Related papers: Poissonian pair correlation for $\alpha n^{\theta}…
Let $x(n):=\alpha n^d \mod 1$ for integer $d >1$ and non-zero real $\alpha$. We show that $\{x(n)\}_{n>0}$ has Poissonian $\ell$-point correlations for almost all choices of $\alpha$ when $d$ is large (depending on $\ell$). This falls in…
A sequence $(x_n)_{n=1}^{\infty}$ on the torus $\mathbb{T}$ exhibits Poissonian pair correlation if for all $s\geq0$, \begin{equation*} \lim_{N\to\infty} \frac{1}{N}\#\left\{1\leq m\neq n \leq N : |x_m-x_n| \leq \frac{s}{N}\right\} = 2s.…
In this note we study the $\sigma$-pair correlation density \begin{equation*}R_2^\sigma([a,b], \{ \theta_n \}_n, N)= \frac{1}{N^{2-\sigma}} \# \big \{ 1 \leq j \neq k \leq N \, \big| \, \theta_{j} - \theta_{k} \in \big […
We prove an abstract result on the correlations of pairs of elements in an exponentially growing discrete subset $\mathcal E$ of $[0,+\infty[\,$ endowed with a weight function. Assume that there exist $\alpha\in\mathbb R$, $c,\delta>0$ such…
We show that any sequence $(x_n)_{n \in \mathbb{N}} \subseteq [0,1]$ that has Poissonian correlations of $k$-th order is uniformly distributed, also providing a quantitative description of this phenomenon. Additionally, we extend…
It is shown that short-range correlations between large prime numbers (~10^5 and larger) have a Poissonian nature. Correlation length \zeta = 4.5 for the primes ~10^5 and it is increasing logarithmically according to the prime number…
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…
Z. Rudnick and P. Sarnak have proved that the pair correlation for the fractional parts of $n^2 \alpha$ is Poissonian for almost all $\alpha$. However, they were not able to find a specific $\alpha$ for which it holds. We show that the…
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…
We say that a sequence $\{x_n\}_{n \geq 1}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \rightarrow \infty} \frac{1}{N} \# \left\{ 1 \leq l \neq m \leq N \, : \, \left\lVert x_l-x_m \right\rVert < \frac{s}{N}…
Let $\{ a(x) \}_{x=1}^{\infty}$ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations $\alpha a(x)$ is Poissonian for Lebesgue almost every $\alpha\in…
Let $\left(a_{n}\right)_{n=1}^{\infty}$ be a lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all $\alpha\in\mathbb{R}$, the pair correlation of $\left(\alpha a_{n}\right)_{n=1}^{\infty}$ mod 1 is…
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…
A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More…
Using a standard definition of fractional powers on the universal cover $\exp:S\to \mathbb{C}^*$ seen as an infinite helicoid embedded in $\mathbb{R}^3$, we study the statistics of pairs from the countable family $\{n^\alpha \, : \, n \in…
We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery's conjecture for the pair correlation of zeros of the Riemann zeta-function which are…
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…
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…
We evaluate the triple correlation of eigenvalues of the Laplacian on generic flat tori in an averaged sense. As two consequence we show that (a) the limit inferior (resp. limit superior) of the triple correlation is almost surely at most…
Let $(f_n)_{n=1}^{\infty}$ be a sequence of polynomials and $\alpha>1$. In this paper we study the distribution of the sequence $(f_n(\alpha))_{n=1}^{\infty}$ modulo one. We give sufficient conditions for a sequence $(f_n)_{n=1}^{\infty}$…