Rational matrix digit systems

TL;DR

This paper develops a theoretical framework for constructing digit systems based on rational matrices that allow finite expansions of elements, with methods for explicitly computing small digit sets and unifying finiteness results across different algebraic structures.

Contribution

It introduces a new framework for digit systems with rational matrices, providing explicit methods for computing digit sets and unifying finiteness results in various algebraic contexts.

Findings

Framework for digit systems with rational matrices

Explicit methods for small digit set computation

Unified proof of finiteness properties

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.