Non-Expansive Matrix Based number Systems

Abstract

An integer $d$-dimensional vector is usually represented by $d$ strings, which express the value of each component separately. This article is devoted to number systems that allow representing integer vectors using a single string of digits. This system is given by an integer square matrix $M$ and a finite set $\mathcal{D}$ of symbols, called digits. We focus on full number systems, in which every vector $\vec{x}\in {\mathbb{Z}}^n$ can be written in the form $\vec{x}={\sum }_{i=0}^{k-1}{M}^i{d}_i$ with $d_i\in \mathcal{D}$ and thus represented by the string $d_{k-1}{d}_{k-2}\cdots d_1{d}_0$. In contrast to the very well-described case where the base $M$ is expansive, we focus on systems in which $M$ is similar to a Jordan block with the eigenvalue $\pm 1$. We follow the work of Caldwell, Hare, and Vávra who initiated the study of such systems. First, we address the question of choosing a minimal sized digit set $\mathcal{D}$. Then we describe optimal representations for vectors from ${\mathbb{Z}}^2$, both with respect to their length and the occurrence of non-zero digits.