On the Pair Correlation Statistic of Sequences with Finite Gap Property

TL;DR

This paper investigates the pair correlation function of sequences on the torus with finitely many gap lengths, showing that the limiting function cannot be continuous in such cases, thus contributing to understanding the possible forms of pair correlation functions.

Contribution

It provides a partial answer to which functions can appear as pair correlation limits, specifically proving that the limit cannot be continuous for sequences with finitely many gap lengths.

Findings

The pair correlation limit function cannot be continuous if the sequence has finitely many gap lengths.

Sequences with finitely many gaps have restrictions on their pair correlation functions.

The result advances understanding of the structure of sequences with prescribed pair correlation properties.

Abstract

The limiting function $f(s)$ of the pair correlation \[ \frac{1}{N} \# \left\{ 1 \leq i\neq j\leq N \middle\vert \left\lVert x_i - x_j \right\rVert \leq \frac{s}{N} \right\} \] for a sequence $(x_N)_{N \in \mathbb{N}}$ on the torus $\mathbb{T}^1$ is said to be Poissonian if it exists and equals $2s$ for all $s \geq 0$. For instance, independent, uniformly distributed random variables generically have this property. Obviously $f(s)$ is always a monotonic function if existent. There are only few examples of sequences where $f(s) \neq 2s$, but where the limit can still be explicitly calculated. Therefore, it is an open question which types of functions $f(s)$ can or cannot appear here. In this note, we give a partial answer on this question by addressing the case that the number of different gap lengths in the sequence is finite and showing that $f$ cannot be continuous then.