On positional representation of integer vectors

TL;DR

This paper demonstrates that for any invertible integer matrix, there exists a finite digit set allowing every integer vector to be represented through a finite sum involving powers of the matrix, and characterizes when this representation is complete.

Contribution

It introduces a method to represent all integer vectors using finite digit sets and matrix powers, extending positional representation theory.

Findings

Existence of finite digit sets for matrix-based representations

Characterization of matrices with complete positional representations

Representation includes vectors in scaled integer lattices

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.