{
  "id": "erdos-similarity-for-geometric-progressions",
  "title": "The Erdős similarity conjecture for geometric progressions",
  "description": "For every q in (0, 1), a compact subset of [0, 1] of measure as close to 1 as we like contains no affine copy x + s{q^n : n ≥ 1}, s ≠ 0, so geometric progressions are not measure universal. Random tables on nested dyadic grids route every point through a finite ordered tree; the first default choice on the route of a center offers many independent chances to hit a copy, windows of indices with gaps keep earlier decisions unchanged, and finitely many dilations represent all of them. The course is an exposition of OpenAI's proof (2026) of the geometric-progression case of the Erdős similarity conjecture.",
  "units": [
    {
      "id": "measure-universal-sets-and-periodic-hitting-sets",
      "title": "Measure-universal sets and periodic hitting sets",
      "reader": "measure-universal-sets-and-periodic-hitting-sets.html",
      "source": "src/measure-universal-sets-and-periodic-hitting-sets.md",
      "source_sha256": "61c52875f17dddb339768096e387a7e13d02dbbe5b14c835af827f59cbe0df8b",
      "status": "author_self_checked"
    },
    {
      "id": "dyadic-grids-and-windows-on-a-tree",
      "title": "Dyadic grids and windows on a tree",
      "reader": "dyadic-grids-and-windows-on-a-tree.html",
      "source": "src/dyadic-grids-and-windows-on-a-tree.md",
      "source_sha256": "75bcc246c390bda070c15c3d4c814b3126a3016c135e72877db1f1c40ab6a70c",
      "status": "author_self_checked"
    },
    {
      "id": "random-routing-and-independent-tests",
      "title": "Random routing and independent tests",
      "reader": "random-routing-and-independent-tests.html",
      "source": "src/random-routing-and-independent-tests.md",
      "source_sha256": "512f33e58050072cef1e13b2d82ec559dd4aed7a9c7a5cbdeff96cb73303249e",
      "status": "author_self_checked"
    },
    {
      "id": "every-scale-and-every-center",
      "title": "Every scale and every center",
      "reader": "every-scale-and-every-center.html",
      "source": "src/every-scale-and-every-center.md",
      "source_sha256": "693288e158d04b39aaa9cdb64b82f97849245d51e7cb1e6ba0b20d9e1fa2b6e1",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Measure and Integration (Fremlin, Measure Theory, Volumes 1 and 2): countable additivity and continuity of measures (112C), translation invariance of Lebesgue measure (134A), outer regularity of Lebesgue outer measure (134F(a)), linear changes of variables (263A)",
    "core course Real Analysis I (Lebl, Basic Analysis I): the Bolzano-Weierstrass theorem (Theorem 2.3.8)",
    "finite probability spaces: independence and conditioning on events of positive probability"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}