Stolarsky's invariance principle for projective spaces, II

TL;DR

This paper provides an alternative analytic proof of Stolarsky's invariance principle extended to various projective spaces, emphasizing the role of spherical functions in the proof.

Contribution

It introduces an analytic proof of Stolarsky's invariance principle for projective spaces using spherical functions, complementing the geometric approach.

Findings

Extension of Stolarsky's principle to projective spaces

Analytic proof leveraging spherical functions

Reinforcement of geometric results with harmonic analysis

Abstract

It was proved in the first part of this work \cite{0} that Stolarsky's invariance principle, known previously for point distributions on the Euclidean spheres \cite{33}, can be extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane. The geometric features of these spaces have been used very essentially in the proof. In the present paper, relying on the theory of spherical functions on such spaces, we give an alternative analytic proof of Stolarsky's invariance principle.