Papers › Rational matrix digit systems
Rational matrix digit systems
Jonas Jankauskas, Jörg M. Thuswaldner
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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 + A 𝐝₁ + ⋯+ A^(ℓ-1)𝐝_(ℓ-1) (ℓ∈ℕ, 𝐝₀,…,𝐝_(ℓ-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.
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