Repetitions in beta-integers

TL;DR

This paper investigates the maximum repetitions of motifs in beta-integers, which model quasicrystals, focusing on cases where only finitely many distances occur, and provides solutions for quadratic non-simple Parry numbers.

Contribution

It analyzes the repetition structure of beta-integers with finite distances, offering new insights into their combinatorial properties and solving a specific case for quadratic non-simple Parry numbers.

Findings

Characterized maximal repetitions in beta-integers

Connected repetition properties to combinatorics on words

Solved a specific case for quadratic non-simple Parry numbers

Abstract

Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this paper, we study the maximal possible repetition of the same motif occurring in beta-integers -- one dimensional models of quasicrystals. We are interested in beta-integers realizing only a finite number of distinct distances between neighboring elements. In such a case, the problem may be reformulated in terms of combinatorics on words as a study of the index of infinite words coding beta-integers. We will solve a particular case for beta being a quadratic non-simple Parry number.