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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\C$ be a bounded domain whose boundary $\\partial\\Omega$ is a Jordan curve.\nLet $z \\in \\Omega$, and let $\\omega$ be the [[harmonic measure|CFD7BPM51W]] of $\\Omega$ at $z$.\nThen:\n\n1. $\\omega(E) = 0$ for every Borel set $E \\subseteq \\partial\\Omega$ of [[Hausdorff dimension|SSNFTHZ6YJ]] less than $1$.\n2. There is a Borel set $F \\subseteq \\partial\\Omega$ of Hausdorff dimension $1$ with $\\omega(\\partial\\Omega \\setminus F) = 0$.\n\n## Remarks\n\nIn short, the harmonic measure of a [[simply connected|0QW64ARSHB]] planar domain has dimension exactly $1$, whatever the dimension of the boundary.\nLet $f$ be a conformal map from the unit disc onto $\\Omega$ with $f(0) = z$.\nBy Carathéodory's theorem $f$ extends to the circle, and $\\omega$ is the image of normalised arc length under this extension.\nIdea of the proof: the growth of $\\log f'$, a Bloch function, is controlled by a law of the iterated logarithm.\nThis shows that $f$ distorts the size of small arcs at almost every point only by subpower factors, which gives both parts.\nExample: for the domain bounded by the Koch snowflake, the boundary has dimension $\\log 4 / \\log 3 \u003e 1$, yet $\\omega$ is carried by a subset of dimension $1$.\nSimple connectivity matters for part 1: let $K$ be the middle-thirds Cantor set, of dimension $\\log 2 / \\log 3 \u003c 1$ and of positive logarithmic capacity.\nThe 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 $1$.\nThe theorem was proved by N. G. Makarov in 1985.\n\n## Explanations\n\n### L1 human\n\nRelease a random walker inside a planar region bounded by a single closed curve, and record where it first hits the curve.\nEven if the curve is wildly wiggly and has dimension larger than $1$, the hitting places concentrate on a part of dimension exactly $1$.\nThe walker almost never reaches the deep fjords of the curve.\n\n### L2 human\n\nMakarov's theorem started the study of the dimension of harmonic measure.\nFor general planar domains, Jones and Wolff showed that harmonic measure is always carried by a set of dimension at most $1$.\nIn higher dimensions the picture is different.\nBourgain proved a bound strictly below the dimension of the space, and Wolff built domains in $\\R^3$ whose harmonic measure has dimension larger than $2$.\nThe proof links harmonic measure with the boundary behaviour of conformal maps and with probabilistic limit theorems.\n",
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  "name": "Makarov's theorem on harmonic measure",
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