A Matrix Analogue of Rational Number Systems

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 $