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Poissonian correlation of higher order differences
Journal of Number Theory ( IF 0.6 ) Pub Date : 2020-12-23 , DOI: 10.1016/j.jnt.2020.11.017
Alex Cohen

A sequence (xn)n=1 on the torus T exhibits Poissonian pair correlation if for all s>0,limN1N#{1mnN:|xmxn|sN}=2s. It is known that this condition implies equidistribution of (xn). We generalize this result to four-fold differences: if for all s>0 we havelimN1N2#{1m,n,k,lN{m,n}{k,l}:|xm+xnxkxl|sN2}=2s then (xn)n=1 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.



中文翻译:

高阶差分的泊松相关

一个序列 (Xn)n=1 在环面上 表现出泊松对相关,如果对所有 >0,N1N#{1nN|X-Xn|N}=2. 众所周知,这个条件意味着 (Xn). 我们将此结果概括为四倍差异:如果对于所有>0 我们有N1N2#{1,n,,N{,n}{,}|X+Xn-X-X|N2}=2 然后 (Xn)n=1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。

更新日期:2020-12-23
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