Related papers: Pair correlation of real-valued vector sequences
We give a survey on the concept of Poissonian pair correlation (PPC) of sequences in the unit interval, on existing and recent results and we state a list of open problems. Moreover, we present and discuss a quite recent multi-dimensional…
The pair correlation statistic is an important concept in real uniform distribution theory. Therefore, sequences in the unit interval with (weak) Poissonian pair correlations have attracted a lot of attention in recent time. The aim of this…
The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $(a_n…
In this article, we examine the Poissonian pair correlation (PPC) statistic for higher-dimensional real sequences. Specifically, we demonstrate that for $d\geq 3$, almost all $(\alpha_1,\ldots,\alpha_d) \in \mathbb{R}^d$, the sequence…
We study the statistics of pairs from the sequence $(n^\alpha)_{n\in\mathbb{N}^*}$, for every parameter $\alpha \in \, ]0,1[$. We prove the convergence of the empirical pair correlation measures towards a measure with an explicit density.…
We establish new conditions under which a sequence of real numbers has metric Poissonian pair correlation. These conditions strengthen results of Aistleitner, El-Baz and Munsch (2021) and resolve one of their open problems under a mild…
In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any $d\geq 2$, strictly increasing sequences $(a_n^{(1)}),\ldots, (a_n^{(d)})$ of natural numbers have metric Poissonian pair…
The paper deals with real valued sequences and its distribution on real line.
We study the statistical convergence of metric valued sequences and of their subsequences. The interplay between the statistical and usual convergences in metric spaces is also studied.
We consider Poissonian pair correlations (PPC) for uniformly distributed sequences of random numbers with a dependency structure. More specifically, we treat two classes of dependent random variables which have widely been studied in the…
The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting,…
A generic uniformly distributed random sequence on the unit interval has Poissonian pair correlations. At the same time, there are only very few explicitly known examples of sequences with this property. Moreover, many types of…
We study the pair correlations of the logarithms of the integral values of quadratic norm forms at various scalings, proving the existence of pair correlation measures. We describe a surprising set of asymptotic behaviours when the scaling…
Although the existence of sequences in the p-adic integers with Poissonian pair correlations has already been shown, no explicit examples had been found so far. In this note we discuss how to transfer real sequences with Poissonian pair…
$k$-level correlation is a local statistic of sequences modulo 1, describing the local spacings of $k$-tuples of elements. For $k = 2$ this is also known as pair correlation. We show that there exists a well spaced increasing sequence of…
We study the notion of inhomogeneous Poissonian pair correlations, proving several properties that show similarities and differences to its homogeneous counterpart. In particular, we show that sequences with inhomogeneous Poissonian pair…
We study the value distribution of diagonal forms in k variables and degree d with random real coefficients and positive integer variables, normalized so that mean spacing is one. We show that the l-correlation of almost all such forms is…
A deterministic sequence of real numbers in the unit interval is called \emph{equidistributed} if its empirical distribution converges to the uniform distribution. Furthermore, the limit distribution of the pair correlation statistics of a…
Poissonian pair correlations have sparked interest within the mathematical community, because of their number theoretic properties, and their connections to quantum physics and probability theory, particularly uniformly distributed random…
For $0<\theta<1$, we show that for almost all $\alpha$, the pair correlation function of the sequence of fractional parts of $\{\alpha n^\theta:n\geq 1 \}$ is Poissonian.