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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2505.07679 |
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| _version_ | 1866909667739828224 |
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| author | Schinina, Vincent |
| author_facet | Schinina, Vincent |
| 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 $hk$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Sumset of Sets of Size $k$ Schinina, Vincent Combinatorics Number Theory 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 $hk$. |
| title | On the Sumset of Sets of Size $k$ |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2505.07679 |