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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $E \\subseteq \\R^n$ and let $x \\in \\R^n$.\nThe set $E$ is *thin at* $x$ if one of the following holds.\n\n1. The point $x$ is not a [[limit point|07EGCNHAVW]] of $E$.\n2. The point $x$ is a limit point of $E$, and there is a [[superharmonic function|X4V2S56YQQ]] $u$ on an open neighbourhood of $x$ such that\n$$\\liminf_{y \\to x,\\ y \\in E \\setminus \\set{x}} u(y) \u003e u(x).$$\n\n## Remarks\n\nThe left side of clause 2 is a [[limit inferior|033RZ32T4Q]] along points of $E$ other than $x$.\nExample: in $\\R^3$ let $x_k = 2^{-k} e_1$ for $k \\ge 1$ and $E = \\set{x_k : k \\ge 1}$, with $x = 0$.\nPut $u(y) = \\sum_{k \\ge 1} 4^{-k} \\abs{y - x_k}^{-1}$.\nThis is a series of positive Newtonian potentials with $u(0) = \\sum_k 4^{-k} 2^{k} = 1$, so $u$ is superharmonic on $\\R^3$.\nSince $u(x_k) = +\\infty$ for every $k$, clause 2 holds, and $E$ is thin at $0$.\nNon-example: a closed ball $B$ is not thin at a point $x \\in \\partial B$.\nIf $u$ were as in clause 2, then $u(x)$ is finite and $u \\ge u(x) + \\varepsilon$ on $B$ near $x$ for some $\\varepsilon \u003e 0$, while $u \\ge u(x) - \\delta$ near $x$ for any $\\delta \u003e 0$ by lower semicontinuity.\nA small sphere about $x$ has nearly half of its area in $B$, so its mean value of $u$ exceeds $u(x)$, against the super-mean value inequality.\nFor $n \\ge 3$, Wiener's criterion says that $E$ is thin at a limit point $x$ if and only if $\\sum_{k} 2^{k(n-2)} c(E_k) \u003c \\infty$, where $E_k = \\set{y \\in E : 2^{-k-1} \\le \\abs{y - x} \\le 2^{-k}}$ and $c$ is Newtonian capacity.\n",
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