Attractors of sequences coding $\beta$-integers

TL;DR

This paper investigates minimal string attractors for prefixes of simple Parry sequences, confirming their minimality and extending the analysis to certain binary non-simple Parry sequences, advancing understanding of $eta$-integer codings.

Contribution

It proves that attractors of alphabet size are minimal for simple Parry sequences and provides attractors for specific binary non-simple Parry sequences.

Findings

Confirmed minimality of attractors of alphabet size for simple Parry sequences.

Provided attractors for prefixes of certain binary non-simple Parry sequences.

Enhanced understanding of $eta$-integer codings and their attractors.

Abstract

In this paper, we describe minimal string attractors of prefixes of simple Parry sequences. These sequences form a coding of distances between consecutive $\beta$-integers in numeration systems with a real base $\beta$. Simple Parry sequences have been recently studied from this point of view and attractors of prefixes have been described. However, the authors themselves had doubts about their minimality and conjectured that attractors of alphabet size should be sufficient. We confirm their conjecture. Moreover, we provide attractors of prefixes of some particular form of binary non-simple Parry sequences.