Square roots and lattices
Jens Marklof
University of Bristol, Bristol, UK

Abstract
We construct a point set in the Euclidean plane that elucidates the relationship between the fine-scale statistics of the fractional parts of 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 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 -action on .
Cite this article
Jens Marklof, Square roots and lattices. Enseign. Math. (2025), published online first
DOI 10.4171/LEM/1092