Poissonian correlation of higher order differences

Abstract

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. \end{equation*} It is known that this condition implies equidistribution of $(x_n)$. We generalize this result to four-fold differences: if for all $s> 0$ we have \begin{equation*} \lim_{N\to\infty} \frac{1}{N^2}\#\left\{\substack{1\leq m,n,k,l\leq N\\\{m,n\}\neq\{k,l\}} : |x_m+x_n-x_k-x_l| \leq \frac{s}{N^2}\right\} = 2s \end{equation*} then $(x_n)_{n=1}^{\infty}$ is equidistributed. This notion generalizes to higher orders, and for any $k$ we show that a sequence exhibiting $2k$-fold Poissonian correlation is equidistributed. In the course of this investigation we obtain a discrepancy bound for a sequence in terms of its closeness to $2k$-fold Poissonian correlation. This result refines earlier bounds of Grepstad & Larcher and Steinerberger in the case of pair correlation, and resolves an open question of Steinerberger.