On the spectra of Pisot-cyclotomic numbers

TL;DR

This paper explores the spectral properties of complex sets generated by Pisot-cyclotomic numbers, analyzing their geometric structure, conditions for quasicrystal modeling, and connections to cut-and-project quasilattices.

Contribution

It provides new insights into the spectral structure and geometric properties of spectra associated with quadratic and cubic Pisot-cyclotomic numbers, including conditions for Delone sets and their relation to quasilattices.

Findings

Spectra can be discrete aperiodic structures with forbidden symmetries.

Conditions for spectra to have the Delone property are discussed.

Connections between spectra, Voronoi tilings, and cut-and-project quasilattices are established.

Abstract

We investigate the complex spectra \[ X^{\mathcal A}(\beta)=\left\{\sum_{j=0}^na_j\beta^j : n\in{\mathbb N},\ a_j\in{\mathcal A}\right\} \] where $\beta$ is a quadratic or cubic Pisot-cyclotomic number and the alphabet $\mathcal A$ is given by $0$ along with a finite collection of roots of unity. Such spectra are discrete aperiodic structures with crystallographically forbidden symmetries. We discuss in general terms under which conditions they possess the Delone property required for point sets modeling quasicrystals. We study the corresponding Voronoi tilings and we relate these structures to quasilattices arising from the cut and project method.