Measurable tilings by abelian group actions

TL;DR

This paper studies measurable tilings of measure spaces by abelian group actions, establishing a dilation lemma and a structure theorem, with applications to classifying random tilings and deforming tilings on tori.

Contribution

It introduces a dilation lemma and a structure theorem for measurable tilings by abelian groups, advancing understanding of their structure and applications.

Findings

Classified factors of iid random tilings of finitely generated abelian groups.

Proved measurable tilings of tori can be linearly deformed into rational shift tilings.

Resolved a conjecture for tilings of the 1-dimensional torus.

Abstract

Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a "dilation lemma" that asserts, roughly speaking, that $F \odot A = X$ implies $F^r \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are "factors of iid", and show that measurable tilings of a torus $\mathbb{T}^d$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).