A tight structure theorem for sumsets

Abstract

Let A = { 0 = a 0 > a 1 > ⋯ > a ℓ + 1 = b } A = \{0 = a_0 > a_1 > \cdots > a_{\ell + 1} = b\} be a finite set of non-negative integers. We prove that the sumset N A NA has a certain easily-described structure, provided that N ⩾ b − ℓ N \geqslant b-\ell , as recently conjectured (see A. Granville and G. Shakan [Acta Math. Hungar. 161 (2020), pp. 700–718]). We also classify those sets A A for which this bound cannot be improved.