Browse › Number theory › 015 Torus-packet equidistribution in prime, quartic, and sextic degrees
Equidistribution of primitive quartic torus packets for arbitrary orders
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
Let K range over totally real quartic fields with no proper intermediate field, and let $\mathcal O$ be any order in K. We prove that the packets of periodic diagonal orbits attached to invertible $\mathcal O$-ideal classes, weighted by orbit volume, equidistribute with no escape of mass as $|\mathop{\mathrm{Disc}}\nolimits (\mathcal O)|$ tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
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 math release, Lean scope note 015: Torus-packet equidistribution in prime, quartic, and sextic degrees (lean/docs/015.md, release of 2026-10-06). The family's Lean scope note lists only the prime-degree companion paper as formalized and gives no Lean statement for the quartic theorem. [family-level]
Related prior literature mentioned by others
- Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel and Akshay Venkatesh, Distribution of periodic torus orbits and Duke’s theorem for cubic fields (Annals of Mathematics 173, 2011). The VibeMathed family entry identifies this cubic equidistribution theorem and its higher-degree discussion as background to the family’s packet-equidistribution claims. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Ilya Khayutin (joint work with A. Wieser), Two-step equidistribution for bi-quadratic torus packets (New England Dynamics and Number Theory Seminar talk, 2021-12-02). The talk is about torus packets on PGL_4(Z)\PGL_4(R) whose limits involve intermediate homogeneous subspaces. These are the proper homogeneous limits in degree four that the paper excludes for primitive fields via its cubic-resolvent count.
Digests and expositions
- Latent Seminar, Arithmetic torus packets and equidistribution (2026-10-06). This exposition distinguishes the three packet theorems and outlines tightness, entropy, and the additional quartic and sextic arguments, explicitly without independently verifying the complete proofs. [family-level]
- Orply / Perplexity, Primitive Quartic Torus Packets Equidistribute Across Arbitrary Orders (2026-10-08). This AI-assisted explainer describes the quartic paper’s exterior-square obstruction, cubic-resolvent rectangle estimate, and uniform treatment of growing order index. [AI-generated per source]
Elsewhere
- Emergent Mind: family 015
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026,
author = {{OpenAI}},
title = {{Equidistribution of primitive quartic torus packets for arbitrary orders}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf}{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/015.md. The Lean statements have not been checked by people against the paper's claims.
Family 015: Torus-packet equidistribution in prime, quartic, and sextic degrees
Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.
Other papers in this family:
- Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type · September 24, 2026
- Equidistribution of Primitive Sextic Torus Packets · October 5, 2026