Pair correlations and equidistribution

JOURNAL OF NUMBER THEORY

卷:182

Article

Aistleitner, Christoph¹  Lachmann, Thomas¹  Pausinger, Florian² 

[1] Graz Univ Technol, Inst Anal & Number Theory, Graz, Austria

[2] Tech Univ Munich, Chair Geometry & Visualizat, Munich, Germany

关键词: Equidistribution;    Pair correlations;    Pseudorandomness;    Fejer kernel;   

DOI  :  10.1016/j.jnt.2017.06.009

来源: Elsevier

【 摘 要 】

A deterministic sequence of real numbers in the unit interval is called equidistributed if its empirical distribution converges to the uniform distribution. Furthermore, the limit distribution of the pair correlation statistics of a sequence is called Poissonicin if the number of pairs x(k), x(l) is an element of (x(n))i <= n <= N which are within distance s/N of each other is asymptotically similar to 2sN. A randomly generated sequence has both of these properties, almost surely. There seems to be a vague sense that having Poissonian pair correlations is a finer property than being equidistributed. In this note we prove that this really is the case, in a precise mathematical sense: a sequence whose asymptotic distribution of pair correlations is Poissonian must necessarily be equidistributed. Furthermore, for sequences which are not equidistributed we prove that the square integral of the asymptotic density of the sequence gives a lower bound for the asymptotic distribution of the pair correlations. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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