ON THE MINIMUM CARDINALITY OF GENERALIZED SUMSETS IN FINITE CYCLIC GROUPS

摘要

For a nonempty subset A of an abelian group G, the generalized sumset h (r)A consists of all sums of h elements of A with at most r repetitions for each element. In this paper, we generalize an earlier result of Bajnok on restricted sumsets h (1)A in Zn to generalized sumsets h (r)A in Zn for 1 ≤ r ≤ h. More precisely, given positive integers h, r, k, we prove an upper bound for the minimum cardinality of h (r)A when A runs through all k-subsets of Zn. This is done by exactly calculating |h (r)A| for a very specific k-subset A = Ad(n, k) of Zn.