Spherical functions and Stolarski's invariance principle

TL;DR

This paper provides a new analytic proof of Stolarsky's invariance principle extended to various projective spaces, using spherical functions on compact symmetric Riemannian manifolds of rank one.

Contribution

It introduces a purely analytic proof of the extended invariance principle, leveraging the theory of spherical functions on rank one symmetric spaces.

Findings

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

Developed a new analytic proof based on spherical functions.

Enhanced understanding of geometric and algebraic structures in these spaces.

Abstract

In the previous paper [25], Stolarsky's invariance principle, known for point distributions on the Euclidean spheres [27], has been extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane. Geometric features of these spaces as well as their models in terms of Jordan algebras have been used very essentially in the proof. In the present paper, we give a new pure analytic proof of the extended Stolarsky's invariance principle, relying on the theory of spherical functions on compact symmetric Riemannian manifolds of rank one.