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,xl∈(xn)1≤n≤N which are within distance s/N of each other is asymptotically ∼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 squareintegral of the asymptotic density of the sequence gives a lower bound for the asymptotic distribution of the pair correlations.
Original language  English 

Pages (fromto)  206220 
Journal  Journal of Number Theory 
Volume  182 
Early online date  18 Jul 2017 
DOIs  
Publication status  Early online date  18 Jul 2017 
Externally published  Yes 
Keywords
 Equidistribution
 Fejér kernel
 Pair correlations
 Pseudorandomness
ASJC Scopus subject areas
 Algebra and Number Theory
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Florian Pausinger
Person: Academic