Abstract
We say that a sequence \(\left( x_n\right) _{n \ge 1}\) in [0, 1) has Poissonian pair correlations if
for all \(s>0\). In this note we show that if the convergence in the above expression is—in a certain sense—fast, then this implies a small discrepancy for the sequence \(\left( x_n\right) _{n \ge 1}\). As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in [0, 1).
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Acknowledgements
Open access funding provided by Johannes Kepler University Linz. The authors thank an anonymous reviewer who pointed out an inaccuracy in the first version of the paper. His helpful comments led to the current, slightly stronger version of Theorem 1.1.
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The authors are supported by the Austrian Science Fund (FWF): Projects F5505-N26 and F5507-N26, which are both part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
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Grepstad, S., Larcher, G. On pair correlation and discrepancy. Arch. Math. 109, 143–149 (2017). https://doi.org/10.1007/s00013-017-1060-1
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DOI: https://doi.org/10.1007/s00013-017-1060-1