Every finite set of integers is an asymptotic approximate group

Abstract

A set $A$ is an $(r,\ell )$-approximate group in the additive abelian group $G$ if $A$ is a nonempty subset of $G$ and there exists a subset $X$ of $G$ such that $|X|\le \ell$ and $rA\subseteq X+A$. The set $A$ is an asymptotic $(r,\ell )$-approximate group if the sumset $hA$ is an $(r,\ell )$-approximate group for all sufficiently large integers $h$. It is proved that every finite set of integers is an asymptotic $(r,r+1)$-approximate group for every integer $r\ge 2$.

arXiv comment7 pages; minor corrections and improvements

References in corpus (1)

• On the cardinality of general $h$-fold sumsets

Cited by in corpus (1)

• On Asymptotic Approximate Groups of Integers