Rational matrix digit systems

Abstract

Let $A$ be a $d \times d$ matrix with rational entries which has no eigenvalue $\lambda \in \mathbb{C}$ of absolute value $|\lambda| < 1$ and let $\mathbb{Z}^d[A]$ be the smallest nontrivial $A$-invariant $\mathbb{Z}$-module. We lay down a theoretical framework for the construction of digit systems $(A, \mathcal{D})$, where $\mathcal{D}\subset \mathbb{Z}^d[A]$ finite, that admit finite expansions of the form \[ \mathbf{x}= \mathbf{d}_0 + A \mathbf{d}_1 + \cdots + A^{\ell-1}\mathbf{d}_{\ell-1} \qquad(\ell\in \mathbb{N},\;\mathbf{d}_0,\ldots,\mathbf{d}_{\ell-1} \in \mathcal{D}) \] for every element $\mathbf{x}\in \mathbb{Z}^d[A]$. We put special emphasis on the explicit computation of small digit sets $\mathcal{D}$ that admit this property for a given matrix $A$, using techniques from matrix theory, convex geometry, and the Smith Normal Form. Moreover, we provide a new proof of general results on this finiteness property and recover analogous finiteness results for digit systems in number fields a unified way.