The two-point correlation function of the fractional parts of $\sqrt {n}$ is Poisson
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- by Daniel El-Baz, Jens Marklof and Ilya Vinogradov PDF
- Proc. Amer. Math. Soc. 143 (2015), 2815-2828 Request permission
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.References
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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
- MR Author ID: 882723
- Email: ilya.vinogradov@bristol.ac.uk
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3336607