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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.06756 |
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| _version_ | 1866912318176100352 |
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| author | Imsong, Kilangbenla Paul, Ram Krishna |
| author_facet | Imsong, Kilangbenla Paul, Ram Krishna |
| contents | There are several notions of largeness in a semigroup. N. Hindman and D. Strauss established that if $u,v \in \mathbb{N}$, $A$ is a $u \times v$ matrix with entries from $\mathbb{Q}$ and $ψ$ is a notion of a large set in $\mathbb{N}$, then $\{\vec{x} \in \mathbb{N}^v: A\vec{x} \in ψ^u \}$ is large in $\mathbb{N}^v$. Among the several notions of largeness, C sets occupies an important place of study because they exhibit strong combinatorial properties. The analogous notion of C set appears for a dense subsemigroup $S$ of $((0, \infty),+)$ called a C-set near zero. These sets also have very rich combinatorial structure. In this article, we investigate the above result for C sets near zero in $\mathbb{R}^+$ when the matrix has real entries. We also develop a new characterisation of C-sets near zero in $\mathbb{R}^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06756 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Preservation of notion of C sets near zero over reals Imsong, Kilangbenla Paul, Ram Krishna Combinatorics There are several notions of largeness in a semigroup. N. Hindman and D. Strauss established that if $u,v \in \mathbb{N}$, $A$ is a $u \times v$ matrix with entries from $\mathbb{Q}$ and $ψ$ is a notion of a large set in $\mathbb{N}$, then $\{\vec{x} \in \mathbb{N}^v: A\vec{x} \in ψ^u \}$ is large in $\mathbb{N}^v$. Among the several notions of largeness, C sets occupies an important place of study because they exhibit strong combinatorial properties. The analogous notion of C set appears for a dense subsemigroup $S$ of $((0, \infty),+)$ called a C-set near zero. These sets also have very rich combinatorial structure. In this article, we investigate the above result for C sets near zero in $\mathbb{R}^+$ when the matrix has real entries. We also develop a new characterisation of C-sets near zero in $\mathbb{R}^+$. |
| title | Preservation of notion of C sets near zero over reals |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.06756 |