Every finite subset of an abelian group is an asymptotic approximate group

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.

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