Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

TL;DR

This paper characterizes when spectra of quadratic Pisot units form Delone sets that are bounded distance to a lattice, using cut-and-project methods and combinatorial coding techniques.

Contribution

It provides a new characterization of spectra related to quadratic Pisot units as lattice-bounded Delone sets via combinatorial and cut-and-project methods.

Findings

Spectra of certain quadratic Pisot units are bounded distance to a lattice.

Use of cut-and-project method to analyze spectra.

Application of combinatorial coding to characterize spectra.

Abstract

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.