The set R₆ (h, k) consists of all possible sizes for the h-fold sumset of sets containing k elements from an additive abelian group G. The exact makeup of this set is still unknown, but there has been progress towards determining which integers are present. We know that R₆ (h, k) -h+1, h+k-1h, where the right side is an interval of integers that includes the endpoints. These endpoints are known to be attained. We will prove that the integers in -h+2, hk-1 are not possible sizes for the h-fold sumset of a set containing k 4 elements of a torsion-free additive abelian group G. Furthermore, we will confirm that this interval can't be made larger by exhibiting a subset of G whose h-fold sumset has size hk.
Vincent Schinina (Mon,) studied this question.