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

By Daniel El-Baz, Jens Marklof, Ilya Vinogradov. 2013

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Elkies and McMullen [Duke Math. J. 123 (2004) 95{139] have shown that the gaps between the fractional parts of p n for n = 1;:::;N, have a limit distribution as N tends to innity. The limit distribution is non-standard and diers distinctly from the exponential distribution expected for independent, uniformly distributed random variables on the unit interval. We complement this result by proving that the twopoint 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 nding 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/043a20d99bf741adf7afe21254b99da159b9cde2