Spectral notions of aperiodic order

TL;DR

This paper unifies different spectral approaches to understanding the structure of non-periodic point sets, especially Delone sets in Euclidean space, and clarifies their interrelations.

Contribution

It provides a unified framework for spectral notions of aperiodic order and explores their connections, with generalizations and examples.

Findings

Unified spectral framework for aperiodic order

Clarified relations between different spectral notions

Extended concepts to broader classes of point sets

Abstract

Various spectral notions have been employed to grasp the structure of point sets, in particular non-periodic ones. In this article, we present them in a unified setting and explain the relations between them. For the sake of readability, we use Delone sets in Euclidean space as our main object class, and give generalisations in the form of further examples and remarks.