Poissonian pair correlation on manifolds via the heat kernel

TL;DR

This paper introduces a new concept of Poissonian pair correlation for Riemannian manifolds and proves that PPC ensures uniform distribution, extending previous work in the field.

Contribution

It generalizes the notion of PPC from Euclidean spaces to Riemannian manifolds and establishes its implication for uniform distribution.

Findings

PPC implies uniform distribution on Riemannian manifolds.

Extension of PPC concept from Euclidean spaces to manifolds.

Connects pair correlation with distribution properties in geometric settings.

Abstract

We define a notion of Poissonian pair correlation (PPC) for Riemannian manifolds without boundary and prove that PPC implies uniform distribution in this setting. This extends earlier work by Grepstad and Larcher, Aistleitner, Lachmann, and Pausinger, Steinerberger, and Marklof.