Number of bounded distance equivalence classes in hulls of repetitive Delone sets

TL;DR

This paper investigates the classification of repetitive Delone sets based on bounded distance equivalence, showing that their hulls contain either a single class or uncountably many, revealing a dichotomy in their structure.

Contribution

It establishes a dichotomy for the number of bounded distance equivalence classes in the hulls of repetitive Delone sets with finite local complexity, extending previous results.

Findings

Hulls of repetitive Delone sets have either one or uncountably many equivalence classes.

The result parallels a similar theorem in prior work (arXiv:2011.00106).

The paper clarifies the structure of Delone set hulls under bounded distance equivalence.

Abstract

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many. A very similar result is proven in arXiv:2011.00106 [math.MG].