Poissonian Pair Correlation in Higher Dimensions

Abstract

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 $s>0$, $$\underset{N\to \infty }{\lim }\frac{1}{N}\#\left\{1\le m\ne n\le N:|x_m-x_n|\le \frac{s}{N}\right\}=2s.$$ It is known that this implies uniform distribution of the sequence $(x_n)$. Hinrichs, Kaltenböck, Larcher, Stockinger \& Ullrich extended this result to higher dimensions and showed that sequences $(x_n)$ in $[0,1{]}^d$ that satisfy, for all $s>0$, $$\underset{N\to \infty }{\lim }\frac{1}{N}\#\left\{1\le m\ne n\le N:\Vert x_m-x_n{\Vert }_{\infty }\le \frac{s}{N}\right\}=(2s{)}^d.$$ are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence $(x_n)$ in ${\mathbb{T}}^d$ satisfies, for all $s>0$, $$\underset{N\to \infty }{\lim }\frac{1}{N}\#\left\{1\le m\ne n\le N:\Vert x_m-x_n{\Vert }_2\le \frac{s}{N}\right\}={\omega }_d{s}^d$$ where ${\omega }_d$ is the volume of the unit ball, then $(x_n)$ is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.