{
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  "body": "\n## Statement\n\n$R\\subseteq\\mathbb{N}^d$ is Presburger-definable if and only if:\n\nif $d \u003e 1$ then all sections of $R$ are Presburger-definable and\n\nthere exists $s\\in\\mathbb{N}$ such that, for every $k\\in\\mathbb{N}$, there exists $t\\in\\mathbb{N}$ such that for all $(x_0,\\dots,x_{d-1})\\in\\mathbb{N}^d$ with\n\n$$\\sum_{i=0}^{d-1}x_i\u003et,$$\n\n$R$ is $s$-periodic in $C(k,(x_0,\\dots,x_{d-1}))$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
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    "kind": "statement",
    "name": "Muchnik's Theorem",
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      {
        "locator": "Presburger arithmetic, section Presburger-definable integer relation \u003e Muchnik's characterization",
        "mention": "wikipedia://en/23756#X6QZ",
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        "wikipedia": "en:23756@r1374594859"
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  "markdown": "https://math2.tamnd.com/node/718DQ1KH8W.md",
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  "name": "Muchnik's Theorem",
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  "tag": "718DQ1KH8W",
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  "url": "https://math2.tamnd.com/wiki/muchniks-theorem/",
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    "tag": "718DQ1KH8W",
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