  If $A$ is a nonempty subset of an additive group $G$, then the $h$-fold
sumset is \[ hA = \{x_1 + \cdots + x_h : x_i \in A_i \text{ for } i=1,2,\ldots,
h\}. \] The set $A$ is an $(r,\ell)$-approximate group in $G$ if $A$ is a
nonempty subset of a group $G$ and there exists a subset $X$ of $G$ such that
$|X| \leq \ell$ and $rA \subseteq XA$. We do not assume that $A$ contains the
identity, nor that $A$ is symmetric, nor that $A$ is finite. The set $A$ is an
asymptotic $(r,\ell)$-approximate group if the sumset $hA$ is an
$(r,\ell)$-approximate group for all sufficiently large $h$.
  It is proved that every polytope in a real vector space is an asymptotic
$(r,\ell)$-approximate group, that every finite set of lattice points is an
asymptotic $(r,\ell)$-approximate group, and that every finite subset of an
abelian group is an asymptotic $(r,\ell)$-approximate group.
