On the Sumset of Sets of Size k

The set RG(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 RG(h,k)⊆[hk-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 [hk-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 $hk$.