{
  "abstract": "An integer $d$-dimensional vector is usually represented by $d$ strings, which express the value of each component separately. This article is devoted to number systems that allow representing integer vectors using a single string of digits. This system is given by an integer square matrix $M$ and a finite set $\\mathcal{D}$ of symbols, called digits. We focus on full number systems, in which every vector $\\vec{x} \\in \\mathbb{Z}^n$ can be written in the form $\\vec{x}=\\sum_{i=0}^{k-1} M^i d_i$ with $d_i \\in \\mathcal{D}$ and thus represented by the string $d_{k-1}d_{k-2}\\cdots d_1d_0$. In contrast to the very well-described case where the base $M$ is expansive, we focus on systems in which $M$ is similar to a Jordan block with the eigenvalue $\\pm 1$. We follow the work of Caldwell, Hare, and Vávra who initiated the study of such systems. First, we address the question of choosing a minimal sized digit set $\\mathcal{D}$. Then we describe optimal representations for vectors from $\\mathbb{Z}^2$, both with respect to their length and the occurrence of non-zero digits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.04990",
  "author_total": 3,
  "authors": [
    {
      "name": "Adam Blažek"
    },
    {
      "name": "Kevin G. Hare"
    },
    {
      "name": "Edita Pelantová"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aast6mt1g6rrv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.04990",
  "fulltext_url": null,
  "id": "aast6mt1g6rrv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.04990",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7160535746",
  "pdf_url": "https://arxiv.org/pdf/2605.04990",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aast6mt1g6rrv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2605.04990",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.04990"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Non-Expansive Matrix Based number Systems",
  "title_text": "Non-Expansive Matrix Based number Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/aast6mt1g6rrv",
  "venue": "arXiv",
  "versions": []
}