A concept of largeness of combinatorially rich sets
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913397792047104 |
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| author | Debnath, Pintu |
| author_facet | Debnath, Pintu |
| contents | In \cite[Proposition 8.21 Page-169]{F} Using the methods of topological dynamics, H. Furstenberg introduced the notion of central set and proved the famous Central Sets Theorem. Later, in \cite{DHS}, D. De, H. Hindman and D. Struss established a strong Central Sets Theorem, where they introduced the notion of $J$-set. Like $J$-set, in \cite{BG} V. Bergelson and D. Glasscock introduced the notion of combinatorially rich set ( $CR$-set). Let $u,v\in\mathbb{N}$ and $A$ be a $u\times v$ matrix with rational entries. In \cite{HS23}, N. Hindman and D. Strauss established that whenever $B$ is a piecewise syndetic set (resp. $J$-set) in $\mathbb{Z}$, $\left\{ \vec{x}\in\mathbb{Z}^{v}:A\vec{x}\in B^{u}\right\} $ is a piecewise syndetic set ( resp. $J$-set) in $\mathbb{Z}^{v}$. In this article, we prove the same result for $CR$-sets using an equivalent definition of $CR$-set in \cite{HHST} by N. Hindman, H. Hosseini, D. Strauss and M. Tootkaboni. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14732 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A concept of largeness of combinatorially rich sets Debnath, Pintu Combinatorics 05D10, 22A15, 54D35 In \cite[Proposition 8.21 Page-169]{F} Using the methods of topological dynamics, H. Furstenberg introduced the notion of central set and proved the famous Central Sets Theorem. Later, in \cite{DHS}, D. De, H. Hindman and D. Struss established a strong Central Sets Theorem, where they introduced the notion of $J$-set. Like $J$-set, in \cite{BG} V. Bergelson and D. Glasscock introduced the notion of combinatorially rich set ( $CR$-set). Let $u,v\in\mathbb{N}$ and $A$ be a $u\times v$ matrix with rational entries. In \cite{HS23}, N. Hindman and D. Strauss established that whenever $B$ is a piecewise syndetic set (resp. $J$-set) in $\mathbb{Z}$, $\left\{ \vec{x}\in\mathbb{Z}^{v}:A\vec{x}\in B^{u}\right\} $ is a piecewise syndetic set ( resp. $J$-set) in $\mathbb{Z}^{v}$. In this article, we prove the same result for $CR$-sets using an equivalent definition of $CR$-set in \cite{HHST} by N. Hindman, H. Hosseini, D. Strauss and M. Tootkaboni. |
| title | A concept of largeness of combinatorially rich sets |
| topic | Combinatorics 05D10, 22A15, 54D35 |
| url | https://arxiv.org/abs/2402.14732 |