{
  "abstract": "A closed rectifiable curve on the torus $\\mathbb T^d=\\mathbb R^d/\\mathbb Z^d$ is a $t$-design curve if normalized arclength integration is exact for trigonometric polynomials $e^{2πi \\langle m,x\\rangle}$ with frequencies $m\\in\\mathbb Z^d$ satisfying $\\|m\\|_\\infty\\le t$. For the arclength $L$ of such curves, we establish the sharp lower bound \\begin{equation}\\label{eq:abs} L\\geq \\Big(\\frac{(t+1)^{2d}-1}{t(t+2)}\\Big)^{1/2}. \\end{equation} Equality is attained by the family of Korobov curves described in \\cite{Kunis,Lindblad}. For even $t$, we also derive an explicit lower bound on the length of total-degree $t$-design curves, defined by the frequency condition $\\|m\\|_1\\leq t$. This bound is sharp for $t=2$ in every dimension and for every even $t$ in dimension two. We determine the minimum length of hyperbolic cross $t^2$-design curves in dimension two. For $d\\geq3$, we establish lower bounds of optimal order for the length of hyperbolic cross $t^d$-design curves. Finally, we obtain lower bounds of optimal order for the length of $t$-design curves associated with sparse frequency sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03362",
  "author_total": 1,
  "authors": [
    {
      "name": "Martin Ehler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55tmw4pg4hb3w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03362",
  "fulltext_url": null,
  "id": "55tmw4pg4hb3w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03362",
  "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.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03362",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55tmw4pg4hb3w.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03362",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03362"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Optimal bounds for design curves on the torus",
  "title_text": "Optimal bounds for design curves on the torus",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/55tmw4pg4hb3w",
  "venue": "arXiv",
  "versions": []
}