Rational matrix digit systems

2021/07/29 by Jonas Jankauskas, Jankauskas, Jonas, Jörg Μ. Thuswaldner +1 · 1 citation
Computer Science · Mathematics · #11A63 #11C20 #11H06 #11P21 #15A30 #15B10 #52A40 #90C05 #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2107.14168

openalex publication_date 2021/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be a d × d matrix with rational entries which has no eigenvalue λ∈ ℂ of absolute value |λ| < 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⊂ ℤ^d[A] finite, that admit finite expansions of the form x= d₀ + A d₁ + ⋯ + A^ℓ-1d_ℓ-1 (ℓ∈ ℕ, d₀,…,d_ℓ-1 ∈ D) for every element x∈ 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.

Cited by

• Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices

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