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

A study by Elkies and McMullen in 2004 showed that the gaps between the fractional parts of root 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 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.