Fixed-strength spherical designs

TL;DR

This paper investigates fixed-degree spherical designs in high dimensions, providing new constructions, exact size estimates, and methods for dimension projection, advancing understanding of their asymptotic behavior.

Contribution

It introduces explicit constructions linking Gaussian and spherical designs, determines the order of minimal signed $t$-designs, and develops a dimension projection technique.

Findings

Smaller spherical designs constructed via Gaussian design connection

Exact order of magnitude for minimal signed $t$-designs established

New methods for projecting designs between dimensions

Abstract

A spherical $t$-design is a finite subset $X$ of the unit sphere such that every polynomial of degree at most $t$ has the same average over $X$ as it does over the entire sphere. Determining the minimum possible size of spherical designs, especially in a fixed dimension as $t \to \infty$, has been an important research topic for several decades. This paper presents results on the complementary asymptotic regime, where $t$ is fixed and the dimension tends to infinity. The main results in this paper are (1) a construction of smaller spherical designs via an explicit connection to Gaussian designs and (2) the exact order of magnitude of minimal-size signed $t$-designs, which is significantly smaller than predicted by a typical degrees-of-freedom heuristic. We also establish a method to ``project'' spherical designs between dimensions, prove a variety of results on approximate designs, and construct new $t$-wise independent subsets of $\{1,2,\dots,q\}^d$ which may be of independent interest. To achieve these results, we combine techniques from algebra, geometry, probability, representation theory, and optimization.