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

TL;DR

This paper proves that the two-point correlation function of the fractional parts of square roots converges to that of independent random variables, contrasting with their non-standard gap distribution.

Contribution

It establishes the convergence of the two-point correlation function and moments for the fractional parts of aaaa n, showing they behave like independent uniform variables.

Findings

Two-point correlation function converges to the independent case.

Moments for points in shifted intervals converge.

Gap distribution remains non-standard, but correlation matches independence.

Abstract

Elkies and McMullen [Duke Math.J.~123 (2004) 95--139] have shown 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.