On regulated partitions

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 ${\hat{I}}_c^3$, ${\hat{I}}_B^3$ 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 ${\hat{I}}_c^3={\hat{I}}_B^3=3$, and for $n\ge 3$ that $n+2\le {\hat{I}}_B^3\le {\hat{I}}_c^3\le 3\cdot 2^{n-2}$. For $n=3$ we improve this to ${\hat{I}}_c^3={\hat{I}}_B^3=5$. This shows a striking difference between the Borel combinatorics of dimension $n=2$ and dimensions $n>2$.