Number representation using generalized $(-\beta)$-transformation

TL;DR

This paper explores generalized negative base number systems using a new transformation approach, aiming to improve upon existing systems by analyzing digit string admissibility and periodicity.

Contribution

It introduces a generalized $(-\beta)$-transformation with an arbitrary interval, broadening the scope of negative base number systems beyond the Ito-Sadahiro definition.

Findings

Characterization of admissible digit strings in the generalized system

Conditions for periodicity of digit expansions

Potential advantages over traditional Ito-Sadahiro systems

Abstract

We study non-standard number systems with negative base $-\beta$. Instead of the Ito-Sadahiro definition, based on the transformation $T_{-\beta}$ of the interval $\big[-\frac{\beta}{\beta+1},\frac{1}{\beta+1}\big)$ into itself, we suggest a generalization using an interval $[l,l+1)$ with $l\in(-1,0]$. Such generalization may eliminate certain disadvantages of the Ito-Sadahiro system. We focus on the description of admissible digit strings and their periodicity.