On the existence and estimates of nested spherical designs

TL;DR

This paper proves the existence of nested spherical designs by adding points to existing sets and provides estimates on the number of points needed, advancing understanding of spherical design construction.

Contribution

It introduces a method to construct nested spherical designs from arbitrary point sets and provides bounds on the number of points required for such designs.

Findings

Existence of nested spherical designs established.

Upper bounds on the number of points needed are derived.

Discussion on optimal order of nested spherical designs included.

Abstract

In this paper, we prove the existence of a spherical $t$-design formed by adding extra points to an arbitrarily given point set on the sphere and, subsequently, deduce the existence of nested spherical designs. Estimates on the number of required points are also given. For the case that the given point set is a spherical $t_1$-design such that $t_1 < t$ and the number of points is of optimal order $t_1^d$, we show that the upper bound of the total number of extra points and given points for forming nested spherical $t$-design is of order $t^{2d+1}$. A brief discussion concerning the optimal order in nested spherical designs is also given.