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  "body": "\n## Statement\n\nFor a real number $x$, let $\\norm{x}$ denote the distance from $x$ to the nearest integer.\nFor every pair of real numbers $\\alpha$ and $\\beta$,\n$$\\liminf_{n \\to \\infty} \\, n \\, \\norm{n\\alpha} \\, \\norm{n\\beta} = 0,$$\nwhere $n$ runs over the positive integers.\n\n## Remarks\n\nThe problem is open; Littlewood posed it around 1930.\nIt holds if $\\alpha$ is not badly approximable, that is, if $\\liminf_n n \\norm{n\\alpha} = 0$.\nIndeed $\\norm{n\\beta} \\le 1/2$ gives $n \\norm{n\\alpha} \\norm{n\\beta} \\le n \\norm{n\\alpha} / 2$.\nRational numbers are not badly approximable, and the same holds with the roles of $\\alpha$ and $\\beta$ exchanged.\nBy the theory of [[continued fractions|04P2RJ4WPS]], a number is badly approximable exactly when its partial quotients are bounded, so the open case is a pair of such [[irrational numbers|0DZXAXEJQ9]].\nCassels and Swinnerton-Dyer proved it in 1955 when $1, \\alpha, \\beta$ span a totally real cubic number field.\nEinsiedler, Katok and Lindenstrauss proved in 2006 that the set of pairs $(\\alpha, \\beta)$ for which it fails has Hausdorff dimension $0$.\nTheir proof uses the action of the diagonal group on $\\mathrm{SL}_3(\\R)/\\mathrm{SL}_3(\\Z)$ and the rigidity of measures invariant under it.\n\n## Explanations\n\n### L1 human\n\nEvery real number can be approximated by fractions, and some numbers, like the golden ratio, resist this as much as possible.\nThe conjecture says that two such numbers cannot resist at the same time: some multiple $n\\alpha$ and $n\\beta$ are both unusually close to integers.\nHere unusually close means that the product of the two errors beats $1/n$ by any factor you like.\nIt is a simple-looking statement about pairs of numbers that has been open for nearly a century.\n",
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    "history": {
      "first_stated": "1930",
      "posed_by": "John Edensor Littlewood"
    },
    "kind": "conjecture",
    "msc": [
      "11J13",
      "37A17",
      "22E40"
    ],
    "name": "Littlewood conjecture",
    "same_as": {
      "wikidata": "Q6653177",
      "wikipedia": "Littlewood conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Littlewood_conjecture"
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    ],
    "tag": "P8D6T5A1QD",
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/P8D6T5A1QD/",
  "links": [
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  "markdown": "https://tamnd.github.io/math2/node/P8D6T5A1QD.md",
  "msc": [
    "11J13",
    "37A17",
    "22E40"
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  "name": "Littlewood conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P8D/P8D6T5A1QD.md",
  "status": null,
  "tag": "P8D6T5A1QD",
  "uri": "m2://P8D6T5A1QD",
  "url": "https://tamnd.github.io/math2/wiki/littlewood-conjecture/",
  "writing": {
    "tag": "P8D6T5A1QD",
    "kind": "conjecture",
    "level": "W0",
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        "note": "the row of the inventory that the node accounts for",
        "have": 0,
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