A tight structure theorem for sumsets

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