Density of shapes of periodic tori in the cubic case

[2502.12754] Density of shapes of periodic tori in the cubic case Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2502.12754 (math) [Submitted on 18 Feb 2025 ( v1 ), last revised 26 Aug 2026 (this version, v2)] Title: Density of shapes of periodic tori in the cubic case Authors: Nguyen-Thi Dang , Nihar Gargava , Jialun Li View PDF HTML (experimental) Abstract: 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. Comments: 19 pages, 2 figures. This project has now been formalized Subjects: Dynamical Systems (math.DS) ; Number Theory (math.NT) MSC classes: 37PXX, 37A44 Cite as: arXiv:2502.12754 [math.DS] (or arXiv:2502.12754v2 [math.DS] for this version) https://doi.org/10.48550/arXiv.2502.12754 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Nihar Prakash Gargava Mr. [ view email ] [v1] Tue, 18 Feb 2025 11:15:58 UTC (2,230 KB) [v2] Wed, 26 Aug 2026 21:11:54 UTC (446 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2025-02 Change to browse by: math math.NT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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