Characterization of rational matrices that admit finite digit representations

Let A be an n×n matrix with rational entries and let Zn[A]:=⋃k=1∞(Zn+AZn+…+Ak−1Zn) be the minimal A-invariant Z-module containing the lattice Zn. If D⊂Zn[A] is a finite set we call the pair (A,D)a digit system. We say that (A,D) has the finiteness property if each z∈Zn[A] can be written in the form z=d0+Ad1+…+Akdk, with k∈N and digitsdj∈D for 0≤j≤k. We prove that for a given matrix A∈Mn(Q) there is a finite set D⊂Zn[A] such that (A,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∈Z[x] that admit digit systems having the finiteness property in the quotient ring Z[x]/(P).