On the cardinality of general hh-fold sumsets

Let A={a0,a1,…,ak−1}A={a0,a1,…,ak−1} be a set of kk integers. For any integer h≥1h≥1 and any ordered kk-tuple of positive integers r=(r0,r1,…,rk−1), we define a general hh-fold sumset, denoted by h(r)A, which is the set of all sums of hh elements of AA, where aiai appearing in the sum can be repeated at most riri times for i=0,1,…,k−1i=0,1,…,k−1. In this paper, we give the best lower bound for |h(r)A| in terms of r and hh and determine the structure of the set AA when |h(r)A| is minimal. This generalizes results of Nathanson, and recent results of Mistri and Pandey and also solves a problem of Mistri and Pandey.