To represent real $m$-dimensional vectors, a positional vector system given by a non-singular matrix $M \in \mathbb{Z}^{m \times m}$ and a digit set $\mathcal{D} \subset \mathbb{Z}^m$ is used. If $m = 1$, the system coincides with the well known numeration system used to represent real numbers. We study some properties of the vector systems which are transformable from the case $m = 1$ to higher dimensions. We focus on algorithm for parallel addition and on systems allowing an eventually periodic representation of vectors with rational coordinates.

翻译：为表示实数 $m$ 维向量，采用由非奇异矩阵 $M \in \mathbb{Z}^{m \times m}$ 和数字集 $\mathcal{D} \subset \mathbb{Z}^m$ 给出的位置向量系统。当 $m = 1$ 时，该系统即等同于用于表示实数的经典记数系统。本文研究可从一维情形推广到高维的向量系统特性，重点探讨并行加法算法及允许有理坐标向量具有最终周期表示的系统。