Approximate groups

Abstract

In this chapter, we describe a recent application of nonstandard methods to multiplicative combinatorics, namely to the structure theorem for finite approximate groups. The general story is much more complicated than the rest of the material in this book and there are already several good sources for the complete story (see Breuillard et al. (Publications mathématiques de l’IHÉS 116:115–221, 2012) or van den Dries (Astérisque 367: Exp. No. 1077, vii, 79–113, 2015)), so we content ourselves to a summary of some of the main ideas. Our presentation will be similar to the presentation from van den Dries (Astérisque 367: Exp. No. 1077, vii, 79–113, 2015).