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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2412.18598 |
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| _version_ | 1866912177443569664 |
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| author | Kravitz, Noah |
| author_facet | Kravitz, Noah |
| contents | Let $hA$ denote the $h$-fold sumset of a subset $A$ of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations $σ_1, \ldots, σ_H \in \mathfrak{S}_n$, there exist finite subsets $A_1, \ldots, A_n \subseteq \mathbb{Z}$ such that for each $1 \leq h \leq H$, the relative order of the quantities $|h A_1|, \ldots, |h A_n|$ is given by $σ_h$. We also establish extensions where $\mathbb{Z}$ is replaced by any other infinite abelian group or where one prescribes some equalities (not only inequalities) among the sumset sizes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18598 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative sizes of iterated sumsets Kravitz, Noah Combinatorics Let $hA$ denote the $h$-fold sumset of a subset $A$ of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations $σ_1, \ldots, σ_H \in \mathfrak{S}_n$, there exist finite subsets $A_1, \ldots, A_n \subseteq \mathbb{Z}$ such that for each $1 \leq h \leq H$, the relative order of the quantities $|h A_1|, \ldots, |h A_n|$ is given by $σ_h$. We also establish extensions where $\mathbb{Z}$ is replaced by any other infinite abelian group or where one prescribes some equalities (not only inequalities) among the sumset sizes. |
| title | Relative sizes of iterated sumsets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.18598 |