On the paucity of lattice triangles
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918407635468288 |
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| author | Angdinata, David Kurniadi Chen, Evan Ono, Ken Zhang, Jiaxin Zhang, Jujian |
| author_facet | Angdinata, David Kurniadi Chen, Evan Ono, Ken Zhang, Jiaxin Zhang, Jujian |
| contents | A rational triangle $T$ (one whose angles are rational multiples of $π$) unfolds to a translation surface $(X_T,ω_T)$. The lattice triangle problem asks to classify those $T$ for which $(X_T,ω_T)$ is a Veech (lattice) surface, which means that the $\operatorname{SL}_2(\mathbb R)$-orbit of $(X_T,ω_T)$ 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 $(π/2,2π/3]$), 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 density 0 subset of the triangles in this window. The main engine in this paper was autoformalized by AxiomProver in Lean (using mathlib). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23928 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the paucity of lattice triangles Angdinata, David Kurniadi Chen, Evan Ono, Ken Zhang, Jiaxin Zhang, Jujian Dynamical Systems Combinatorics Number Theory 37E35, 30F60 A rational triangle $T$ (one whose angles are rational multiples of $π$) unfolds to a translation surface $(X_T,ω_T)$. The lattice triangle problem asks to classify those $T$ for which $(X_T,ω_T)$ is a Veech (lattice) surface, which means that the $\operatorname{SL}_2(\mathbb R)$-orbit of $(X_T,ω_T)$ 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 $(π/2,2π/3]$), 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 density 0 subset of the triangles in this window. The main engine in this paper was autoformalized by AxiomProver in Lean (using mathlib). |
| title | On the paucity of lattice triangles |
| topic | Dynamical Systems Combinatorics Number Theory 37E35, 30F60 |
| url | https://arxiv.org/abs/2603.23928 |