Makarov's theorem on harmonic measure

Remarks

In short, the harmonic measure of a simply connected planar domain has dimension exactly $1$, whatever the dimension of the boundary. Let $f$ be a conformal map from the unit disc onto $\Omega$ with $f(0) = z$. By Carathéodory's theorem $f$ extends to the circle, and $\omega$ is the image of normalised arc length under this extension. Idea of the proof: the growth of $\log f'$, a Bloch function, is controlled by a law of the iterated logarithm. This shows that $f$ distorts the size of small arcs at almost every point only by subpower factors, which gives both parts. Example: for the domain bounded by the Koch snowflake, the boundary has dimension $\log 4 / \log 3 > 1$, yet $\omega$ is carried by a subset of dimension $1$. Simple connectivity matters for part 1: let $K$ be the middle-thirds Cantor set, of dimension $\log 2 / \log 3 < 1$ and of positive logarithmic capacity. The harmonic measure of the complement of $K$ in the Riemann sphere, at $\infty$, is carried by $K$ and so by a set of dimension less than $1$. The theorem was proved by N. G. Makarov in 1985.

Going deeper

For a student

Makarov's theorem started the study of the dimension of harmonic measure. For general planar domains, Jones and Wolff showed that harmonic measure is always carried by a set of dimension at most $1$. In higher dimensions the picture is different. Bourgain proved a bound strictly below the dimension of the space, and Wolff built domains in $\R^3$ whose harmonic measure has dimension larger than $2$. The proof links harmonic measure with the boundary behaviour of conformal maps and with probabilistic limit theorems.