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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.07679 |
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Table of Contents:
- 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 $hk$.