Fixed-strength spherical designs

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. We combine techniques from algebra, geometry, and probability to prove upper bounds on the size of designs, including an optimal bound for signed spherical designs. We also prove lower and upper bounds for approximate designs and establish an explicit connection between spherical and Gaussian designs.