{
  "id": "planar-halving-lines",
  "title": "A power saving for planar halving lines",
  "description": "OpenAI's theorem (2026) that n points in the plane with no three on a line have at most C n^(4/3 - eps) halving lines for an absolute eps > 0, the first improvement of Dey's exponent 4/3; the same bound for the switches of rank sweeps uniformly in the rank, and the bound C n (k+1)^(1/3 - eps0) for k-sets. Dey's bound through convex chains and crossings, the induction from a packing estimate to a power saving, and the packing estimate by contradiction: rank coordinates, weak limits of measures, monotone functions of two variables, three transverse foliations, a zoom with radial comparison, a spatial measure and a count of boundary activity.",
  "units": [
    {
      "id": "halving-lines-and-rank-sweeps",
      "title": "Halving lines and rank sweeps",
      "reader": "halving-lines-and-rank-sweeps.html",
      "source": "src/halving-lines-and-rank-sweeps.md",
      "source_sha256": "fbe9d3a8f1950a790ea909d6d34aa9cdd413d775c4d0390c514c0126b0ec163e",
      "status": "author_self_checked"
    },
    {
      "id": "chains-crossings-and-deys-bound",
      "title": "Chains, crossings and Dey's bound",
      "reader": "chains-crossings-and-deys-bound.html",
      "source": "src/chains-crossings-and-deys-bound.md",
      "source_sha256": "3ba72f7059a19626999560f81632f880d75dc4f1d124f128af12d4168cb83989",
      "status": "author_self_checked"
    },
    {
      "id": "from-packing-to-a-power-saving",
      "title": "From packing to a power saving",
      "reader": "from-packing-to-a-power-saving.html",
      "source": "src/from-packing-to-a-power-saving.md",
      "source_sha256": "e2aca37f7554f79538ef3d4fc29e7a675a8f8e6c38f7021a84771275cc1e5695",
      "status": "author_self_checked"
    },
    {
      "id": "measures-on-compact-metric-spaces",
      "title": "Measures on compact metric spaces",
      "reader": "measures-on-compact-metric-spaces.html",
      "source": "src/measures-on-compact-metric-spaces.md",
      "source_sha256": "0de72496735bf43e9b8ae17186e2cd40bb13aa30f1968c4c673d59886831ca7a",
      "status": "author_self_checked"
    },
    {
      "id": "rank-coordinates",
      "title": "Rank coordinates",
      "reader": "rank-coordinates.html",
      "source": "src/rank-coordinates.md",
      "source_sha256": "6b64f2355bb7e9e697b0028b167a2cacd40fb0f86b8fba51b0fe1d13bec7d227",
      "status": "author_self_checked"
    },
    {
      "id": "limiting-measures",
      "title": "Limiting measures",
      "reader": "limiting-measures.html",
      "source": "src/limiting-measures.md",
      "source_sha256": "4e8c0638bf4055b71973f9149acd1d5962ba170fed21b10d5c04b9766aadd807",
      "status": "author_self_checked"
    },
    {
      "id": "monotone-functions-of-two-variables",
      "title": "Monotone functions of two variables",
      "reader": "monotone-functions-of-two-variables.html",
      "source": "src/monotone-functions-of-two-variables.md",
      "source_sha256": "8e5a7fe7b2a528d4b0473a74f4771c3fac505370e0f3ead85f2ef1b2a35fabb5",
      "status": "author_self_checked"
    },
    {
      "id": "independent-gradients-and-tangent-directions",
      "title": "Independent gradients and tangent directions",
      "reader": "independent-gradients-and-tangent-directions.html",
      "source": "src/independent-gradients-and-tangent-directions.md",
      "source_sha256": "20db0b0fbe81de53eec0fe7249da15523473af162ba19ffcdd783dd58c96bafb",
      "status": "author_self_checked"
    },
    {
      "id": "three-transverse-foliations",
      "title": "Three transverse foliations",
      "reader": "three-transverse-foliations.html",
      "source": "src/three-transverse-foliations.md",
      "source_sha256": "0e7b1a4783051ce80ee5c0dd246a5e234b283e347b1ab4f5e47c5b23cfbcdf05",
      "status": "author_self_checked"
    },
    {
      "id": "the-zoom-at-a-rank-point",
      "title": "The zoom at a rank point",
      "reader": "the-zoom-at-a-rank-point.html",
      "source": "src/the-zoom-at-a-rank-point.md",
      "source_sha256": "6ccef29cd88c8ffaf5a489e075240a652b390bfb029f0bc49042788b4552bb1a",
      "status": "author_self_checked"
    },
    {
      "id": "a-spatial-measure",
      "title": "A spatial measure",
      "reader": "a-spatial-measure.html",
      "source": "src/a-spatial-measure.md",
      "source_sha256": "9b910b3fea97e13126d70963875b13e889e1cbfae4ff51f9f3061939b1ee9330",
      "status": "author_self_checked"
    },
    {
      "id": "boundary-activity",
      "title": "Boundary activity",
      "reader": "boundary-activity.html",
      "source": "src/boundary-activity.md",
      "source_sha256": "93052a43f2b07eb1ff71c1222e30252f33f62e731322584d0bdfc61ce434bb89",
      "status": "author_self_checked"
    },
    {
      "id": "levels-and-k-sets",
      "title": "Levels and k-sets",
      "reader": "levels-and-k-sets.html",
      "source": "src/levels-and-k-sets.md",
      "source_sha256": "307f11e6dd5f774a127c670ba5d65976793a80a21117e1fdba5c298fcf1bfb27",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "real analysis in one and several variables: core courses Real Analysis I and II (Lebl, Basic Analysis I and II, version 6.3)",
    "measure theory: core course Measure and Integration (Fremlin, Measure Theory, Volumes 1 and 2): the monotone class theorem (136B), the Radon-Nikodym theorem (232E), the Lebesgue decomposition (232I), density points (261D(b)), Lipschitz extensions (262B(b)) and Rademacher's theorem (262Q)",
    "the Riesz representation theorem: Theorem 2.2 of the programme's lesson Haar measure on locally compact groups (course AN-01), used in the fourth lesson",
    "Brouwer's fixed-point theorem: Theorem 30.1 of the core course Algebraic Topology, used in the ninth lesson",
    "planar graphs with v >= 3 vertices have at most 3v - 6 edges: Theorem 5.34 of the core course Applied Combinatorics (Keller and Trotter), used in the second lesson",
    "a nonzero polynomial does not vanish on a nonempty open set: Lemma 3.1 of the programme's lesson Drawings and crossing numbers (course crossing-numbers-of-complete-graphs), used in the first lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}