Convolutions of sets with bounded VC-dimension are uniformly continuous

We study a notion of VC-dimension for subsets of groups, defining this for a set A to be the VC-dimension of the family {(chi A)boolean AND A : x is an element of A . A(-1)}. We show that if a finite subset A of an abelian group has bounded VC-dimension, then the convolution 1(A) * 1(-A) is Bohr uniformly continuous, in a quantitatively strong sense. This generalises and strengthens a version of the stable arithmetic regularity lemma of Terry and Wolf [25] in various ways. In particular, it directly implies that the Polynomial Bogolyubov-Ruzsa Conjecture - a strong version of the Polynomial Freiman-Ruzsa Conjecture - holds for sets with bounded VC-dimension. We also prove some results in the non-abelian setting. In some sense, this gives a structure theorem for translation-closed set systems with bounded (classical) VC-dimension: if a VC-bounded family of subsets of an abelian group is closed under translation, then each member has a simple description in terms of Bohr sets, up to a small error.