Abstract
Let A be a matrix with rational entries which has no eigenvalue of absolute value and let be the smallest nontrivial A-invariant -module. We lay down a theoretical framework for the construction of digit systems , where finite, that admit finite expansions of the form for every element . We put special emphasis on the explicit computation of small digit sets 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.
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