Spherical $t$-Designs on $\mathbb S^2$ with $54t^2$ Points

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 Ro\c{s}ca 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 $\Sph$.