论文标题
高阶差异的泊松相关性
Poissonian correlation of higher order differences
论文作者
论文摘要
序列$（x_n）_ {n = 1}^{\ infty} $上的torus $ \ m athbb {t} $展示poissonian对的相关性，如果所有$ s \ geq0 $，\ begin \ begin {equination*} \ lim_ {n \ to \ infty} \ frac {1} {n} \＃\ left \ {1 \ leq m \ neq n \ leq n \ leq n：| | x_m-x_n | \ leq \ frac {s} {n} \ right \} = 2s。 \ end {equation*} 众所周知，这种条件意味着$（x_n）$的等式分布。我们将此结果推广到四倍的区别：如果对于所有$ s> 0 $，我们都有 \ begin {equation*} \ lim_ {n \ to \ to \ infty} \ frac {1} {n^2} \＃\ left \ left \ {\ ordack {1 \ leq m，n，n，k，k，l \ leq n \\\\\\\\\\\\\\\\\\\\\\ {m，n \} \ leq \ frac {s} {n^2} \ right \} = 2S \ end {equation*} 然后$（x_n）_ {n = 1}^{\ infty} $是均衡的。这个概念概括为更高的订单，对于任何$ k $，我们表明的序列表现出$ 2K $倍的泊松相关性。在这项调查过程中，我们获得了一个差异，即其接近$ 2K $倍的泊松相关性。在配对相关性的情况下，该结果完善了Grepstad＆Larcher和Steinerberger的早期范围，并解决了Steinerberger的一个空旷问题。
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.
