A generalization of sumset and its applications

Let A be a nonempty finite subset of an additive abelian group G and let r and h be positive integers. The generalized h-fold sumset of A, denoted by h(r) A, is the set of all sums of h elements of A, where each element appears in a sum at most r times. The direct problem for h(r) A is to find a lower bound for | h(r) A| in terms of | A|. The inverse problem for h(r) A is to determine the structure of the finite set A for which | h(r) A| is minimal with respect to some fixed value of | A|. If G = Z, the direct and inverse problems are well studied. In case of G = Z/ pZ, p a prime, the direct problem has been studied very recently by Monopoli (J. Number Theory, 157 (2015) 271-279). In this paper, we express the generalized sumset h(r) A in terms of the regular and restricted sumsets. As an application of this result, we give a new proof of the theorem of Monopoli and as the second application, we present new proofs of direct and inverse theorems for the case G = Z.