Point distribution in compact metric spaces, III. Two-point homogeneous spaces

TL;DR

This paper extends discrepancy and distance sum bounds, including Stolarsky's invariance principle, to all two-point homogeneous spaces, providing optimal bounds for point distributions and applications to t-designs.

Contribution

It generalizes Stolarsky's invariance principle to all projective and octonionic spaces, deriving bounds for discrepancies and distances in these spaces.

Findings

Extended Stolarsky's invariance principle to all projective spaces and octonionic plane

Derived spherical function expansions for discrepancies and distances

Proved optimal bounds for quadratic discrepancies and pairwise distances

Abstract

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known, they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. Our concern is with discrepancies of distributions in metric balls and sums of pairwise distances between points of distributions in such spaces. Using the geometric features of two-point spaces, we show that Stolarsky's invariance principle, well-known for the Euclidean spheres, can be extended to all projective spaces and the octonionic projective plane (Theorem 2.1 and Corollary 2.1). We obtain the spherical function expansions for discrepancies and sums of distances (Theorem 9.1). Relying on these expansions, we prove in all such spaces the best possible bounds for quadratic discrepancies and sums of pairwise distances (Theorem 2.2). Applications to $t$-designs on such two-point homogeneous spaces are also considered. It is shown that the optimal $t$-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances. (Corollaries 3.1 and 3.2).