Sumsets with a minimum number of distinct terms
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929668904452096 |
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| author | Bhanja, Jagannath |
| author_facet | Bhanja, Jagannath |
| contents | For a set $A$ of $k$ elements from an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote this set by $h^{(\geq r)}A$. The set $h^{(\geq r)}A$ generalizes the classical sumsets $hA$ and $h\hat{}A$ for $r=1$ and $r=h$, respectively. As the main result of this article, we give an upper bound for the minimum size of $h^{(\geq r)}A$ over $\mathbb{Z}_m$ for $m \geq 2$. Further, by an observation relating the sumsets $hA$, $h\hat{}A$, and $h^{(\geq r)}A$ we obtain the sharp lower bound on the size of $h^{(\geq r)}A$ and also characterize the set $A$ for which the lower bound on the size of $h^{(\geq r)}A$ is tight over the groups $\mathbb{Z}$ and $\mathbb{Z}_p$, where $p$ is a prime number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03977 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sumsets with a minimum number of distinct terms Bhanja, Jagannath Combinatorics Number Theory 11P70, 11B13, 11B75 For a set $A$ of $k$ elements from an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote this set by $h^{(\geq r)}A$. The set $h^{(\geq r)}A$ generalizes the classical sumsets $hA$ and $h\hat{}A$ for $r=1$ and $r=h$, respectively. As the main result of this article, we give an upper bound for the minimum size of $h^{(\geq r)}A$ over $\mathbb{Z}_m$ for $m \geq 2$. Further, by an observation relating the sumsets $hA$, $h\hat{}A$, and $h^{(\geq r)}A$ we obtain the sharp lower bound on the size of $h^{(\geq r)}A$ and also characterize the set $A$ for which the lower bound on the size of $h^{(\geq r)}A$ is tight over the groups $\mathbb{Z}$ and $\mathbb{Z}_p$, where $p$ is a prime number. |
| title | Sumsets with a minimum number of distinct terms |
| topic | Combinatorics Number Theory 11P70, 11B13, 11B75 |
| url | https://arxiv.org/abs/2307.03977 |