A tight structure theorem for sumsets

Abstract

Let $A=\{0=a_0<a_1<\cdots <a_{\ell +1}=b\}$ be a finite set of non-negative integers. We prove that the sumset $NA$ has a certain easily-described structure, provided that $N⩾b-\ell$, as recently conjectured by Shakan and the first author. We also classify those sets $A$ for which this bound cannot be improved.

8 pages, minor changes to exposition. Accepted version