{
  "abstract": "Let $P,Q \\in \\mathbb{Z}^{d\\times d}$ be invertible coprime matrices such that all the eigenvalues of $Q^{-1}P$ have modulus greater than 1, and $\\mathbb{Z}^d[Q^{-1}P]$ be the smallest non-trivial $Q^{-1}P$-invariant $\\mathbb{Z}$-module containing $\\mathbb{Z}^d$. Suppose there is a finite digit set $\\mathcal{D}\\subseteq \\mathbb{Z}^d + P\\mathbb{Z}^d[Q^{-1}P]$ for which every vector $x \\in \\mathbb{Z}^d + P\\mathbb{Z}^d[Q^{-1}P]$ can be represented in the form \\[ x = \\sum_{i=0}^{\\ell-1} (Q^{-1}P)^i Q^{-1}d_i, \\] where the digits $d_i \\in \\mathcal{D}$ for all $i \\in \\{0,1,\\ldots,\\ell-1\\}$. We call such a representation a $P/Q$-expansion of $x$, and we say that the digit system $(P,Q,\\mathcal{D})$ has the finiteness property. If, in addition, $\\mathcal{D}$ is a complete set of residues of the quotient group $(\\mathbb{Z}^d + P\\mathbb{Z}^d[Q^{-1}P])/P\\mathbb{Z}^d[Q^{-1}P]$, then the digits $d_0, d_1, \\dots, d_{\\ell-1}$ in the $P/Q$-expansion of $x$ are unique whenever $\\ell \\in \\mathbb{Z}^+$ 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 $\\mathcal{D}$ in which $(P,Q,\\mathcal{D})$ has the finiteness property. For $d=2$, we make use of finite automata to construct digit systems $(P,Q,\\mathcal{D})$ having both the finiteness and uniqueness properties.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.08732",
  "author_total": 2,
  "authors": [
    {
      "name": "Anjelo Gabriel R. Cruz"
    },
    {
      "name": "Manuel Joseph C. Loquias"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bnhvecnpdkqxw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.08732",
  "fulltext_url": null,
  "id": "bnhvecnpdkqxw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.08732",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7167927178",
  "pdf_url": "https://arxiv.org/pdf/2607.08732",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bnhvecnpdkqxw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2607.08732",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.08732"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Matrix Analogue of Rational Number Systems",
  "title_text": "A Matrix Analogue of Rational Number Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/bnhvecnpdkqxw",
  "venue": "arXiv",
  "versions": []
}