TY - JOUR
T1 - Square roots and lattices
AU - Marklof, Jens
PY - 2024/12/13
Y1 - 2024/12/13
N2 - 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.
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.
U2 - 10.48550/arXiv.2406.09107
DO - 10.48550/arXiv.2406.09107
M3 - Article (Academic Journal)
SN - 0013-8584
JO - L'Enseignement mathematique
JF - L'Enseignement mathematique
ER -