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  • Christoph Aistleitner (a1), Thomas Lachmann (a2) and Niclas Technau (a3)


We consider sequences of the form $(a_{n}\unicode[STIX]{x1D6FC})_{n}$ mod 1, where $\unicode[STIX]{x1D6FC}\in [0,1]$ and where $(a_{n})_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $\unicode[STIX]{x1D6FC}$ in the sense of Lebesgue measure, we say that $(a_{n})_{n}$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_{n})_{n}$ . Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterizes the metric pair correlation property in terms of the additive energy, similar to Khintchine’s criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_{n})_{n}$ having large additive energy which, however, maintains the metric pair correlation property.



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1. Aistleitner, C., Metric number theory, lacunary series and systems of dilated functions. In Uniform Distribution and Quasi-Monte Carlo Methods (Radon Series on Computational and Applied Mathematics 15 ), De Gruyter (Berlin, 2014), 116.
2. Aistleitner, C., Larcher, G. and Lewko, M., Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems. With an appendix by Jean Bourgain. Israel J. Math. 222(1) 2017, 463485.10.1007/s11856-017-1597-5
3. Berry, M. and Tabor, M., Level clustering in the regular spectrum. Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 356(1686) 1977, 375394.10.1098/rspa.1977.0140
4. Bloom, T. F., Chow, S., Gafni, A. and Walker, A., Additive energy and the metric Poissonian property. Mathematika 64(3) 2018, 679700.10.1112/S0025579318000207
5. Bloom, T. F. and Walker, A., GCD sums and sum-product estimates. Preprint, 2018,arXiv:1806.07849.
6. Bugeaud, Y., Approximation by Algebraic Numbers (Cambridge Tracts in Mathematics 160 ), Cambridge University Press (Cambridge, 2004).10.1017/CBO9780511542886
7. Chow, S., Bohr sets and multiplicative Diophantine approximation. Duke Math. J. 167(9) 2018, 16231642.10.1215/00127094-2018-0001
8. Harman, G., Metric Number Theory (London Mathematical Society Monographs. New Series 18 ), Clarendon Press (Oxford, 1998).
9. Heath-Brown, D. R., Pair correlation for fractional parts of 𝛼n 2 . Math. Proc. Cambridge Philos. Soc. 148(3) 2010, 385407.10.1017/S0305004109990466
10. Lachmann, T. and Technau, N., On exceptional sets in the metric Poissonian pair correlations problem. Monatsh. Math. 189 2019, 137156.10.1007/s00605-018-1199-2
11. Marklof, J., The Berry–Tabor conjecture. In European Congress of Mathematics, Vol. II (Barcelona, 2000) (Progress in Mathematics 202 ), Birkhäuser (Basel, 2001), 421427.10.1007/978-3-0348-8266-8_36
12. Rudnick, Z. and Sarnak, P., The pair correlation function of fractional parts of polynomials. Comm. Math. Phys. 194(1) 1998, 6170.10.1007/s002200050348
13. Rudnick, Z., Sarnak, P. and Zaharescu, A., The distribution of spacings between the fractional parts of n 2𝛼. Invent. Math. 145(1) 2001, 3757.10.1007/s002220100141
14. Rudnick, Z. and Zaharescu, A., The distribution of spacings between fractional parts of lacunary sequences. Forum Math. 14(5) 2002, 691712.10.1515/form.2002.030
15. Truelsen, J. L., Divisor problems and the pair correlation for the fractional parts of n 2𝛼. Int. Math. Res. Not. IMRN 2010(16) 2010, 31443183.
16. Walker, A., The primes are not metric Poissonian. Mathematika 64(1) 2018, 230236.10.1112/S002557931700050X
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