2018/01/05 by Jonas Jankauskas, Jankauskas, Jonas, Jörg Μ. Thuswaldner +1 · 1 citation
Computer Science · Mathematics · #11A63 (Primary) 11K16 #11C20 #11R04 #13G05 #15B36 #37A45 (Secondary) #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1801.01839
openalex publication_date 2018/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be an n \× n matrix with rational entries and let n
mathbbZn[A] :=
bigcupk=1
infty
left(
mathbbZn + A
mathbbZn +\n
dots + Ak-1
mathbbZn
right) be the minimal A-invariant\n\ℤ-module containing the lattice \ℤn. If\n\D\⊂\ℤn[A] is a finite set we call the pair\n(A,\D) a digit system. We say that (A,\D) has the\nfiniteness property if each \z \∈ \ℤn[A] can be written in\nthe form \
mathbfz =
mathbfd0 + A
mathbfd1 +
dots +\nAk
mathbfdk, with k\∈\ℕ and digits \dj \∈\n\D for 0\≤ j\≤ k. We prove that for a given matrix A \∈\nMn(\ℚ) there is a finite set \D\⊂\ℤn[A] such\nthat (A, \D) has the finiteness property if and only if A has no\neigenvalue of absolute value < 1. This result is the matrix analogue of the\nheight reducing property of algebraic numbers. In proving this result we also\ncharacterize integer polynomials P \∈ \ℤ[x] that admit digit systems\nhaving the finiteness property in the quotient ring \ℤ[x]/(P).\n