Square roots and lattices

TL;DR

This paper explores the connection between the fractional parts of square roots and lattice point distributions, constructing a point set that reveals how these statistics relate to shifted lattices and their transformations.

Contribution

It introduces a novel point set construction linking fractional square root statistics with lattice point processes, highlighting a non-invariant limit process under standard group actions.

Findings

Converges to a lattice-like random point process after rotation and stretching.

Limit process is not invariant under the standard SL(2,R) action.

Extends previous work on gap statistics of sqrt(n) mod 1.

Abstract

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 $\sqrt{n}$ mod 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 $\mathrm{SL}(2,\mathbb{R})$-action on $\mathbb{R}^2$.