Substitution Delone Sets

TL;DR

This paper investigates substitution Delone sets, a class of self-similar aperiodic structures, providing conditions for their existence and exploring their properties through examples.

Contribution

It establishes necessary conditions for solutions to substitution equations involving Delone sets and analyzes cases where solutions are Delone sets.

Findings

Necessary conditions for substitution Delone sets are identified.

The paper characterizes when solutions are Delone sets.

Examples illustrate limitations of the theoretical results.

Abstract

This paper addresses the problem of describing aperiodic discrete structures that have a self-similar or self-affine structure. Substitution Delone set families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an inflation functional equation under the action of an expanding integer matrix in R^d. This paper studies such functional equation in which each X_i is a discrete multiset (a set whose elements are counted with a finite multiplicity). It gives necessary conditions on the coefficients of the functional equation for discrete solutions to exist. It treats the case where the equation has Delone set solutions. Finally, it gives a large set of examples showing limits to the results obtained.