{
  "abstract": "The Delange function of base $3$ is the continuous function $T(x)=\\sum_{j\\ge0}3^{-j}τ(3^jx)$ of Takagi type, where $τ(x)=2\\min\\{\\mathrm{dist}(x,\\mathbb{Z}),1/3\\}$. Multiplied by $3^n$, its values at the points $m/3^n$ are the smallest numbers $ξ_n(m)=2(nm-S_3(m))$ of edges leaving a set of $m$ vertices in the ternary Hamming cube $Q_n^3$, where $S_3(m)=\\sum_{j<m}s_3(j)$ and $s_3$ is the ternary digit sum. Both facts are known. We show that the local minima of $T$ are exactly the ternary rationals $m/3^n$, and that the local maxima are exactly the points $(m+1/2)/3^n+k$ with integers $k$, $m$, $n$ such that $n\\ge0$, $0\\le m<3^n$ and $s_3(m)=n$. We also characterize, in terms of the ternary digits $ξ_1,ξ_2,\\ldots$ of $x$, when $T$ has an infinite derivative at $x$ from one or both sides. At a ternary rational the right derivative is $+\\infty$ and the left derivative is $-\\infty$. At every other point, the right derivative is $+\\infty$ exactly when $\\sum_{j\\le n}(1-ξ_j)\\to+\\infty$. The condition for the left derivative to be $+\\infty$ is stronger and also involves the lengths of the maximal strings of zeros among the digits. For the value $-\\infty$ the two sides are exchanged and the roles of zeros and twos are swapped. In particular, $T$ has an infinite derivative at every rational point that is not a ternary rational and whose period has different numbers of zeros and twos. We also determine the maximum of $ξ_n(m)$ over every block of sizes $a3^k\\le m\\le(a+1)3^k$, describe by the ternary digits of $h$ the sizes $h$ for which the $h$-extra edge connectivity of $Q_n^3$ equals $ξ_n(h)$, and compute the Fourier coefficients of $T$ and its variation at each level. Finally, we determine the level set at height $3/4$, which has dimension $\\log2/\\log9$, and show that no level set has a larger upper box dimension and that $\\{x\\in[0,1]:T(x)=y\\}$ is finite for almost every $y$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21265",
  "author_total": 1,
  "authors": [
    {
      "name": "Chengyi Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e072ys5dcqjzt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21265",
  "fulltext_url": null,
  "id": "e072ys5dcqjzt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21265",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7222555683",
  "pdf_url": "https://arxiv.org/pdf/2610.21265",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e072ys5dcqjzt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.21265",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21265"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Local Extrema and Infinite Derivatives of the Delange Function of Base 3",
  "title_text": "Local Extrema and Infinite Derivatives of the Delange Function of Base 3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/e072ys5dcqjzt",
  "venue": "arXiv",
  "versions": []
}