From positional representation of numbers to positional representation of vectors

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.