{
  "id": "the-dimension-of-self-similar-measures",
  "title": "The dimension of self-similar measures",
  "description": "Entropy of random variables with finitely many values: the chain rule, conditioning, mutual information, mixtures, and estimates for the binary entropy. Entropy of a law on the line at a scale, averaged over the positions of the grid: comparison of scales, concavity, convolution, uniform steps. Self-similar measures on the line; entropy at small scales and dimension; uniform nonsaturation. The two-point gain; pair capacities; the gain of a convolution from fair pairs. The pair bound for a finite law, by random nested partitions. The entropy rate of the walk of maps; the upper bound for the dimension. Block types; entropy of translations given the types; pair capacity at fine scales. The dimension formula d = min(1, h/chi); the formula without exact overlaps. The course is an exposition, with a complete proof, of OpenAI's theorem (2026) on the dimension of self-similar measures on the line; the proof assumes that the measure is exact dimensional.",
  "units": [
    {
      "id": "entropy-of-finite-random-variables",
      "title": "Entropy of finite random variables",
      "reader": "entropy-of-finite-random-variables.html",
      "source": "src/entropy-of-finite-random-variables.md",
      "source_sha256": "e44436761f813ef1bf6db83f6f69933f26f61d34f1faf666c3ee7da10323c7bf",
      "status": "author_self_checked"
    },
    {
      "id": "entropy-at-a-scale",
      "title": "Entropy at a scale",
      "reader": "entropy-at-a-scale.html",
      "source": "src/entropy-at-a-scale.md",
      "source_sha256": "229cf12d7f22c90aa875e8d4ef1d91debef1f27844b989a63291e78feed378c9",
      "status": "author_self_checked"
    },
    {
      "id": "the-one-bit-deficit",
      "title": "The one-bit deficit of a self-similar measure",
      "reader": "the-one-bit-deficit.html",
      "source": "src/the-one-bit-deficit.md",
      "source_sha256": "56299a80ca88dfba972621911cf141e7feef92335189988f66a0a70a92158173",
      "status": "author_self_checked"
    },
    {
      "id": "the-gain-from-a-pair-of-points",
      "title": "The gain from a pair of points",
      "reader": "the-gain-from-a-pair-of-points.html",
      "source": "src/the-gain-from-a-pair-of-points.md",
      "source_sha256": "4d3ea2cc56cf3dcab8b2d85546a928ef856d0fba9c542a02d19043b5e2442a6c",
      "status": "author_self_checked"
    },
    {
      "id": "the-pair-bound-for-a-finite-law",
      "title": "The pair bound for a finite law",
      "reader": "the-pair-bound-for-a-finite-law.html",
      "source": "src/the-pair-bound-for-a-finite-law.md",
      "source_sha256": "92d9c347f850b495d2ac82b69417992bff912bf5d94fd791b70d52d300fec08b",
      "status": "author_self_checked"
    },
    {
      "id": "the-random-walk-of-maps-and-the-upper-bound",
      "title": "The random walk of maps and the upper bound for the dimension",
      "reader": "the-random-walk-of-maps-and-the-upper-bound.html",
      "source": "src/the-random-walk-of-maps-and-the-upper-bound.md",
      "source_sha256": "b481c5b85f702550289d9461bbe60c30e0cc47ff6ee9d832efb4fe4117c7e36c",
      "status": "author_self_checked"
    },
    {
      "id": "block-types-and-pairs-at-fine-scales",
      "title": "Block types and pairs at fine scales",
      "reader": "block-types-and-pairs-at-fine-scales.html",
      "source": "src/block-types-and-pairs-at-fine-scales.md",
      "source_sha256": "b1aea92b7ffc7d27837738b6a7f5a2a08a9ffc470fc882c49e1cc064f8a918a7",
      "status": "author_self_checked"
    },
    {
      "id": "disjoint-windows-and-the-dimension-formula",
      "title": "Disjoint windows and the dimension formula",
      "reader": "disjoint-windows-and-the-dimension-formula.html",
      "source": "src/disjoint-windows-and-the-dimension-formula.md",
      "source_sha256": "7e64b0ffe483706fd046e090f1b3bd675338a4e6536b6464dce030a93ee83be2",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Measure and Integration (D10): the Lebesgue integral of bounded measurable functions, measurable families of measures, exchange of two integrals of nonnegative functions",
    "finite sums, the natural logarithm and the inequality ln x <= x - 1; the lessons prove every other fact they use"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}