Approximate subgroups with bounded VC-dimension

We combine the fundamental results of Breuillard, Green, and Tao (Publ Math Inst Hautes Études Sci 116:115–221, 2012) on the structure of approximate groups, together with “tame” arithmetic regularity methods based on work of the authors and Terry (J Eur Math Soc (JEMS) 24(2):583–621, 2022), 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, Palacín, and Wolf (Selecta Math (N.S.) 27(4):Paper No. 53, 19,2021) on finite stable sets of small tripling.