What question did this study set out to answer?

This research aims to explore the pair correlation statistics of non-uniform sequences, specifically identifying characteristics that contrast with the Poissonian property.

June 20, 2026

An Explicit non-Poissonian Pair Correlation Function

Key Points

This research aims to explore the pair correlation statistics of non-uniform sequences, specifically identifying characteristics that contrast with the Poissonian property.
Investigated pair correlation statistics in a non-uniform distribution context.
Analyzed the gap structure of a standard low-dispersion sequence.
Compared findings with existing weak pair correlation functions.
Identified the first explicit example of a non-Poissonian pair correlation function.
Highlighted key differences between non-Poissonian and weak pair correlations, indicating a linear relationship.
Demonstrated that many deterministic sequences fail to exhibit Poissonian pair correlation properties.

Abstract

A generic uniformly distributed random sequence on the unit interval has Poissonian pair correlations. Usually, the pair correlations statistic is therefore studied for equidistributed sequences. At the same time, there are only very few explicitly known examples of sequences with this property and many types of deterministic sequences have been proven to fail having the Poissonian pair correlation property. In this paper we study the pair correlation statistic in the non-uniform case and analyze the first elementary example of such a sequence, namely Formula: see text, which is a standard low-dispersion sequence. The proof heavily relies on a full understanding of the gap structure of Formula: see text. Furthermore, we discuss differences to the weak pair correlation function which turns out to be linear.

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