{
  "abstract": "As is known, the $A_2$-continued representation of numbers is not topologically equivalent to the classical binary representation; therefore, the digit projector of such representations is a discontinuous function. In this paper, we introduce a continuous function that serves as an analogue of the digit projector of the classical binary representation of numbers into the digits of the $A_2$-continued representation with zero redundancy, namely a function of the form \\[f(Δ^2_{α_1α_2...α_{2n-1}α_{2n}...})= Δ^{A_2}_{(\\frac{1}{2})^{1-α_1}(\\frac{1}{2})^{α_2}... (\\frac{1}{2})^{1-α_{2n-1}}(\\frac{1}{2})^{α_{2n}}...}, α_n\\in \\{0,1\\}.\\] It is proved that the function $f$ is well-defined, continuous, and monotone. Using the normal properties of numbers with respect to their binary representation and Lebesgue's theorem asserting the existence of a finite derivative for a continuous monotone function almost everywhere, the singularity of the function $f$ is established. The paper also establishes a relationship between the considered function, the right-shift operator on digits, and the inversor of the continued representation of numbers. This relationship is then used to establish the singularity of the inversor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15158",
  "author_total": 2,
  "authors": [
    {
      "name": "O. O. Nikorak"
    },
    {
      "name": "S. P. Ratushniak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1nntg6s6ykfzw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15158",
  "fulltext_url": null,
  "id": "1nntg6s6ykfzw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15158",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7203718356",
  "pdf_url": "https://arxiv.org/pdf/2608.15158",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1nntg6s6ykfzw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.15158",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15158"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Continuous digit projector from binary representations of numbers onto $A_2$-representation",
  "title_text": "A Continuous digit projector from binary representations of numbers onto A_2-representation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/1nntg6s6ykfzw",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.31861/bmj2025.02.05",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}