A sequence ${(x_n)}_{n=1}^{\infty }$ on the torus $\mathbb{T}$ exhibits Poissonian pair correlation if for all $s>0$,$\underset{N\to \infty }{\lim }\frac{1}{N}\mathrm{\#}\{1\le m\ne n\le N:|x_m-x_n|\le \frac{s}{N}\}=2s.$ 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$\underset{N\to \infty }{\lim }\frac{1}{N^2}\mathrm{\#}\{\begin{array}{c}\hfill 1\le m,n,k,l\le N\hfill \\ \hfill \{m,n\}\ne \{k,l\}\hfill \end{array}:|x_m+x_n-x_k-x_l|\le \frac{s}{N^2}\}=2s$ 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.

一个序列 ${(X_n)}_{n=1}^{\infty }$ 在环面上 $\mathbb{吨}$ 表现出泊松对相关，如果对所有 $秒>0$,$\underset{N\to \infty }{\mathrm{林}}\frac{1}{N}\mathrm{\#}\{1\le 米\ne n\le N：|X_{米}-X_n|\le \frac{秒}{N}\}=2秒.$ 众所周知，这个条件意味着 $(X_n)$. 我们将此结果概括为四倍差异：如果对于所有$秒>0$ 我们有$\underset{N\to \infty }{\mathrm{林}}\frac{1}{N^2}\mathrm{\#}\{\begin{array}{c}\hfill 1\le 米,n,克,升\le N\hfill \\ \hfill \{米,n\}\ne \{克,升\}\hfill \end{array}：|X_{米}+X_n-X_{克}-X_{升}|\le \frac{秒}{N^2}\}=2秒$ 然后 ${(X_n)}_{n=1}^{\infty }$是均匀分布的。这个概念推广到更高阶，对于任何k，我们表明表现出 2 k 倍泊松相关的序列是等分布的。在这项调查的过程中，我们根据序列与 2 k倍泊松相关性的接近程度获得了序列的差异界限。该结果在对相关的情况下改进了 Grepstad & Larcher 和 Steinerberger 的早期边界，并解决了 Steinerberger 的一个悬而未决的问题。