On the paucity of lattice triangles

[2603.23928] On the paucity of lattice triangles Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2603.23928 (math) [Submitted on 25 Mar 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: On the paucity of lattice triangles Authors: David Kurniadi Angdinata , Evan Chen , Ken Ono , Jesse Thorner , Jiaxin Zhang , Jujian Zhang View PDF HTML (experimental) Abstract: A rational triangle $T$ (one whose angles are rational multiples of $\pi$) unfolds to a translation surface ${X_T}$. The lattice triangle problem asks to classify those $T$ for which ${X_T}$ is a Veech (lattice) surface, which means that the $\operatorname{SL}_2(\mathbb R)$-orbit of ${X_T}$ is closed in its stratum (so its projection to moduli space is a Teichmüller curve). The most mysterious regime is the "hard obtuse window" (largest angle in $(\pi/2,2\pi/3]$), where it is conjectured that no lattice triangles exist. Using an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, we prove a quantitative theorem that rules out all but a proportion $n^{-1+o(1)}$ of the triangles in this window with denominator $n$. The main technical result in our proof was autoformalized by AxiomProver in Lean (using mathlib). Comments: The paper is significantly revised based on referee comments. And, Jesse Thorner is added as a co-author Subjects: Dynamical Systems (math.DS) ; Combinatorics (math.CO); Number Theory (math.NT) MSC classes: 37E35, 30F60 Cite as: arXiv:2603.23928 [math.DS] (or arXiv:2603.23928v2 [math.DS] for this version) https://doi.org/10.48550/arXiv.2603.23928 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Evan Chen [ view email ] [v1] Wed, 25 Mar 2026 04:37:17 UTC (19 KB) [v2] Thu, 27 Aug 2026 01:09:57 UTC (15 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2026-03 Change to browse by: math math.CO 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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