A TIGHT STRUCTURE THEOREM FOR SUMSETS

Let A = {0 = a(0) < a(1) < center dot center dot center dot < a(l+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 - l, as recently conjectured (see A. Granville and G. Shakan [Acta Math. Hungar. 161 (2020), pp. 700-718]). We also classify those sets A for which this bound cannot be improved.