On positional representation of integer vectors

Abstract

We show that any $m\times m$ matrix $M$ with integer entries and $\det M=\Delta \ne 0$ 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 $${\mathrm{Fin}}_{\mathcal{D}}(M)=\{\sum _{k\in I}{M}^k{d}_k:\varnothing \ne I\text{ finite subset of }\mathbb{Z}\text{ and }{d}_k\in \mathcal{D}\text{ for each }k\in I\}\subset \bigcup _{k\in \mathbb{N}}\frac{1}{{\Delta }^k}{\mathbb{Z}}^m.$$ We also characterize the matrices $M$ for which the sets ${\mathrm{Fin}}_{\mathcal{D}}(M)$ and $\bigcup _{k\in \mathbb{N}}\frac{1}{{\Delta }^k}{\mathbb{Z}}^m$ coincide.