  We give a structural description of the finite subsets $A$ of an arbitrary
group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for
some bounded $d$ and sufficiently large $n$, showing that such sets are
controlled by (a bounded number of translates of) a coset nilprogression in a
certain precise sense. This description recovers some previous results of
Breuillard-Green-Tao and Breuillard-Tointon concerning sets of polynomial
growth; we are also able to describe the subsequent growth of $|A^m|$ fairly
explicitly for $m \geq n$, at least when $A$ is a symmetric neighbourhood of
the identity. We also obtain an analogous description of symmetric probability
measures $\mu$ whose $n$-fold convolutions $\mu^{*n}$ obey the condition $\|
\mu^{*n} \|_{\ell^2}^{-2} \leq n^d \|\mu \|_{\ell^2}^{-2}$. In the abelian
case, this description recovers the inverse Littlewood-Offord theorem of
Nguyen-Vu, and gives a variant of a recent nonabelian inverse Littlewood-Offord
theorem of Tiep-Vu.
  Our main tool to establish these results is the inverse theorem of
Breuillard, Green, and the author that describes the structure of approximate
groups.
