Square roots and lattices

We construct a point set in the Euclidean plane that elucidates the relationship between the fine-scale statistics of the fractional parts of \sqrt{n} and directional statistics for a shifted lattice. We show that the randomly rotated, and then stretched, point set converges in distribution to a lattice-like random point process. This follows closely the arguments in Elkies and McMullen’s original analysis for the gap statistics of \sqrt{n}\bmod 1 in terms of random affine lattices [Duke Math. J. 123 (2004), 95–139]. There is, however, a curious subtlety: the limit process emerging in our construction is not invariant under the standard \operatorname{SL}(2,\mathbb{R}) -action on {\mathbb{R}}^{2} .