{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be a nonempty [[bounded set|09QSZVVM26]] in $\\R^n$.\nFor $\\varepsilon \u003e 0$, let $N(A, \\varepsilon)$ be the smallest number of sets of [[diameter|0KW3PN547K]] at most $\\varepsilon$ that cover $A$.\nThe *upper* and *lower box-counting dimensions* of $A$ are\n$$\\overline{\\dim}_B A = \\limsup_{\\varepsilon \\to 0^+} \\frac{\\log N(A, \\varepsilon)}{\\log (1/\\varepsilon)}, \\qquad \\underline{\\dim}_B A = \\liminf_{\\varepsilon \\to 0^+} \\frac{\\log N(A, \\varepsilon)}{\\log (1/\\varepsilon)}.$$\nIf they are equal, the common value $\\dim_B A$ is the *Minkowski–Bouligand dimension*, or box-counting dimension, of $A$.\n\n## Remarks\n\nOne gets the same values if $N(A, \\varepsilon)$ counts the cubes of an $\\varepsilon$-mesh grid that meet $A$, or the largest number of disjoint balls of radius $\\varepsilon$ centred in $A$.\nFor bounded $A$, $\\overline{\\dim}_B A = \\inf \\set{ s : \\mathcal{M}^{*s}(A) = 0 }$, where $\\mathcal{M}^{*s}$ is the upper [[Minkowski content|HS2NTCMZ4T]].\nAlways $\\dim_H A \\le \\underline{\\dim}_B A \\le \\overline{\\dim}_B A$, where $\\dim_H$ is the [[Hausdorff dimension|SSNFTHZ6YJ]].\n\nExample: let $C$ be the middle-thirds [[Cantor set|0BXNDZFQME]].\nIt is covered by $2^k$ intervals of length $3^{-k}$, and each set of diameter at most $3^{-k}$ meets at most two of them.\nSo $2^{k-1} \\le N(C, 3^{-k}) \\le 2^k$, and the ratio tends to $\\log 2 / \\log 3$, so $\\dim_B C = \\log 2 / \\log 3$.\n\nThe box dimension does not ignore countable sets.\nThe set $\\Q \\cap [0, 1]$ is dense, so $N(\\Q \\cap [0, 1], \\varepsilon) = N([0, 1], \\varepsilon)$, and its box dimension is $1$.\nIts Hausdorff dimension is $0$.\n\n## Explanations\n\n### L1 human\n\nCount how many small sets of size $\\varepsilon$ are needed to cover the set.\nIf the count grows like $\\varepsilon^{-d}$ as $\\varepsilon$ shrinks, the box dimension is $d$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "WYRKJ412FT",
    "kind": "definition",
    "msc": [
      "28A80",
      "28A75"
    ],
    "name": "Minkowski–Bouligand dimension",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Minkowski%E2%80%93Bouligand_dimension"
      }
    ],
    "tag": "F7G7CJY143",
    "uses": [
      "09QSZVVM26",
      "0KW3PN547K",
      "0T3RZQXDYD",
      "0BXNDZFQME",
      "SSNFTHZ6YJ",
      "HS2NTCMZ4T"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/F7G7CJY143/",
  "links": [
    {
      "field": "defines",
      "target": "WYRKJ412FT"
    },
    {
      "field": "uses",
      "target": "09QSZVVM26"
    },
    {
      "field": "uses",
      "target": "0KW3PN547K"
    },
    {
      "field": "uses",
      "target": "0T3RZQXDYD"
    },
    {
      "field": "uses",
      "target": "0BXNDZFQME"
    },
    {
      "field": "uses",
      "target": "SSNFTHZ6YJ"
    },
    {
      "field": "uses",
      "target": "HS2NTCMZ4T"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/F7G7CJY143.md",
  "msc": [
    "28A80",
    "28A75"
  ],
  "name": "Minkowski–Bouligand dimension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/F7G/F7G7CJY143.md",
  "status": null,
  "tag": "F7G7CJY143",
  "uri": "m2://F7G7CJY143",
  "url": "https://math2.tamnd.com/wiki/minkowski-bouligand-dimension-definition/",
  "writing": {
    "tag": "F7G7CJY143",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}