Asymptotically optimal designs on compact algebraic manifolds

TL;DR

This paper constructs nearly optimal t-designs on compact algebraic manifolds, including Grassmannians, with a number of points comparable to the polynomial space dimension, extending and improving previous sphere results.

Contribution

It generalizes the construction of t-designs from spheres to broader algebraic manifolds, notably enhancing bounds for Grassmannians.

Findings

Constructed t-designs with point counts matching polynomial space dimensions

Extended sphere t-design results to general algebraic manifolds

Improved bounds for t-designs on Grassmannians

Abstract

We find t-designs on compact algebraic manifolds with a number of points comparable to the dimension of the space of polynomials of degree t on the manifold. This generalizes results on the sphere by Bondarenko, Radchenko and Viazovska. Of special interest is the particular case of the Grassmannians where our results improve the bounds that had been proved previously.