Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers

TL;DR

This paper investigates the combinatorial and arithmetical properties of $eta$-integers for quadratic Parry numbers, determining fractional positions, balance of associated infinite words, and their relation to Sturmian sequences.

Contribution

It provides precise bounds on fractional positions and characterizes the balance of infinite words coding $eta$-integers for a class of quadratic Parry numbers, including non-sturmian cases.

Findings

Maximal number of fractional positions determined with ±1 accuracy

Exact balance of the coding infinite words established

Connection between balance properties and arithmetical features of $eta$-integers

Abstract

We study arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of $\beta$-fractional positions, which may arise as a result of addition of two $\beta$-integers. For the infinite word $u_\beta$ coding distances between consecutive $\beta$-integers, we determine precisely also the balance. The word $u_\beta$ is the fixed point of the morphism $A \to A^{m-1}B$ and $B\to A^{m-n-1}B$. In the case $n=1$ the corresponding infinite word $u_\beta$ is sturmian and therefore 1-balanced. On the simplest non-sturmian example with $n\geq 2$, we illustrate how closely the balance and arithmetical properties of $\beta$-integers are related.