关于受控划分
On regulated partitions
AI总结：
本文研究了自由动作$\mathbb{Z}^n$在0维Polish空间上的受控划分，发现对于$n=2$，调节数为3；对于$n≥3$，调节数范围在n+2到3⋅2^{n-2}之间，且n=3时进一步确定为5。
摘要：
本文研究了连续和Borel矩形划分的组合学，特别是自由动作$\mathbb{Z}^n$在0维Polish空间上的自由部分$F(2^{\mathbb{Z}^n})$的移位动作。通过研究相应的受控划分概念来实现这一目标。主要研究的概念是划分的连续和Borel调节数。定义为对应受控划分中能相交于一点的最大矩形数量。连续和Borel调节数$γ_c$，$γ_B$是作为连续（分别Borel）矩形划分范围内的最小可能值。对于$n=2$，有$γ_c=γ_B=3$，对于$n≥3$，有$n+2≤γ_B≤γ_c≤3⋅2^{n-2}$。对于$n=3$，我们将其改进为$γ_c=γ_B=5$。这显示了在$n=2$和$n>2$维度之间Borel组合学的显著差异。
英文摘要：
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 $n>2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
