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Spherical t-Designs on 𝕊2 with 54t2 Points

  • Zhiqiang Xu
  • Zili Xu
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Abstract

We prove that, for every integer t≥ 1, the unit two-sphere admits a spherical t-design consisting of exactly 54t2 points. More generally, such a design exists with exactly 6q2 points for every integer q≥ 3t. The proof builds on the topological degree method of Bondarenko, Radchenko and Viazovska, using an explicit equal-area partition of the sphere based on the map of Roşca and Plonka. By choosing the cell centers to minimize the average squared geodesic distance and deriving sharper sampling estimates, we obtain the stated quadratic bound on the size of spherical t-designs on Sph.

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Cite

BibTeXRIS
BibTeX
@misc{xu2026spherical,
  title = {{Spherical $t$-Designs on $\mathbb S^2$ with $54t^2$ Points}},
  author = {Zhiqiang Xu and Zili Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15016},
  eprint = {2609.15016},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15016},
}
RIS
TY  - UNPB
TI  - Spherical t-Designs on 𝕊^2 with 54t^2 Points
AU  - Zhiqiang Xu
AU  - Zili Xu
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15016
UR  - https://arxiv.org/abs/2609.15016
AB  - We prove that, for every integer t≥ 1, the unit two-sphere admits a spherical t-design consisting of exactly 54t^2 points. More generally, such a design exists with exactly 6q^2 points for every integer q≥ 3t. The proof builds on the topological degree method of Bondarenko, Radchenko and Viazovska, using an explicit equal-area partition of the sphere based on the map of Roşca and Plonka. By choosing the cell centers to minimize the average squared geodesic distance and deriving sharper sampling estimates, we obtain the stated quadratic bound on the size of spherical t-designs on Sph.
ER  -