On positional representation of integer vectors

Abstract: We show that any $m\times m$ matrix $M$ with integer entries and $\det M =\Delta \neq 0$ can be equipped by a finite digit set $\mathcal{D}\subset\mathbb{Z}^m$ such that any integer $m$-dimensional vector belongs to the set $$ {\rm Fin}{\mathcal{D}}(M)= \Bigl{\sum{k\in I}M^k {d}k : \emptyset\neq I \text{ finite subset of } \mathbb{Z} \text{ and } {d}_k \in \mathcal{D} \text{ for each } k \in I\Bigr} \subset \bigcup\limits{k\in \mathbb{N}} \frac{1}{\Delta^k}\mathbb{Z}^{m} \,. $$ We also characterize the matrices $M$ for which the sets $ {\rm Fin}{\mathcal{D}}(M)$ and $ \bigcup\limits{k\in \mathbb{N}} \frac{1}{\Delta^k}\mathbb{Z}^{m}$ coincide.