Bourgain's dimension bound for harmonic measure

Remarks

In short, $\dim \omega^x \le n - \varepsilon_n$: harmonic measure never fills a set of full dimension $n$, even when the boundary itself has dimension $n$. By mutual absolute continuity of harmonic measures, the conclusion does not depend on $x$. Idea of the proof: a multiscale argument looks at cubes that meet the boundary. In each such cube, harmonic measure is spread unevenly over the subcubes by a uniform amount. This loss at each scale accumulates into the dimension drop $\varepsilon_n$. Example: in the plane one can take $\varepsilon_2 = 1$. This is the theorem of Jones and Wolff that planar harmonic measure is carried by a set of dimension at most $1$. For Jordan domains it also follows from Makarov's theorem. The value of the best constant $\varepsilon_n$ is unknown for $n \ge 3$. It cannot be taken equal to $1$ in $\R^3$: Wolff constructed domains in $\R^3$ whose harmonic measure has dimension larger than $2$. The theorem was proved by J. Bourgain in 1987.

Going deeper

For a student

This is the main general result on the dimension of harmonic measure in higher dimensions. In the plane the sharp answer is known: dimension at most $1$ for all domains, and exactly $1$ for simply connected ones. In space, Wolff's examples show that the planar answer $n - 1$ is wrong, and Bourgain's bound is the general upper estimate. The question of the optimal $\varepsilon_n$ connects potential theory with geometric measure theory and the study of rough boundaries.