Pure Discrete Spectrum and Regular Model Sets in Unimodular Substitution Tilings on R^d

TL;DR

This paper investigates primitive unimodular substitution tilings in R^d, establishing conditions under which they have pure discrete spectrum and correspond to regular model sets via a cut-and-project scheme.

Contribution

It demonstrates that under certain algebraic and spectral conditions, substitution tilings can be characterized as regular model sets within a Euclidean cut-and-project framework.

Findings

Pure discrete spectrum implies regular model sets under specified conditions.

Construction of a Euclidean cut-and-project scheme for unimodular substitution tilings.

Eigenvalues of the expansion maps are algebraic conjugates with equal multiplicity.

Abstract

We consider primitive substitution tilings on R^d whose expansion maps are unimodular. We assume that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, we can construct a cut-and-project scheme with a Euclidean internal space. Under some additional condition, we show that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.