{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "8WWTMFR278"
    },
    {
      "field": "uses",
      "target": "FH0Z9VHRH7"
    },
    {
      "field": "defines",
      "target": "NX4V39GJCV"
    }
  ],
  "body": "\n## Summary\n\nA set $E \\subseteq \\R^n$ is thin at a point $x$ if $E$ is very sparse near $x$.\nPrecisely, some [[superharmonic function|X4V2S56YQQ]] is strictly larger along $E$ near $x$ than at $x$ itself.\nThinness measures how visible a set is to Brownian motion and to harmonic functions near a point.\nA boundary point of a domain is irregular for the Dirichlet problem exactly when the complement of the domain is thin there.\n[[Wiener's criterion|STMT1K1865]] decides thinness by a series of capacities.\n\n## Explanations\n\n### L1 human\n\nA set can come arbitrarily close to a point and still be almost invisible from it.\nThinness makes this precise with superharmonic functions, which may jump up along a thin set but not along a fat one.\nA sequence of points converging to $x$ in space $\\R^3$ is thin at $x$, because points have capacity zero.\nA solid ball is never thin at a point of its boundary.\nIn $\\R^3$, a sufficiently sharp spike, as in Lebesgue's example, is thin at its tip even though it reaches that point.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "31C40",
      "31B15"
    ],
    "name": "Thin set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fine_topology_(potential_theory)"
      }
    ],
    "tag": "7DGF6MMEGX"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/7DGF6MMEGX/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/7DGF6MMEGX.md",
  "msc": [
    "31C40",
    "31B15"
  ],
  "name": "Thin set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7DG/7DGF6MMEGX.md",
  "status": null,
  "tag": "7DGF6MMEGX",
  "uri": "m2://7DGF6MMEGX",
  "url": "https://tamnd.github.io/math2/wiki/thin-set/",
  "writing": {
    "tag": "7DGF6MMEGX",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}