{
  "abstract": "A spherical $t$-design is a finite set of points on the $d$-dimensional unit sphere such that averaging any polynomial of degree at most $t$ over these points gives its exact spherical average. We study the corresponding notion for curves, where point averages are replaced by normalized arclength integration. For fixed $d\\ge 2$, the length of such curves is known to be bounded below by a positive multiple of $t^{d-1}$ as $t\\to\\infty$. We show that this asymptotic order is attained: for every $d\\ge 2$, there exists a constant $C_d>0$ such that, for every positive integer $t$ and every prescribed length $L\\ge C_dt^{d-1}$, there is a simple spherical $t$-design curve of length $L$. This establishes the analogue for curves of the theorem of Bondarenko, Radchenko, and Viazovska, which resolved the Korevaar-Meyers conjecture for spherical design points. More generally, we prove the corresponding result on closed, connected Riemannian manifolds of dimension $d\\ge 2$, where eigenfunctions of the Laplace-Beltrami operator substitute for polynomials. The proof combines a geometric construction of curves with continuously adjustable local lengths and an application of Brouwer's degree theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01895",
  "author_total": 1,
  "authors": [
    {
      "name": "Martin Ehler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51ew54a2zpj3t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01895",
  "fulltext_url": null,
  "id": "51ew54a2zpj3t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01895",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01895",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51ew54a2zpj3t.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01895",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01895"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Optimal asymptotic bounds for design curves on manifolds",
  "title_text": "Optimal asymptotic bounds for design curves on manifolds",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/51ew54a2zpj3t",
  "venue": "arXiv",
  "versions": []
}