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 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.