Weak Poissonian Correlations

TL;DR

This paper investigates weak Poissonian pair correlations in sequences, showing they can be detected at small scales for certain parameters and establishing hierarchical relations among different correlation strengths.

Contribution

It introduces the concept of weak Poissonian correlations for sequences, demonstrating their detectability at small scales for 1<1 and their hierarchical strength compared to other correlation notions.

Findings

Weak 1-PPC can be detected at small scales for 1<1.

1-PPC is stronger than 0<1-PPC.

Poissonian 1-correlations of order k+1 imply those of order k.

Abstract

We examine a property of sequences called Poissonian pair correlations with parameter $0\leqslant \beta \leqslant 1$ (abbreviated as $\beta$-PPC). We prove that when $\beta<1,$ the property of $\beta$-PPC, also known as weak Poissonian correlations, can be detected at the behaviour of sequences at small scales, and show that this does not happen for the classical notion of PPC, that is, when $\beta = 1$. Furthermore, we show that whenever $0\leqslant \alpha < \beta \leqslant 1$, $\beta$-PPC is stronger than $\alpha$-PPC. We also include a discussion on weak Poissonian correlations of higher orders, showing that for $\beta < 1$, Poissonian $\beta$-correlations of order $k+1$ imply Poissonian $\beta$-correlations of $k$-th order with the same parameter $\beta$.