A generalization of sumsets modulo a prime

Let A be a subset of an abelian group G  . For integers h,r≥1h,r≥1 the generalized h  -fold sumset, denoted by h(r)Ah(r)A, is the set of sums of h elements of A, where each element appears in the sum at most r   times. If G=ZG=Z, lower bounds for |h(r)A||h(r)A| are known, as well as the structure of the sets of integers for which |h(r)A||h(r)A| is minimal. In this paper, we generalize this result by giving a lower bound for |h(r)A||h(r)A| when G=Z/pZG=Z/pZ for a prime p  , and show new proofs for the direct and inverse problems in ZZ.