{
  "abstract": "We study how many bits are needed to write down an elementary analytic expression that uniformly approximates a Lipschitz function. Expressions are finite trees over the basis $\\{+,-,\\times,÷,\\exp\\}$ with the variables and rational constants, and their size is $\\dc(f)$, the length of the shortest description of $f$ in a fixed binary description language. For functions of one variable we prove that every $L$-Lipschitz $F:[0,1]\\to[0,1]$ has an everywhere-defined approximant of accuracy $Δ$ with $\\dc(f)\\le C\\,(L/Δ+\\lg(1/Δ+2))$ for all $L,Δ>0$, by a linear spline smoothed with logistic ramps in a description language with one repetition construct. We prove a matching lower bound valid for \\emph{every} description language: for $Δ\\le 1/36$ and every $L>0$ some $L$-Lipschitz function needs at least $\\tfrac1{16}(L/Δ+\\lg(1/Δ+2))$ bits. Hence the worst-case description complexity is $Θ(L/Δ+\\lg(1/Δ+2))$. For functions on $[0,1]^d$ the same method gives $Θ_d\\bigl((L/Δ)^d+\\lg(1/Δ+2)\\bigr)$, using a tensor-product spline whose coefficients are mixed second differences and cost $O(d)$ bits each. Without the repetition construct the upper bounds carry an additional logarithmic factor, whose necessity for plain expression trees is left open. A reference encoder, decoder and evaluator confirm the constructions numerically, and figures and tables report the checks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14893",
  "author_total": 1,
  "authors": [
    {
      "name": "Artem Vozmishchev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7bjadg8fbrcpw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14893",
  "fulltext_url": null,
  "id": "7bjadg8fbrcpw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14893",
  "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",
      "cs.CL"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7221825451",
  "pdf_url": "https://arxiv.org/pdf/2610.14893",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7bjadg8fbrcpw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14893",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14893"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Description Complexity Bounds for Analytic Approximations of Lipschitz Functions",
  "title_text": "Description Complexity Bounds for Analytic Approximations of Lipschitz Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/7bjadg8fbrcpw",
  "venue": "arXiv",
  "versions": []
}