标题： 关于大小为$k$的集合的和集 显示英文标题

标题： On the Sumset of Sets of Size $k$

摘要： 集合$\mathcal{R}_{G}(h,k)$包含所有可能的大小，这些大小是包含来自交换阿贝尔群$G$中$k$个元素的集合的$h$-重和集的大小。 这个集合的具体组成仍然未知，但已经取得了一些进展以确定哪些整数存在。 我们知道$\mathcal{R}_{G}(h,k)\subseteq\left[hk-h+1,\binom{h+k-1}{h}\right]$，其中右边是一个包含端点的整数区间。 这些端点已被证实可以达到。 我们将证明，整数$\left[hk-h+2,hk-1\right]$在包含 torsion-free 加法阿贝尔群$G$中的$k\geq 4$个元素的集合的$h$-fold sumset 中是不可能的大小。 此外，我们将确认通过展示一个包含$G$个元素的子集，其$h$-fold sumset 的大小为$hk$，这个区间无法被扩大。 显示英文摘要

摘要： 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 $\mathcal{R}_{G}(h,k)\subseteq\left[hk-h+1,\binom{h+k-1}{h}\right]$, 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 $\left[hk-h+2,hk-1\right]$ are not possible sizes for the $h$-fold sumset of a set containing $k\geq 4$ 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 $hk$.