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arXiv · 2603.23928

On the paucity of lattice triangles

David Kurniadi Angdinata·Evan Chen·Ken Ono·Jesse Thorner·Jiaxin Zhang·Jujian Zhang
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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 Teichm\"uller 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).

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David Kurniadi Angdinata, Evan Chen, Ken Ono, Jesse Thorner, Jiaxin Zhang, Jujian Zhang. 2026-03-25. On the paucity of lattice triangles. https://arxiv.org/abs/2603.23928

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