{
  "abstract": "In this paper, we study a numeration system with a non--integer base $a>1$ over the binary alphabet $A=\\{0,1\\}$: $$\\left[0;\\frac{r}{a-1}\\right]\\ni x=\\sum\\limits_{n=1}^{\\infty}\\frac{α_n}{a^n} \\equiv Δ^{r_a}_{α_1α_2\\ldotsα_n\\ldots}, \\quad α_n\\in A.$$ We investigate the geometry of $r_a$--representation of numbers, including the geometric interpretation of digits, the structure of cylinder overlaps, and the associated metric properties. The main object of our study is a nowhere monotone function of unbounded variation defined by $$f\\left(x=\\sum\\limits_{n=1}^{\\infty}\\frac{α_n}{2^n}\\right) =\\sum\\limits_{n=1}^{\\infty}\\frac{α_n}{a^n},$$ where the classical binary representation of the argument is assumed not to end with the infinite period $(1)$. We derive a system of two functional equations satisfied by $f$. We prove that the function has fractal level sets and a self--affine graph. We also evaluate the integral of $f$ over the interval $[0,1]$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34746",
  "author_total": 3,
  "authors": [
    {
      "name": "Svitlana Vaskevych"
    },
    {
      "name": "Yuliia Vovk"
    },
    {
      "name": "Oleksandr Pratsiovytyi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0m7trecwg690x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34746",
  "fulltext_url": null,
  "id": "0m7trecwg690x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34746",
  "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.DS"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214703819",
  "pdf_url": "https://arxiv.org/pdf/2609.34746",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0m7trecwg690x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34746",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34746"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Some applications of binary numeration systems with nonzero redundancy to the theory of locally complex functions",
  "title_text": "Some applications of binary numeration systems with nonzero redundancy to the theory of locally complex functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/0m7trecwg690x",
  "venue": "arXiv",
  "versions": []
}