  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.
