# Sequences with almost Poissonian Pair Correlations

## TL;DR

This paper investigates the pair correlation properties of specific uniformly distributed sequences, showing that van der Corput and Kronecker sequences for the golden mean nearly have Poissonian pair correlations, but not exactly.

## Contribution

It demonstrates that these classical sequences have almost Poissonian pair correlations, filling a gap in understanding their distributional properties.

## Key findings

- Van der Corput sequences have $eta$-pair correlations for all $0<eta<1$.
- Kronecker sequence for the golden mean has $eta$-pair correlations for all $0<eta<1$.
- Neither sequence has Poissonian pair correlations ($eta=1$).

## Abstract

Although a generic uniformly distributed sequence has Poissonian pair correlations, only one explicit example has been found up to now. Additionally, it is even known that many classes of uniformly distributed sequences, like van der Corput sequences, Kronecker sequences and LS sequences, do not have Poissonian pair correlations. In this paper, we show that van der Corput sequences and the Kronecker sequence for the golden mean are as close to having Poissonian pair correlations as possible: they both have $\alpha$-pair correlations for all $0 < \alpha < 1$ but not for $\alpha = 1$ which corresponds to Poissonian pair correlations.