A generalization of sumsets of set of integers

Let A be a nonempty finite set of integers. The h-fold sumset of A, denoted by hA, is the set of all sums of h elements of A with repetitions allowed. A restricted h-fold sumset of A  , denoted by hˆA, is the set of all sums of h distinct elements of A  . For h≥1h≥1 and r≥1r≥1, we define a generalized h  -fold sumset, denoted by h(r)Ah(r)A, which is the set of all sums of h elements of A, where each element appearing in the sum can be repeated at most r times. Thus the h-fold sumset hA and the restricted h  -fold sumset hˆA are particular cases of the sumset h(r)Ah(r)A for r=hr=h and r=1r=1, respectively. The direct problem for h(r)Ah(r)A is to find a lower bound for |h(r)A||h(r)A| in terms of |A||A|. The inverse problem for h(r)Ah(r)A is to determine the structure of the finite set A   of integers for which |h(r)A||h(r)A| is minimal. In this paper we solve both the problems.