Related papers: Localized Quantitative Criteria for Equidistributi…

Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m…

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.…

A sequence $(x_n)_{n=1}^{\infty}$ on the torus $\mathbb{T} \cong [0,1]$ is said to exhibit Poissonian pair correlation if the local gaps behave like the gaps of a Poisson random variable, i.e. $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \#…

A well known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In this paper we give sufficient conditions for an analogue of this theorem to hold for…

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}…

It is well known that (i) for every irrational number $\alpha$ the Kronecker sequence $m\alpha$ ($m=1,...,M$) is equidistributed modulo one in the limit $M\to\infty$, and (ii) closed horocycles of length $\ell$ become equidistributed in the…

Let \{X_1, X_2, ...\} be a sequence of independent and identically distributed positive random variables of Pareto-type with index \alpha>0 and let \{N(t); t\geq 0\} be a counting process independent of the X_i's. For any fixed t\geq 0,…

In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon, and then investigated the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. We give an exact formula…

This study is motivated by a series of recent papers that show that, if a given deterministic sequence in the unit interval has a Poisson pair correlation function, then the sequence is uniformly distributed. Analogous results have been…

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…

Let $(X,\mathfrak{B},\mu)$ be a Borel probability space. Let $T_n: X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $\nu$ be a probability measure on $X$ such that $\frac{1}{N}\sum_{n=1}^N (T_n)_\ast \nu \rightarrow…

The Thue--Morse sequence $\{t(n)\}_{n\geqslant 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$, where $t(n)=1$ (resp. $=0$) if the binary expansion of $n$ has an odd (resp.…

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 $(f_n)_{n=1}^{\infty}$…

We consider $n\times n$ random matrices $M_{n}=\sum_{\alpha =1}^{m}{\tau _{\alpha }}\mathbf{y}_{\alpha }\otimes \mathbf{y}_{\alpha }$, where $\tau _{\alpha }\in \mathbb{R}$, $\{\mathbf{y}_{\alpha }\}_{\alpha =1}^{m}$ are i.i.d. isotropic…

Elkies and McMullen [Duke Math.J.~123 (2004) 95--139] have shown that the gaps between the fractional parts of \sqrt n for n=1,\ldots,N, have a limit distribution as N tends to infinity. The limit distribution is non-standard and differs…

By a classical theorem of Koksma the sequence of fractional parts $(\{x^n\})_{n \geq 1}$ is uniformly distributed for almost all values of $x$. In the present paper we obtain an exact quantitative version of Koksma's theorem, by calculating…

We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \left \lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \right \rbrace = 2s…

Assume that X is a set of sample statistics which follow a special case Central Limit Theorem, namely: as the sample size n increases the corresponding distribution becomes multivariate Normal with the mean (of each X) equal to zero and…

We prove a quantum ergodicity theorem in position space for the eigenfunctions of a Schr\"odinger operator $-\Delta+V$ on a rectangular torus $\mathbb{T}^2$ for $V\in L^2(\mathbb{T}^2)$ with an algebraic rate of convergence in terms of the…

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…