  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.
