On algebraic sums, trees and ideals in the Cantor space
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911884725190656 |
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| author | Michalski, Marcin Rałowski, Robert Żeberski, Szymon |
| author_facet | Michalski, Marcin Rałowski, Robert Żeberski, Szymon |
| contents | We work in the Cantor space $2^ω$. The results of the paper adhere the following pattern. Let $\mathcal{I}\in \{\mathcal{M}, \mathcal{N}, \mathcal{M}\cap \mathcal{N}, \mathcal{E}\}$ and $T$ be a perfect, uniformly perfect or Silver tree. Then for every $A\in \mathcal{I}$ there exists $T'\subseteq T$ of the same kind as $T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\inω$. We also prove weaker statements for splitting trees. For the case $\mathcal{E}$ we also provide a simple characterization of basis of $\mathcal{E}$. We use these results to prove that the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to $u_0$ and $v_0$, provided that $\mathfrak{c}$ is a regular cardinal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13775 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On algebraic sums, trees and ideals in the Cantor space Michalski, Marcin Rałowski, Robert Żeberski, Szymon Logic General Topology Primary: 03E75, 28A05, Secondary: 03E05, 54H05 We work in the Cantor space $2^ω$. The results of the paper adhere the following pattern. Let $\mathcal{I}\in \{\mathcal{M}, \mathcal{N}, \mathcal{M}\cap \mathcal{N}, \mathcal{E}\}$ and $T$ be a perfect, uniformly perfect or Silver tree. Then for every $A\in \mathcal{I}$ there exists $T'\subseteq T$ of the same kind as $T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\inω$. We also prove weaker statements for splitting trees. For the case $\mathcal{E}$ we also provide a simple characterization of basis of $\mathcal{E}$. We use these results to prove that the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to $u_0$ and $v_0$, provided that $\mathfrak{c}$ is a regular cardinal. |
| title | On algebraic sums, trees and ideals in the Cantor space |
| topic | Logic General Topology Primary: 03E75, 28A05, Secondary: 03E05, 54H05 |
| url | https://arxiv.org/abs/2405.13775 |