The structure of higher sumsets

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 of integers with \min (A)=0 and \gcd (A)=1 there exist two sets, the “head” and the “tail”, such that if m\ge \max (A)-|A|+2, then the m-fold sumset 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 for which the bound \max (A)-|A|+2 cannot be substantially improved.