On Asymptotic Approximate Groups of Integers

Abstract

Let $r$ be a positive integer, and let $A$ be a nonempty finite set of at least two integers. We let ${\tilde{C}}_r(A)$ denote the {\em asymptotic $r$-covering number} of $A$, that is, the smallest integer value of $l$ for which, for all sufficiently large positive integers $h$, the $rh$-fold sumset of $A$ is contained in at most $l$ translates of the $h$-fold sumset of $A$. Nathanson proved that ${\tilde{C}}_r(A)$ is always at most $r+1$; here we extend this result to prove that ${\tilde{C}}_r(A)$ is always at least $r$, and determine all sets $A$ for which ${\tilde{C}}_r(A)=r$.

11 pages