Optimal asymptotic bounds for spherical designs

TL;DR

This paper proves a conjecture that for any number of points above a certain threshold, there exists a spherical t-design on the sphere S^d, with the number of points proportional to t^d, establishing optimal asymptotic bounds.

Contribution

It establishes the existence of spherical t-designs with a number of points proportional to t^d, confirming a long-standing conjecture by Korevaar and Meyers.

Findings

Existence of spherical t-designs for N ≥ c_d t^d

Optimal asymptotic bounds for the number of points in spherical designs

Resolution of the Korevaar-Meyers conjecture

Abstract

In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending only on $d$.