Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772485 | Journal of Number Theory | 2018 | 15 Pages |
Abstract
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 Poissonian if the number of pairs xk,xlâ(xn)1â¤nâ¤N which are within distance s/N of each other is asymptotically â¼2sN. 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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Christoph Aistleitner, Thomas Lachmann, Florian Pausinger,