On the structure of the sumsets

Let A be a set of nonnegative integers. For h≥2, denote by hA the set of all the integers representable by a sum of h elements from A. In this paper, we prove that, if k≥3, and A={a₀,a₁,…,a_k−1} is a finite set of integers such that 0=a₀<a₁a_k−1 and (a₁,…,a_k−1)=1, then there exist integers c and d and sets C⊆[0,c−2] and D⊆[0,d−2] such that hA=C∪[c,ha_k−1−d]∪(ha_k−1−D) for all . The result is optimal. This improves Nathanson’s result: h≥max{1,(k−2)(a_k−1−1)a_k−1}.