Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The two-point correlation function of the fractional parts of $\sqrt {n}$ is Poisson
HTML articles powered by AMS MathViewer

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
Similar Articles
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