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Elkies and McMullen [Duke Math.J.~123 (2004) 95--139] have shown that thegaps between the fractional parts of \sqrt n for n=1,\ldots,N, have a limitdistribution as N tends to infinity. The limit distribution is non-standard anddiffers distinctly from the exponential distribution expected for independent,uniformly distributed random variables on the unit interval. We complement thisresult by proving that the two-point correlation function of the above sequenceconverges to a limit, which in fact coincides with the answer for independentrandom variables. We also establish the convergence of moments for theprobability of finding r points in a randomly shifted interval of size 1/N. Thekey ingredient in the proofs is a non-divergence estimate for translates ofcertain non-linear horocycles.

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