Integers in number systems with positive and negative quadratic Pisot base

TL;DR

This paper compares the combinatorial structures of integer sets in quadratic Pisot base systems with positive and negative bases, revealing when their coding languages coincide or relate through morphic images.

Contribution

It introduces a detailed comparison of the languages coding distances between consecutive integers in quadratic Pisot base systems with positive and negative bases, identifying conditions for their equivalence or morphic relation.

Findings

Languages of $u_\beta$ and $u_{-\beta}$ coincide for roots of $x^2 - m x - m$.

Languages are related via morphic images for other quadratic Pisot numbers.

Studied the group structure of $(-\beta)$-integers.

Abstract

We consider numeration systems with base $\beta$ and $-\beta$, for quadratic Pisot numbers $\beta$ and focus on comparing the combinatorial structure of the sets $\Z_\beta$ and $\Z_{-\beta}$ of numbers with integer expansion in base $\beta$, resp. $-\beta$. Our main result is the comparison of languages of infinite words $u_\beta$ and $u_{-\beta}$ coding the ordering of distances between consecutive $\beta$- and $(-\beta)$-integers. It turns out that for a class of roots $\beta$ of $x^2-mx-m$, the languages coincide, while for other quadratic Pisot numbers the language of $u_\beta$ can be identified only with the language of a morphic image of $u_{-\beta}$. We also study the group structure of $(-\beta)$-integers.