On Asymptotic Approximate Groups of Integers

TL;DR

This paper studies the asymptotic r-covering number of finite sets of integers, establishing bounds and characterizing sets where this number equals r, thus extending Nathanson's earlier results.

Contribution

It extends Nathanson's result by proving the lower bound of the asymptotic r-covering number and characterizes sets where this number equals r.

Findings

The asymptotic r-covering number is always at least r.

It is always at most r+1, as previously shown.

Sets with asymptotic r-covering number exactly r are fully characterized.

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 $\tilde{C}_r(A)=r$.