The higher-dimensional Duke problem for torus packets

The higher-dimensional Duke problem asks whether complete packets of compact orbits of the diagonal group on the space SL_n(Z)\SL_n(R) of unimodular lattices become equidistributed as their discriminants grow. The packets come from ideal classes of orders in totally real number fields of degree n. Duke's theorem on closed geodesics attached to real quadratic fields is the case n = 2. Three manuscripts dated 24 September and 5 October 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim the packet form of the problem in three settings:

every fixed prime degree at least five, for arbitrary local types of lattices;
primitive quartic fields, for arbitrary orders;
primitive sextic fields, for maximal orders.

This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

As recounted in the introductions of the manuscripts:
- Linnik's ergodic method, with Skubenko's work, established the distribution of closed geodesics under an auxiliary splitting condition at a fixed prime. Duke removed this condition for fundamental discriminants by analytic methods. Einsiedler, Lindenstrauss, Michel and Venkatesh later gave a dynamical proof for all positive nonsquare order discriminants.
- Einsiedler, Lindenstrauss, Michel and Venkatesh connected the arithmetic separation of periodic torus orbits with the entropy of their limits (arXiv:math/0607815). Their equidistribution theorem for cubic fields combines subconvexity, local harmonic analysis and measure classification (arXiv:0708.1113). Their discussion of higher degree is conditional on subconvexity in prime degree. In composite degree it also identifies proper intermediate homogeneous limits as a possibility.
- Einsiedler, Katok and Lindenstrauss proved that an ergodic probability measure invariant under the full diagonal group with positive entropy is homogeneous; in prime dimension this forces Haar measure.
- Khayutin obtained entropy bounds for maximal-order packets whose Galois groups act two-transitively, including the primitive quartic types (arXiv:1510.08481). Lemke Oliver, Thorner and Zaman proved equidistribution for orders in totally real fields of each fixed prime degree at least five, outside a sparse exceptional family of fields (arXiv:2012.14422, Theorem 2.3). Wieser and Yang obtained entropy bounds for certain maximal-order quartic packets whose fields have a quadratic subfield, under further hypotheses (doi:10.4171/CMH/613).

The original results

Prime degree, arbitrary local type (24 September 2026). Theorem 1.1 of the first manuscript concerns a fixed prime n ≥ 5 and a sequence of full lattices in totally real fields of degree n. It states that the volume-weighted packet measures of one local homothety type converge weakly to Haar probability measure as the discriminant of the multiplier order tends to infinity, with no escape of mass. Nonmaximal orders of unbounded index are allowed, and the field discriminants may stay bounded. The manuscript states that it makes no assertion about an individual orbit, a rate of convergence or composite degrees. It describes its result as treating every field, where the theorem of Lemke Oliver, Thorner and Zaman excludes an exceptional family.
Primitive quartic fields, arbitrary orders (5 October 2026). Theorem 1.1 of the second manuscript states the same conclusion for totally real quartic fields with no proper intermediate field and any order. The packets are formed from all invertible ideal classes of the order and weighted by orbit volume, and the limit is taken as the discriminant of the order tends to infinity. The field may remain fixed while the order index grows.
Primitive sextic fields (5 October 2026). Theorem 1.1 of the third manuscript states that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate field equidistribute with no escape of mass as the field discriminant tends to infinity. Every ordering of the real embeddings is allowed.
- Method, as described:
  - The prime-degree proof counts lattice vectors with upper bounds relative to the residue of the Dedekind zeta function. Its inputs are Stark's exceptional-zero result and the short-interval bounds of Shiu and Pollack. These counts give tightness and a bound for the mass of small balls, hence positive entropy, and measure classification in prime dimension then gives Haar measure.
  - In composite degree, measure classification leaves homogeneous limits on proper block subgroups, and these must be excluded. The quartic manuscript treats limits with two blocks of size two. It passes to the exterior square, where the products of opposite coordinates lie in a cubic resolvent field, and proves a rectangle count uniformly in the order by local quadratic Fourier analysis and a weighted Selberg sieve.
  - The sextic manuscript bounds the packet mass of two-sided diagonal tubes. It counts an ordinary vector together with a dual vector whose coordinatewise product lies in the codifferent; primitivity of the field supplies the needed separation. The bound is then combined with conditional measures formed at intermediate resolutions before taking the limit, a product rule for them, and an argument showing that every root direction is active.
Sources. Commit 1581d4c of the openai/math repository, folders Equidistribution of prime-degree torus packets with arbitrary local type, Equidistribution of primitive quartic torus packets for arbitrary orders and Equidistribution of primitive sextic torus packets.
Formalization. The collection's Lean scope note for this family, lean/docs/015.md, covers the prime-degree manuscript. It states that the formalization proves equidistribution of volume-weighted torus packets for totally real fields of every fixed prime degree at least five, with tightness and arbitrary local homothety types, and it links the comparator statement DukePrimeDegree.lean. The quartic and sextic manuscripts are not covered by the note.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 18 October 2026)
2026-10-16	J. Jardine, Lattice24, referee reviews rank_418 (prime degree), rank_319 (quartic) and rank_070 (sextic) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	The three reviews report no gap. Each notes that Stark's lemma on exceptional zeros is used in paraphrase and was not checked against the original. The reviews of the prime-degree and sextic manuscripts say the same of the entropy criterion of Einsiedler, Lindenstrauss, Michel and Venkatesh, and the prime-degree review also of the prime-dimension corollary of Einsiedler, Katok and Lindenstrauss. The review of the quartic manuscript checked its conductor-uniform Fourier estimates only for exponent bookkeeping.	The reviews are published under CC BY 4.0, in an audit that Lattice24 describes as complete.

Notes

The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.