THE TWO-POINT CORRELATION FUNCTION OF THE FRACTIONAL PARTS OF √ n IS POISSON

Elkies and McMullen [Duke Math. J. 123 (2004) 95–139] have shown that the gaps between the fractional parts of √ n for n = 1, . . . , 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 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 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.