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The two-point correlation function of the fractional parts of $ \sqrt{n}$ is Poisson


Authors: Daniel El-Baz, Jens Marklof and Ilya Vinogradov
Journal: Proc. Amer. Math. Soc. 143 (2015), 2815-2828
MSC (2010): Primary 11J71; Secondary 11K36, 11P21, 22E40, 37A17, 37A25
DOI: https://doi.org/10.1090/S0002-9939-2015-12489-6
Published electronically: February 16, 2015
MathSciNet review: 3336607
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Abstract: A study by Elkies and McMullen in 2004 showed that the gaps between the fractional parts of $ \sqrt n$ for $ n=1,\ldots ,N$, have a limit distribution as $ N$ tends to infinity. The limit distribution is non-standard and differs distinctly from the exponential distribution expected for independent, uniformly distributed random variables on the unit interval. We complement this result by proving that the two-point correlation function of the above sequence converges to a limit, which in fact coincides with the answer for independent random variables. We also establish the convergence of moments for the probability of finding $ r$ points in a randomly shifted interval of size $ 1/N$. The key ingredient in the proofs is a non-divergence estimate for translates of certain non-linear horocycles.


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Additional Information

Daniel El-Baz
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: daniel.el-baz@bristol.ac.uk

Jens Marklof
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: j.marklof@bristol.ac.uk

Ilya Vinogradov
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: ilya.vinogradov@bristol.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2015-12489-6
Received by editor(s): July 1, 2013
Received by editor(s) in revised form: February 21, 2014
Published electronically: February 16, 2015
Additional Notes: The second author was also supported by a Royal Society Wolfson Research Merit Award
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 291147
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society