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PreprintCC BY 4.0arXiv

Optimal asymptotic bounds for design curves on manifolds

  • Martin Ehler
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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≥ 2, the length of such curves is known to be bounded below by a positive multiple of td−1 as t→∞. We show that this asymptotic order is attained: for every d≥ 2, there exists a constant Cd>0 such that, for every positive integer t and every prescribed length L≥ Cdtd−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≥ 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.

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Cite

BibTeXRIS
BibTeX
@misc{ehler2026optimal,
  title = {{Optimal asymptotic bounds for design curves on manifolds}},
  author = {Martin Ehler},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01895},
  eprint = {2611.01895},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01895},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Optimal asymptotic bounds for design curves on manifolds
AU  - Martin Ehler
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01895
UR  - https://arxiv.org/abs/2611.01895
AB  - 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≥ 2, the length of such curves is known to be bounded below by a positive multiple of t^d−1 as t→∞. We show that this asymptotic order is attained: for every d≥ 2, there exists a constant C_d>0 such that, for every positive integer t and every prescribed length L≥ 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≥ 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.
ER  -