On the structure of ($-\beta$)-integers

TL;DR

This paper explores the structure of $(-eta)$-integers, a generalization of $eta$-integers for negative bases, by characterizing them as fixed points of an anti-morphism when $eta$ is a Parry number analogue.

Contribution

It provides a detailed description of the structure of $(-eta)$-integers using anti-morphisms, extending the understanding of number representations in negative bases.

Findings

$(-eta)$-integers form a structured set characterized by anti-morphisms.

The structure is explicitly described for $eta$ as a Parry number analogue.

The approach generalizes known results for positive base $eta$-integers.

Abstract

The $(-\beta)$-integers are natural generalisations of the $\beta$-integers, and thus of the integers, for negative real bases. When $\beta$ is the analogue of a Parry number, we describe the structure of the set of $(-\beta)$-integers by a fixed point of an anti-morphism.