The distribution of directions in an affine lattice: two-point correlations and mixed moments

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

“…3. The proof of this claim follows closely our discussion in [1], which produced an analogous result for the two-point statistics of directions in an affine lattice. (We note that Sinai [14] has recently proposed an alternative approach to the statistics of √ n mod 1, but will not exploit this here.…”

“…This setting is already discussed in [2, Section 3.5], and the upper bounds obtained in the present paper are sufficient to establish Theorem 3 in this case. Note that the limit process is different for each c; it coincides with the limit process E c (k, I) studied in [1]. As we point out in [1], the second moments of E c (k, I) are Poisson, and hence Theorem 1 holds independently of the choice of c.…”

“…The second moments and two-point correlation function however coincide with those of a Poisson process with intensity 1. This is a consequence of the Siegel integral formula, see [1]. Specifically, we have…”

“…The limiting point process characterised by the probabilities E(k, I) is the same as for the directions of affine lattice points with irrational shift [10,1]; in the notation of [1],…”

“…The case of the second mixed moment implies, by a standard argument, the convergence of the two-point correlation function stated in Theorem 1, cf. [1]. For…”