Weak Poissonian correlations (Q6624152)

For \(N\ge 1\), \(0\le\beta\le 1\) and \(s>0\), define the pair correlation function with parameter \(\beta\) of a sequence \((x_n)_{n\in\mathbb N}\subseteq[0,1]\) to be \N\[\N R_2(\beta;s,N)=\frac{1}{N^{2-\beta}}\#\left\{m,n\le N,\,m\ne n:\|x_m-x_n\|\le\frac{s}{N^\beta}\right\}. \N\]\NThe sequence \((x_n)_{n\in\mathbb N}\) has weak Poissonian pair correlations with parameter \(0<\beta<1\) (abbreviated as \(\beta\)-PPC) if \(\lim_{N\to\infty}R_2(\beta;s,N)=2 s\) for all \(0<s\le 1/2\). It is proved that when \(0\le\beta<1\), if there exists some constant \(s_0>0\) such that \(\lim_{N\to\infty}R_2(\beta;s,N)=2 s\) for all \(s<s_0\), then the sequence \((x_n)_{n\in\mathbb N}\) has the property of \(\beta\)-PPC, and is shown that this does not happen for the classical notion of PPC, that is, when \(\beta=1\). Furthermore, it is shown that whenever \(0\le\alpha<\beta\le 1\), the property of \(\beta\)-PPC is stronger than \(\alpha\)-PPC. A discussion on weak Poissonian correlations of higher orders is also included, showing that for \(\beta<1\), Poissonian \(\beta\)-correlations of order \(k+1\) imply Poissonian \(\beta\)-correlations of \(k\)-th order with the same parameter \(\beta\).