Short··0:21·Number theory
Duke's Theorem in Higher Degree: OpenAI Claims a Proof
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About this video
Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. OpenAI claims the same for the torus packets of totally real fields of any fixed prime degree at least five, for primitive quartic fields with arbitrary orders, and for primitive sextic fields; these are OpenAI's preprints, not yet peer reviewed. The animation shows the degree-2 case: real periodic orbits for discriminants 5, 2669 and 34689 in the space of lattices, drawn as the 3-sphere around a trefoil knot (Ghys).
Papers (OpenAI, catalogue result 015):
- Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type: github.com/openai/math/blob/main/preprints/Equidistribution-o…
- Equidistribution of primitive quartic torus packets for arbitrary orders: github.com/openai/math/blob/main/preprints/Equidistribution-o…
- Equidistribution of Primitive Sextic Torus Packets: github.com/openai/math/blob/main/preprints/Equidistribution-o…
The animation illustrates the result; it is not a proof.
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Every result explained: gradientdamage.com/p/015
About the problem
Number theory·OpenAI Math·Result 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. (Lean)
Topics
torus packetsequidistributionDuke theoremDuke's theoremclosed geodesicsperiodic orbitsspace of latticesmodular surfacetrefoil knotGhys trefoilhomogeneous dynamicsmeasure rigiditytotally real fieldsideal classesclass groupnumber theoryescape of massHaar measureOpenAIOpenAI mathAI mathematicsAI proofmathsmath shorts3D animation
