On dynamical $C^{\star}$-set and its combinatorial consequences
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| Format: | Preprint |
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2024
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| _version_ | 1866909362730041344 |
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| author | Debnath, Pintu Goswami, Sayan |
| author_facet | Debnath, Pintu Goswami, Sayan |
| contents | Using the methods from topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. Later D. De, Neil Hindman, and D. Strauss [Fund. Math.199 (2008), 155-175.] established a stronger version of the Central Sets Theorem and then introduced the notion of $C$-sets satisfying the Central Sets Theorem and studied the properties of these sets. For any weak mixing system $\left(X, \mathcal{B},μ, T\right),$ and $A_{0},A_{1}\in\mathcal{B}$, with $μ\left(A_{0}\right)μ\left(A_{1}\right)>0$, R. Kung and X.Ye [Disc. Cont. Dyn. sys., 18 (2007) 817-827.] proved that the set $N\left(A,B\right)= \left\{n:μ\left(A_{0}\cap T^{-n}A_{1}\right)>0\right\}$ intersects all sets of positive upper Banach density. However, later N. Hindman and D. Strauss [New York J. Math. 26 (2020) 230-260.] proved that there exist $C$-sets having zero upper Banach density. Inspired by this result, in this article, we prove that $N\left(A, B \right)$ intersects with all $C$-sets. Then we introduce the notion of a dynamical $C^{\star}$-set and then we study their combinatorial properties. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_18632 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On dynamical $C^{\star}$-set and its combinatorial consequences Debnath, Pintu Goswami, Sayan Dynamical Systems Combinatorics Using the methods from topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. Later D. De, Neil Hindman, and D. Strauss [Fund. Math.199 (2008), 155-175.] established a stronger version of the Central Sets Theorem and then introduced the notion of $C$-sets satisfying the Central Sets Theorem and studied the properties of these sets. For any weak mixing system $\left(X, \mathcal{B},μ, T\right),$ and $A_{0},A_{1}\in\mathcal{B}$, with $μ\left(A_{0}\right)μ\left(A_{1}\right)>0$, R. Kung and X.Ye [Disc. Cont. Dyn. sys., 18 (2007) 817-827.] proved that the set $N\left(A,B\right)= \left\{n:μ\left(A_{0}\cap T^{-n}A_{1}\right)>0\right\}$ intersects all sets of positive upper Banach density. However, later N. Hindman and D. Strauss [New York J. Math. 26 (2020) 230-260.] proved that there exist $C$-sets having zero upper Banach density. Inspired by this result, in this article, we prove that $N\left(A, B \right)$ intersects with all $C$-sets. Then we introduce the notion of a dynamical $C^{\star}$-set and then we study their combinatorial properties. |
| title | On dynamical $C^{\star}$-set and its combinatorial consequences |
| topic | Dynamical Systems Combinatorics |
| url | https://arxiv.org/abs/2410.18632 |