A dichotomy for bounded displacement equivalence of Delone sets

TL;DR

This paper establishes a dichotomy in the classification of minimal Delone sets in Euclidean space, showing they are either uniformly spread or have many distinct bounded displacement classes, with implications for well-known tiling models.

Contribution

It proves a fundamental dichotomy for minimal Delone sets in Euclidean space, extending to models like cut-and-project sets and substitution tilings, regardless of finite local complexity.

Findings

All minimal Delone sets are either uniformly spread or have infinitely many bounded displacement classes.

The dichotomy applies to various well-studied Delone set constructions.

The results hold under the natural Chabauty--Fell topology, unifying different settings.

Abstract

We prove that in every compact space of Delone sets in $\mathbb{R}^d$ which is minimal with respect to the action by translations, either all Delone sets are uniformly spread, or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty--Fell topology, which is the natural topology on the space of closed subsets of $\mathbb{R}^d$. This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.