A Matrix Analogue of Rational Number Systems

Abstract

Let P,Q ∈ ℤ^{d× d} be invertible coprime matrices such that all the eigenvalues of Q⁻¹P have modulus greater than 1, and ℤ^d[Q⁻¹P] be the smallest non-trivial Q⁻¹P-invariant ℤ-module containing ℤ^d. Suppose there is a finite digit set 𝒟⊆ ℤ^d + Pℤ^d[Q⁻¹P] for which every vector x ∈ ℤ^d + Pℤ^d[Q⁻¹P] can be represented in the form x = ∑_i=0^ℓ−1 (Q⁻¹P)ⁱ Q⁻¹d_i, where the digits d_i ∈ 𝒟 for all i ∈ {0,1,…,ℓ−1}. We call such a representation a P/Q-expansion of x, and we say that the digit system (P,Q,𝒟) has the finiteness property. If, in addition, 𝒟 is a complete set of residues of the quotient group (ℤ^d + Pℤ^d[Q⁻¹P])/Pℤ^d[Q⁻¹P], then the digits d₀, d₁, …, d_ℓ−1 in the P/Q-expansion of x are unique whenever ℓ ∈ ℤ⁺ is minimal, and the resulting digit system is said to have the uniqueness property. We present sufficient conditions for the existence of a digit set 𝒟 in which (P,Q,𝒟) has the finiteness property. For d=2, we make use of finite automata to construct digit systems (P,Q,𝒟) having both the finiteness and uniqueness properties.

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Cite

BibTeX RIS

BibTeX

@misc{cruz2026matrix,
  title = {{A Matrix Analogue of Rational Number Systems}},
  author = {Anjelo Gabriel R. Cruz and Manuel Joseph C. Loquias},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.08732},
  eprint = {2607.08732},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.08732},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A Matrix Analogue of Rational Number Systems
AU  - Anjelo Gabriel R. Cruz
AU  - Manuel Joseph C. Loquias
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2607.08732
UR  - https://arxiv.org/abs/2607.08732
AB  - Let P,Q ∈ ℤ^d× d be invertible coprime matrices such that all the eigenvalues of Q^−1P have modulus greater than 1, and ℤ^d[Q^−1P] be the smallest non-trivial Q^−1P-invariant ℤ-module containing ℤ^d. Suppose there is a finite digit set 𝒟⊆ ℤ^d + Pℤ^d[Q^−1P] for which every vector x ∈ ℤ^d + Pℤ^d[Q^−1P] can be represented in the form x = ∑_i=0^ℓ−1 (Q^−1P)^i Q^−1d_i, where the digits d_i ∈ 𝒟 for all i ∈ {0,1,…,ℓ−1}. We call such a representation a P/Q-expansion of x, and we say that the digit system (P,Q,𝒟) has the finiteness property. If, in addition, 𝒟 is a complete set of residues of the quotient group (ℤ^d + Pℤ^d[Q^−1P])/Pℤ^d[Q^−1P], then the digits d_0, d_1, …, d_ℓ−1 in the P/Q-expansion of x are unique whenever ℓ ∈ ℤ^+ is minimal, and the resulting digit system is said to have the uniqueness property. We present sufficient conditions for the existence of a digit set 𝒟 in which (P,Q,𝒟) has the finiteness property. For d=2, we make use of finite automata to construct digit systems (P,Q,𝒟) having both the finiteness and uniqueness properties.
ER  -

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