# Pair correlation and equidistribution on manifolds

## TL;DR

This paper explains why sequences with Poisson pair correlation are uniformly distributed and extends the understanding to general Euclidean domains and Riemannian manifolds using a simple statistical argument.

## Contribution

It introduces a straightforward statistical approach that generalizes the link between pair correlation and equidistribution to broader geometric settings.

## Key findings

- Poisson pair correlation implies uniform distribution in various settings
- The argument extends to bounded Euclidean domains and Riemannian manifolds
- Provides a unified explanation for recent results in the field

## Abstract

This study is motivated by a series of recent papers that show that, if a given deterministic sequence in the unit interval has a Poisson pair correlation function, then the sequence is uniformly distributed. Analogous results have been proved for point sequences on higher-dimensional tori. The purpose of this paper is to describe a simple statistical argument that explains this observation and furthermore permits a generalisation to bounded Euclidean domains as well as compact Riemannian manifolds.