On positional representation of integer vectors

We show that any $m\times m$ matrix $M$ with integer entries and $\det M =Δ\neq 0$ 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 ${\rm Fin}_{\mathcal{D}}(M)= \Bigl\{\sum_{k\in I}M^k {d}_k : \emptyset\neq I \text{ finite subset of } \mathbb{Z} \text{ and } {d}_k \in \mathcal{D} \text{ for each } k \in I\Bigr\} \subset \bigcup\limits_{k\in \mathbb{N}} \frac{1}{Δ^k}\mathbb{Z}^{m} \,.$ We also characterize the matrices $M$ for which the sets ${\rm Fin}_{\mathcal{D}}(M)$ and $\bigcup\limits_{k\in \mathbb{N}} \frac{1}{Δ^k}\mathbb{Z}^{m}$ coincide.

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