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  "body": "\n## Statement\n\nLet $n \\ge 3$ and let $Q$ be a real [[quadratic form|0RBQS1JJCR]] in $n$ variables.\nAssume that $Q$ is nondegenerate, that $Q$ is [[indefinite|01RBSSGMY5]], and that $Q$ is not a real multiple of a form with rational coefficients.\nThen the set of values $Q(\\Z^n) = \\set{Q(x) : x \\in \\Z^n}$ is [[dense|0BSJSX7ME2]] in $\\R$.\nIn particular, for every $\\varepsilon \u003e 0$ there is $x \\in \\Z^n \\setminus \\set{0}$ with $0 \u003c \\abs{Q(x)} \u003c \\varepsilon$.\n\n## Remarks\n\nMargulis proved the statement in 1987, so the name \"conjecture\" is historical.\nFollowing an idea of Raghunathan, it reduces to the case $n = 3$ and to the orbits of the group $\\mathrm{SO}(Q)$ on the space $\\mathrm{SL}_3(\\R)/\\mathrm{SL}_3(\\Z)$ of unimodular lattices in $\\R^3$.\nThe heart of the proof is Margulis's theorem that a relatively compact orbit of $\\mathrm{SO}(2,1)$ in this space is compact, while the irrationality of $Q$ rules out a compact orbit.\n[[Ratner's theorems|3MXR03096V]] later gave a general picture of such orbit closures, as predicted by [[Raghunathan's conjecture|ME4ZSRAT5Z]].\nAn example: $Q(x, y, z) = x^2 + y^2 - \\sqrt{2}\\, z^2$ satisfies the hypotheses, so its values at integer points are dense in $\\R$.\nEach hypothesis is needed.\nIf $Q$ is definite, the values on $\\Z^n$ form a discrete set.\nIf $Q$ is a multiple $c Q_0$ of a rational form $Q_0$ with denominators dividing $d$, its values lie in the discrete set $(c/d)\\Z$.\nFor $n = 2$ the form $x^2 - \\varphi^2 y^2 = (x - \\varphi y)(x + \\varphi y)$, with $\\varphi$ the golden ratio, is irrational and indefinite.\nBut $\\abs{x - \\varphi y} \\ge c/\\abs{y}$ for a constant $c \u003e 0$ and all integers $x$ and $y \\ne 0$.\nThis shows that the nonzero values of the form at integer points stay away from $0$.\n\n## Explanations\n\n### L1 human\n\nA quadratic form is a polynomial like $x^2 + y^2 - \\sqrt{2}\\, z^2$.\nFeed it only whole numbers and look at the outputs.\nThe theorem says that, with at least three variables, mixed signs and an irrational coefficient that cannot be scaled away, the outputs come arbitrarily close to every real number.\nThis is a statement about integers that was proved with the dynamics of group actions.\n\n### L2 human\n\nThe theorem was a turning point for the use of homogeneous dynamics in number theory.\nThe key move is to replace integer points by the lattice $\\Z^3$.\nThen one moves this lattice by elements of the orthogonal group of $Q$, which do not change the values of $Q$.\nThen density of values becomes a statement about the closure of an orbit of $\\mathrm{SO}(2,1)$ in the space of lattices.\nThis viewpoint led to Ratner's classification of orbit closures for unipotent flows and to quantitative versions counting integer points with $a \u003c Q(x) \u003c b$.\n",
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  "msc": [
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  "name": "Oppenheim conjecture",
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  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4S5/4S50G4PS26.md",
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