On the Delone property of (-\beta)-integers

TL;DR

This paper investigates the properties of (-β)-integers, revealing that unlike classical integers, they are not always uniformly discrete or relatively dense, which impacts their mathematical structure.

Contribution

It demonstrates that (-β)-integers can lack uniform discreteness and density, providing new insights into their geometric and combinatorial properties.

Findings

(-β)-integers are not necessarily uniformly discrete.

(-β)-integers are not necessarily relatively dense.

They can be described by fixed points of anti-morphisms.

Abstract

The (-\beta)-integers are natural generalisations of the \beta-integers, and thus of the integers, for negative real bases. They can be described by infinite words which are fixed points of anti-morphisms. We show that they are not necessarily uniformly discrete and relatively dense in the real numbers.