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 $\mathbb{ Z}[x]/(P)$.
Categories: math.NT