Approximate subgroups with bounded VC-dimension

We combine the fundamental results of Breuillard, Green, and Tao on the structure of approximate groups, together with "tame" arithmetic regularity methods based on work of the authors and Terry, to give a structure theorem for finite subsets $A$ of arbitrary groups $G$ where $A$ has "small tripling" and bounded VC-dimension: Roughly speaking, up to a small error, $A$ will be a union of a bounded number of translates of a coset nilprogression of bounded rank and step (see Theorem 2.1). We also prove a stronger result in the setting of bounded exponent (see Theorem 2.2). Our results extend recent work of Martin-Pizarro, Palac\'{i}n, and Wolf on finite stable sets of small tripling.