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

TL;DR

The paper proves that every finite subset of an abelian group, as well as polytopes and lattice points, forms an asymptotic approximate group, expanding understanding of sumset structures in additive groups.

Contribution

It establishes that all finite subsets of abelian groups are asymptotic approximate groups, generalizing previous results to broader classes of sets.

Findings

Finite subsets of abelian groups are asymptotic approximate groups.

Polytopes in real vector spaces are asymptotic approximate groups.

Finite lattice point sets are asymptotic approximate groups.

Abstract

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.