Density of shapes of periodic tori in the cubic case
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909499968716800 |
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| author | Dang, Nguyen-Thi Gargava, Nihar Li, Jialun |
| author_facet | Dang, Nguyen-Thi Gargava, Nihar Li, Jialun |
| contents | Consider the compact orbits of the $\mathbb{R}^2$ action of the diagonal group on $\operatorname{SL}(3,\mathbb{R})/\operatorname{SL}(3,\mathbb{Z})$, the so-called periodic tori. For any periodic torus, the set of periods of the orbit forms a lattice in $\mathbb{R}^2$. Such a lattice, re-scaled to covolume one, gives a shape point in $\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$.
We prove that the shapes of all periodic tori are dense in $\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$. This implies the density of shapes of the unit groups of totally real cubic orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Density of shapes of periodic tori in the cubic case Dang, Nguyen-Thi Gargava, Nihar Li, Jialun Dynamical Systems Number Theory 37PXX, 37A44 Consider the compact orbits of the $\mathbb{R}^2$ action of the diagonal group on $\operatorname{SL}(3,\mathbb{R})/\operatorname{SL}(3,\mathbb{Z})$, the so-called periodic tori. For any periodic torus, the set of periods of the orbit forms a lattice in $\mathbb{R}^2$. Such a lattice, re-scaled to covolume one, gives a shape point in $\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$. We prove that the shapes of all periodic tori are dense in $\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$. This implies the density of shapes of the unit groups of totally real cubic orders. |
| title | Density of shapes of periodic tori in the cubic case |
| topic | Dynamical Systems Number Theory 37PXX, 37A44 |
| url | https://arxiv.org/abs/2502.12754 |