Related papers: Poissonian Pair Correlation in Higher Dimensions

The purpose of this note is to present a construction of sequences which do not have metric Poissonian pair correlations (MPPC) and whose additive energies grow at rates that come arbitrarily close to a threshold below which it is believed…

A symmetric subset of the reals is one that remains invariant under some reflection x --> c-x. Given 0 < x < 1, there exists a real number D(x) with the following property: if 0 < d < D(x), then every subset of [0,1] with measure x contains…

The idea behind Poisson approximation to the binomial distribution was used in [J. de la Cal, F. Luquin, J. Approx. Theory, 68(3), 1992, 322-329] and subsequent papers in order to establish the convergence of suitable sequences of positive…

Let $X = [0,1]$, and let $T:X\to X$ be an expanding piecewise linear map sending each interval of linearity to $[0,1]$. For $\psi:\mathbb N\to\mathbb R_{\geq 0}$, $x\in X$, and $N\in\mathbb N$ we consider the recurrence counting function \[…

We have developed a heuristic showing that in the Dirichlet divisor problem for the almost all $n \in \mathbb{N}^{+}$: $$ R(n) \leq O(\psi(n)n^{\frac{1}{4}}) $$ where $$ R(n) = \Big\lvert \sum_{x=1}^{n}\Big\lfloor\frac{n}{x}\Big\rfloor -…

Let $(m, n, k)$ be a tuple of integers with the property that if $i \leq k$, then $m + i$ and $n + i$ have the same radical. Using a result on the abc Conjecture, we bound $k$ from above, improving a result of Balasubramanian, Shorey, and…

Let $X_1,\ldots,X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density $f$. Under some conditions on $f$, we obtain a Poisson limit theorem, as $n \to \infty$, for the number of large probability…

A $d$-dimensional Ising model on a lattice torus is considered. As the size $n$ of the lattice tends to infinity, a Poisson approximation is given for the distribution of the number of copies in the lattice of any given local configuration,…

Let $X_1,\dots,X_n$ be independent centered random vectors in $\mathbb{R}^d$. This paper shows that, even when $d$ may grow with $n$, the probability $P(n^{-1/2}\sum_{i=1}^nX_i\in A)$ can be approximated by its Gaussian analog uniformly in…

Let S_{n,k} denote the random geometric graph obtained by placing points inside a square of area n according to a Poisson point process of intensity 1 and joining each such point to the k=k(n) points of the process nearest to it. In this…

Distance correlation has become an increasingly popular tool for detecting the nonlinear dependence between a pair of potentially high-dimensional random vectors. Most existing works have explored its asymptotic distributions under the null…

The determination of the cross correlation between an $m$-sequence and its decimated sequence has been a long-standing research problem. Considering a ternary $m$-sequence of period $3^{3r}-1$, we determine the cross correlation…

We demonstrate the level statistics in the vicinity of the Anderson transition in $d>2$ dimensions to be universal and drastically different from both Wigner-Dyson in the metallic regime and Poisson in the insulator regime. The variance of…

We systematically analyse the coherence length in even-even nuclei. The pairing coherence length in the spin-singlet channel for the effective density dependent delta (DDD) and Gaussian interaction is estimated. We consider in our…

Given a family $\mathcal{F}$ of subsets of $[n]$, we say two sets $A, B \in \mathcal{F}$ are comparable if $A \subset B$ or $B \subset A$. Sperner's celebrated theorem gives the size of the largest family without any comparable pairs. This…

Let $\nu\in M^1([0,\infty[)$ be a fixed probability measure. For each dimension $p\in \mathbb{N}$, let $(X_n^{p})_{n\geq1}$ be i.i.d. $\mathbb{R}^p$-valued random variables with radially symmetric distributions and radial distribution…

Given $n$ pairwise disjoint sets $X_1,\ldots, X_n$, we call the elements of $S=X_1\times\ldots\times X_n$ strings. A nonempty set of strings $W\subseteq S$ is said to be well-connected if for every $v\in W$ and for every $i\, (1\le i\le…

The classical Erd\H{o}s-Szekeres theorem dating back almost a hundred years states that any sequence of $(n-1)^2+1$ distinct real numbers contains a monotone subsequence of length $n$. This theorem has been generalised to higher dimensions…

The maximal correlation coefficient is a well-established generalization of the Pearson correlation coefficient for measuring non-linear dependence between random variables. It is appealing from a theoretical standpoint, satisfying…

Let $X_1, \ldots , X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda_1, \ldots , \lambda_n > 0$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate…