two free generators of a free subgroup that can be expressed as products of boundedly many generators. The third main ingredient is a result, due independently to Sanders and to Croot and Sisask, that if A is an arbitrary approximate group then there is a relatively large neighbourhood of the identity S with the property that a large power of S is contained in a small power of A; we prove this in full.

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.

We present various applications of Breuillard, Green and Tao’s rough classification of finite approximate groups to groups of polynomial growth. We define polynomial, exponential and intermediate growth, and show that these concepts are stable under changes of generating set and passing to subgroups of finite index. We prove Breuillard, Green and Tao’s result that if a ball of large enough radius in a Cayley graph is of size polynomial in the radius then the underlying group is virtually nilpotent. We deduce that all larger balls also have polynomial bounds on their sizes. We guide the reader in the exercises to Breuillard and Tointon’s results that a finite group of large diameter admits large virtually nilpotent and virtually abelian quotients. We also prove the same authors’ result that a finite simple group has diameter bounded by a small power of the size of the group. We prove an isoperimetric inequality for finite groups due to Breuillard, Green and Tao. Finally, we give a brief high-level introduction to applications of approximate groups to the construction of expanders.

We present Tointon’s proof of Freiman’s theorem in an arbitrary nilpotent group. More specifically, we show that a finite K-approximate subgroup of a nilpotent group of bounded step is contained in a relatively small coset progression of rank bounded by a polynomial in K. We start by treating the torsion-free case, where the details are easiest. As part of our proof of the general case we show that if X is a union of subgroups in an abelian p-group of rank r then the subgroup generated by X has diameter at most r with respect to X. We also show that if H is a subgroup of a nilpotent group G generated by a K-approximate group A, and H is contained in a bounded power of A, then the normal closure of H in G is also contained in a bounded power of A.

We introduce coset progressions and Bohr sets, and show that the two notions are roughly equivalent up to Freiman homomorphism. To facilitate the proof of this we introduce lattices and convex bodies and their basic properties, and prove Minkowski’s second theorem from the geometry of numbers.

We introduce nilpotent groups, define nilprogressions and nilpotent progressions, and present some of their basic properties. We start by introducing the Heisenberg group. We present some specific examples of nilprogressions and nilpotent progressions in the Heisenberg group and show that they have small tripling. We then define general nilpotent groups and present their basic properties. Next, we introduce commutators, the collecting process and basic commutators. Finally, we define nilprogressions and nilpotent progressions in general, show that they have small tripling and show that the notions of nilprogression and nilpotent progression are roughly equivalent.