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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.04693 |
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| _version_ | 1866908867234889728 |
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| author | Gao, Su Jackson, Steve |
| author_facet | Gao, Su Jackson, Steve |
| contents | This paper considers the combinatorics of continuous and Borel rectangular partitions of free actions of $\mathbb{Z}^n$ on $0$-dimensional Polish spaces, specifically the free part $F(2^{\mathbb{Z}^n})$ of the shift action of $\mathbb{Z}^n$ on the space $2^{\mathbb{Z}^n}$. This is done through the study of a corresponding notion of regulated partitions of $\mathbb{R}^n$. The main concepts studied are the continuous and Borel {\em regulation} numbers of the partition. This is defined as the maximum number of rectangles in the corresponding regulated partition that can intersect in a point. The continuous and Borel regulation numbers $γ_c$, $γ_B$ are the minimum possible values of these numbers as we range over continuous (respectively Borel) rectangular partitions of $F(2^{\mathbb{Z}^n})$. It is shown that for $n=2$ that $γ_c=γ_B=3$, and for $n \geq 3$ that $n+2\leq γ_B \leq γ_c \leq 3\cdot 2^{n-2}$. For $n=3$ we improve this to $γ_c=γ_B=5$. This shows a striking difference between the Borel combinatorics of dimension $n=2$ and dimensions $n>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04693 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On regulated partitions Gao, Su Jackson, Steve Logic 03E15 52C22 This paper considers the combinatorics of continuous and Borel rectangular partitions of free actions of $\mathbb{Z}^n$ on $0$-dimensional Polish spaces, specifically the free part $F(2^{\mathbb{Z}^n})$ of the shift action of $\mathbb{Z}^n$ on the space $2^{\mathbb{Z}^n}$. This is done through the study of a corresponding notion of regulated partitions of $\mathbb{R}^n$. The main concepts studied are the continuous and Borel {\em regulation} numbers of the partition. This is defined as the maximum number of rectangles in the corresponding regulated partition that can intersect in a point. The continuous and Borel regulation numbers $γ_c$, $γ_B$ are the minimum possible values of these numbers as we range over continuous (respectively Borel) rectangular partitions of $F(2^{\mathbb{Z}^n})$. It is shown that for $n=2$ that $γ_c=γ_B=3$, and for $n \geq 3$ that $n+2\leq γ_B \leq γ_c \leq 3\cdot 2^{n-2}$. For $n=3$ we improve this to $γ_c=γ_B=5$. This shows a striking difference between the Borel combinatorics of dimension $n=2$ and dimensions $n>2$. |
| title | On regulated partitions |
| topic | Logic 03E15 52C22 |
| url | https://arxiv.org/abs/2603.04693 |