Year: 2026
Type: preprint
Abstract: We prove that, for every integer $t\geq 1$, 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 $q\geq 3t$. 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 $\Sph$.
Source: arXiv (Cornell University)
Authors: Xu, Zhiqiang, Xu, Zili
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213335103