Rational matrix digit systems
Jonas Jankauskas, Jörg Μ. Thuswaldner · Linear and Multilinear Algebra · 2022
Let A be a d×d matrix with rational entries which has no eigenvalue λ∈C of absolute value |λ|<1 and let Zd[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⊂Zd[A] finite, that admit finite expansions of the form x=d0+Ad1+⋯+Aℓ−1dℓ−1(ℓ∈N,d0,…,dℓ−1∈D)for every element x∈Zd[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.