  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 $\{ (xA) \cap A : x \in A\cdot
A^{-1} \}$. 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 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.
