On the structure of the sumsets

Let A be a set of nonnegative integers. For h ≥ 2 , denote by h A 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 = { a 0 , a 1 , … , a k − 1 } is a finite set of integers such that 0 = a 0 < a 1 < ⋯ < a k − 1 and ( a 1 , … , a k − 1 ) = 1 , then there exist integers c and d and sets C ⊆ [ 0 , c − 2 ] and D ⊆ [ 0 , d − 2 ] such that h A = C ∪ [ c , h a k − 1 − d ] ∪ ( h a k − 1 − D ) for all h ≥ ∑ i = 2 k − 1 a i − k + 1 . The result is optimal. This improves Nathanson’s result: h ≥ max { 1 , ( k − 2 ) ( a k − 1 − 1 ) a k − 1 } .