The dimension of self-similar measures

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.

Eight lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

  1. Entropy of finite random variables
  2. Entropy at a scale
  3. The one-bit deficit of a self-similar measure
  4. The gain from a pair of points
  5. The pair bound for a finite law
  6. The random walk of maps and the upper bound for the dimension
  7. Block types and pairs at fine scales
  8. Disjoint windows and the dimension formula

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites