A Matrix Analogue of Rational Number Systems

Abstract

Let $P,Q \in \mathbb{Z}^{d\times d}$ be invertible coprime matrices such that all the eigenvalues of $Q^{-1}P$ have modulus greater than 1, and $\mathbb{Z}^d[Q^{-1}P]$ be the smallest non-trivial $Q^{-1}P$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^d$. Suppose there is a finite digit set $\mathcal{D}\subseteq \mathbb{Z}^d + P\mathbb{Z}^d[Q^{-1}P]$ for which every vector $x \in \mathbb{Z}^d + P\mathbb{Z}^d[Q^{-1}P]$ can be represented in the form \[ x = \sum_{i=0}^{\ell-1} (Q^{-1}P)^i Q^{-1}d_i, \] where the digits $d_i \in \mathcal{D}$ for all $i \in \{0,1,\ldots,\ell-1\}$. We call such a representation a $P/Q$-expansion of $x$, and we say that the digit system $(P,Q,\mathcal{D})$ has the finiteness property. If, in addition, $\mathcal{D}$ is a complete set of residues of the quotient group $(\mathbb{Z}^d + P\mathbb{Z}^d[Q^{-1}P])/P\mathbb{Z}^d[Q^{-1}P]$, then the digits $d_0, d_1, \dots, d_{\ell-1}$ in the $P/Q$-expansion of $x$ are unique whenever $\ell \in \mathbb{Z}^+$ is minimal, and the resulting digit system is said to have the uniqueness property. We present sufficient conditions for the existence of a digit set $\mathcal{D}$ in which $(P,Q,\mathcal{D})$ has the finiteness property. For $d=2$, we make use of finite automata to construct digit systems $(P,Q,\mathcal{D})$ having both the finiteness and uniqueness properties. We also obtain the $P/Q$-expansion of a vector $x$ in $\mathbb{R}^d$ by means of the so-called expansion tree of the digit system $(P,Q,\mathcal{D})$.