Rational matrix digit systems

Let A be a d x d matrix with rational entries which has no eigenvalue lambda is an element of C of absolute value vertical bar lambda vertical bar < 1 and let Z(d) [A] be the smallest nontrivial A-invariant Z-module. We lay down a theoretical framework for the construction of digit systems (A,D), where D subset of Z(d) [A] finite, that admit finite expansions of the form x = d(0) + Ad(1) + ... +A(l-1) d(l-1) (l is an element of N, d(0), ... ,d(l-1) is an element of D) for every element x is an element of Z(d) [A]. We put special emphasis on the explicit computation of small digit sets 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.