Square roots and lattices

Jens Marklof

doi:10.48550/arXiv.2406.09107

Square roots and lattices

Jens Marklof^*

^*Corresponding author for this work

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Abstract

We construct a point set in the Euclidean plane that elucidates the relationship between the fine-scale statistics of the fractional parts of √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 Elkiesand McMullen’s original analysis for the gap statistics of √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 SL(2,R)-action on R².

Original language	English
Pages (from-to)	175-190
Number of pages	16
Journal	L'Enseignement mathematique
Volume	72
Issue number	1/2
DOIs	https://doi.org/10.48550/arXiv.2406.09107 https://doi.org/10.4171/LEM/1092
Publication status	Published - 12 Feb 2025

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© 2025 Fondation L’Enseignement Mathématique.

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10.4171/LEM/1092Licence: CC BY

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AB - We construct a point set in the Euclidean plane that elucidates the relationship between the fine-scale statistics of the fractional parts of √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 Elkiesand McMullen’s original analysis for the gap statistics of √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 SL(2,R)-action on R2.

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Square roots and lattices

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