  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.
