Weak Poissonian correlations

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 \(\beta\).