Stolarsky's invariance principle for projective spaces

TL;DR

This paper extends Stolarsky's invariance principle from Euclidean spheres to real, complex, quaternionic, and octonionic projective spaces, providing explicit formulas for constants involved.

Contribution

It introduces new explicit formulas for invariance principles in various projective spaces, expanding the known scope of Stolarsky's principle.

Findings

Extended invariance principle to multiple projective spaces

Derived explicit formulas for constants in the invariance principles

Confirmed the applicability of the principle beyond Euclidean spheres

Abstract

We show that Stolarsky's invariance principle, known for point distributions on the Euclidean spheres, can be extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane. A part of the results (Theorem~1.1 ana Corollary~1.1) was given early in the previous paper [22], while the explicit formulas for the constants in the invariance principles for projective spaces (Theorem~1.2 and Corollary~1.2) are new.