The structure of higher sumsets

Abstract

Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set A A of integers with min ( A ) = 0 \min (A)=0 and gcd ( A ) = 1 \gcd (A)=1 there exist two sets, the “head” and the “tail”, such that if m ≥ max ( A ) − | A | + 2 m\ge \max (A)-|A|+2 , then the m m -fold sumset m A mA consists of the union of the head, the appropriately shifted tail, and a long block of consecutive integers separating them. We give sharp estimates for the length of the block, and find all those sets A A for which the bound max ( A ) − | A | + 2 \max (A)-|A|+2 cannot be substantially improved.