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Ergodicity of triangular billiards with an irrational angle
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Abstract
We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is ergodic for normalized area times uniform angular measure. No genericity or Diophantine condition is required. The flow is considered outside the null set of trajectories that hit a vertex.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Lean scope notes for family 150 (release repository, lean/docs/150.md, October 2026). States that the Lean development proves the ergodicity theorem of this paper, constructing the flow off a null set, proving uniqueness of the flight chain, measure preservation and the flow law a.e. [family-level]
- OpenAI, IrrationalTriangleBilliard.lean (openai/math, lean/ComparatorChallenges, October 2026). Comparator statement
irrational_triangle_billiardfor a triangle with an angle irrational relative to π; its conclusion bundles a.e. flow construction, measure preservation and that invariant sets have measure 0 or 1 (the proof is left assorryin this statement file). [family-level]
Related prior literature mentioned by others
- Giovanni Forni and Nelson Moll, Cohomological equation for geodesic flows on flat surfaces (arXiv:2510.18128v1, 2025-10-20). Azmat’s exposition identifies this work as the analytic framework behind the family’s treatment of invariant functions and highlights the horizontal Sobolev regularity assumption that this manuscript seeks to remove. [family-level]
Possibly relevant literature (found by the agent, not checked)
- G. Casati, T. Prosen, Mixing property of triangular billiards (arXiv:chao-dyn/9908022; Phys. Rev. Lett. 83, 1999). Numerical evidence that triangles with all angles irrational are mixing; complements the numerical studies [WCP, ZSBJ] the paper cites as raising doubts about ergodicity.
- J. Seaward, Ergodicity in Golden Billiards (arXiv:1910.14350, 2019). Comment on the Wang–Casati–Prosen numerics (cited as [WCP]) arguing that "strongly irrational" right triangles are likely ergodic; directly relevant to the paper's discussion of [WCP].
- Z. Hwang, C. A. Marx, J. Seaward, S. Jitomirskaya, M. Olshanii, Growth of the Wang-Casati-Prosen counter in an integrable billiard (SciPost Phys. 14, 017 (2023); arXiv:2011.09467). Studies the counter underlying the Wang–Casati–Prosen non-ergodicity evidence for irrational right triangles, which the paper cites as [WCP].
- S. Troubetzkoy, Approximation and billiards (arXiv:math/0611599, 2006). Survey of ergodicity and weak-mixing results via the Katok–Zemlyakov approximation method, i.e. the genericity results (KMS, Vorobets) the paper contrasts with its every-triangle statement.
- Č. Lozej, G. Casati, T. Prosen, Quantum chaos in triangular billiards (arXiv:2110.04168; Phys. Rev. Research 4, 013138, 2022). Numerical study of eigenvalues, spectral statistics and eigenfunction localization in irrational and one-rational-angle triangles; related to the paper's quantum-ergodicity consequences in its spectral section.
Digests and expositions
- Azmat, Irrational Triangular Billiards: How OpenAI’s Proof Tackles a Decades-Old Ergodicity Mystery (Binary Verse AI, 2026-10-15). Explains this manuscript’s Fourier-coefficient estimates, successive limits and quantum consequences, and identifies the October companion’s stronger weak-mixing claim while disclaiming independent verification of every analytic step. [family-level]
- heyprof, Triangular billiards — irrational angles mix (3D) (interactive exposition, undated; accessed 2026-10-21). Illustrates family 150 using trajectories, direction distributions and paired trajectories, noting that floating-point simulation cannot distinguish exact rationality from irrationality. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 150
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026,
author = {{OpenAI}},
title = {{Ergodicity of triangular billiards with an irrational angle}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf}{OAI:Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/150.md. The Lean statements have not been checked by people against the paper's claims.
Family 150: Weak mixing of triangular billiards with an irrational angle
Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.
Other papers in this family:
- Weak mixing of triangular billiards with an irrational angle · October 5, 2026