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.
- Entropy of finite random variables
- Entropy at a scale
- The one-bit deficit of a self-similar measure
- The gain from a pair of points
- The pair bound for a finite law
- The random walk of maps and the upper bound for the dimension
- Block types and pairs at fine scales
- Disjoint windows and the dimension formula
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Finite sums and the natural logarithm; the Lebesgue integral of bounded functions (core course Measure and Integration); the lessons prove the other facts they use.