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 Granville and Shakan. We also classify those sets $$A$$ for which this bound cannot be improved.