Lattice Substitution Systems and Model Sets

TL;DR

This paper investigates conditions under which lattice partition sets form regular model sets, demonstrating that certain substitution systems produce pure point diffractive sets, exemplified by the chair and sphinx tilings.

Contribution

It provides a main theorem linking matrix substitution systems on lattices to regular model sets with pure point diffraction, using internal space methods.

Findings

Main theorem establishes equivalence conditions for regular model sets

Chair and sphinx tilings are proven to be pure point diffractive

Methods connect substitution systems to diffraction properties

Abstract

The paper studies ways in which the sets of a partition of a lattice in $\RR^n$ become regular model sets. The main theorem gives equivalent conditions which assure that a matrix substitution system on a lattice in $\RR^n$ gives rise to regular model sets (based on $p$-adic-like internal spaces), and hence to pure point diffractive sets. The methods developed here are used to show that the $n-$dimensional chair tiling and the sphinx tiling are pure point diffractive.