On the Sumset of Sets of Size $k$

The set ${\mathcal{R}}_G(h,k)$ 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 ${\mathcal{R}}_G(h,k)\subseteq [hk-h+1,\left(\genfrac{}{}{0px}{}{h+k-1}{h}\right)]$, 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 $k\ge 4$ 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 $hk$.