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  "body": "\n## Statement\n\nLet $n \\ge 2$.\nThere is a number $\\varepsilon_n \u003e 0$, depending only on $n$, with the following property.\nLet $\\Omega \\subseteq \\R^n$ be a bounded domain, let $x \\in \\Omega$, and let $\\omega^x$ be the [[harmonic measure|CFD7BPM51W]] of $\\Omega$ at $x$.\nThen there is a Borel set $F \\subseteq \\partial\\Omega$ with $\\omega^x(\\partial\\Omega \\setminus F) = 0$ and [[Hausdorff dimension|SSNFTHZ6YJ]] at most $n - \\varepsilon_n$.\n\n## Remarks\n\nIn 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 $n$.\nBy mutual absolute continuity of harmonic measures, the conclusion does not depend on $x$.\nIdea of the proof: a multiscale argument looks at cubes that meet the boundary.\nIn each such cube, harmonic measure is spread unevenly over the subcubes by a uniform amount.\nThis loss at each scale accumulates into the dimension drop $\\varepsilon_n$.\nExample: in the plane one can take $\\varepsilon_2 = 1$.\nThis is the theorem of Jones and Wolff that planar harmonic measure is carried by a set of dimension at most $1$.\nFor Jordan domains it also follows from [[Makarov's theorem|CM9BDNEA10]].\nThe value of the best constant $\\varepsilon_n$ is unknown for $n \\ge 3$.\nIt cannot be taken equal to $1$ in $\\R^3$: Wolff constructed domains in $\\R^3$ whose harmonic measure has dimension larger than $2$.\nThe theorem was proved by J. Bourgain in 1987.\n\n## Explanations\n\n### L1 human\n\nStart a random path in a domain in space and record where it first hits the boundary.\nEven if the boundary is so rough that it has full dimension, the hitting points concentrate on a set of strictly smaller dimension.\nThe drop in dimension is at least a fixed amount that depends only on the dimension of the space.\n\n### L2 human\n\nThis is the main general result on the dimension of harmonic measure in higher dimensions.\nIn the plane the sharp answer is known: dimension at most $1$ for all domains, and exactly $1$ for simply connected ones.\nIn space, Wolff's examples show that the planar answer $n - 1$ is wrong, and Bourgain's bound is the general upper estimate.\nThe question of the optimal $\\varepsilon_n$ connects potential theory with geometric measure theory and the study of rough boundaries.\n",
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