Non-Expansive Matrix Based number Systems
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 $d_{k-1}d_{k-2}\cdots d_1d_0$. 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 V\'avra 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.