On regulated partitions

TL;DR

This paper investigates the combinatorics of regulated partitions in free actions of bZ^n on Polish spaces, revealing dimension-dependent properties and precise regulation numbers for different dimensions.

Contribution

It introduces the concepts of continuous and Borel regulation numbers for partitions and determines their exact values for specific dimensions, highlighting a dimension-dependent combinatorial behavior.

Findings

For n=2, bg_c=bg_B=3.

For n=3, bg_c=bg_B=5.

For n, the regulation numbers satisfy n+2 bg_B bg_c bg bg_c bg 3 bg 2^{n-2}.

Abstract

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 $\gamma_c$, $\gamma_B$ 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 $\gamma_c=\gamma_B=3$, and for $n \geq 3$ that $n+2\leq \gamma_B \leq \gamma_c \leq 3\cdot 2^{n-2}$. For $n=3$ we improve this to $\gamma_c=\gamma_B=5$. This shows a striking difference between the Borel combinatorics of dimension $n=2$ and dimensions $n>2$.