Square roots and lattices
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909429300985856 |
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| author | Marklof, Jens |
| author_facet | Marklof, Jens |
| contents | 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 $\mathbb{R}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09107 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Square roots and lattices Marklof, Jens Number Theory Dynamical Systems Probability 11K06, 37D40, 60G55 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 $\mathbb{R}^2$. |
| title | Square roots and lattices |
| topic | Number Theory Dynamical Systems Probability 11K06, 37D40, 60G55 |
| url | https://arxiv.org/abs/2406.09107 |