Abstract
For a sequence of integers {a(x)} x≥1 we show that the distribution of the pair correlations of the fractional parts of {〈αa(x)〉} x≥1 is asymptotically Poissonian for almost all α if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of α such that {〈αx d〉} fails to have Poissonian pair correlation is at most \(\frac{{d + 2}}{{d + 3}} < 1\). This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least \(\frac{2}{{d + 1}}\).
An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix two problems raised in the paper are solved.
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With an appendix by Jean Bourgain Dedicated to the memory of Professor Steven A. Gaal (1924–2016)
The first author is supported by the Austrian Science Fund (FWF) by an FWF Schrödinger scholarship, FWF project I1751-N26 and FWF START project Y-901. The first and the second author are supported by the FWF project F5507-N26, which is part of the Special Research Program Quasi-Monte Carlo Methods: Theory and Applications.
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Aistleitner, C., Larcher, G. & Lewko, M. Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems. Isr. J. Math. 222, 463–485 (2017). https://doi.org/10.1007/s11856-017-1597-5
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DOI: https://doi.org/10.1007/s11856-017-1597-5