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 $N \geqslant b-\ell$, as recently conjectured by Shakan and the first author. We also classify those sets $A$ for which this bound cannot be improved.