Characterization of rational matrices that admit finite digit representations

TL;DR

This paper characterizes rational matrices that allow finite digit representations, showing such representations exist if and only if the matrix has no eigenvalues with absolute value less than one, extending height reducing properties.

Contribution

It provides a complete characterization of matrices over rationals that admit finite digit systems, linking eigenvalues to the existence of such representations.

Findings

Finite digit systems exist iff eigenvalues have absolute value ≥ 1.

Characterization of polynomials admitting finite digit systems in quotient rings.

Extension of height reducing properties to matrix settings.

Abstract

Let $A$ be an $n \times n$ matrix with rational entries and let \[ \mathbb{Z}^n[A] := \bigcup_{k=1}^{\infty} \left( \mathbb{Z}^n + A\mathbb{Z}^n + \dots + A^{k-1}\mathbb{Z}^n\right) \] be the minimal $A$-invariant $\mathbb{Z}$-module containing the lattice $\mathbb{Z}^n$. If $\mathcal{D}\subset\mathbb{Z}^n[A]$ is a finite set we call the pair $(A,\mathcal{D})$ a digit system. We say that $(A,\mathcal{D})$ has the finiteness property if each $\mathbf{z} \in \mathbb{Z}^n[A]$ can be written in the form \[ \mathbf{z} = \mathbf{d}_0 + A\mathbf{d}_1 + \dots + A^k\mathbf{d}_k, \] with $k\in\mathbb{N}$ and digits $\mathbf{d}_j \in \mathcal{D}$ for $0\le j\le k$. We prove that for a given matrix $A \in M_n(\mathbb{Q})$ there is a finite set $\mathcal{D}\subset\mathbb{Z}^n[A]$ such that $(A, \mathcal{D})$ has the finiteness property if and only if $A$ has no eigenvalue of absolute value $< 1$. This result is the matrix analogue of the height reducing property of algebraic numbers. In proving this result we also characterize integer polynomials $P \in \mathbb{Z}[x]$ that admit digit systems having the finiteness property in the quotient ring $\mathbb{Z}[x]/(P)$.