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  "body": "\n## Statement\n\nLet $n \\ge 1$, and let $\\mathcal{L}_n$ be the set of [[lattices|WN9V7WSJKF]] in $\\R^n$.\nIdentify $\\mathcal{L}_n$ with the [[quotient space|0BR3FY5Q8M]] $\\mathrm{GL}_n(\\R)/\\mathrm{GL}_n(\\Z)$, by sending $g$ to the lattice $g\\Z^n$, and give it the quotient topology.\nLet $A \\subseteq \\mathcal{L}_n$.\nThen $A$ has [[compact|0R6EWFKRCF]] closure in $\\mathcal{L}_n$ if and only if there are constants $C \u003e 0$ and $\\varepsilon \u003e 0$ such that for every $\\Lambda \\in A$:\n\n1. the covolume of $\\Lambda$ is at most $C$;\n2. every nonzero vector $v \\in \\Lambda$ has $\\norm{v} \\ge \\varepsilon$.\n\n## Remarks\n\nThe proof uses reduction theory.\nUnder the two bounds, each lattice of $A$ has a reduced basis whose vectors have length bounded above and below, with angles bounded away from $0$.\nSuch bases form a compact set of matrices, whose image in $\\mathcal{L}_n$ contains $A$.\nThe converse holds because covolume and the length of a shortest nonzero vector are continuous on $\\mathcal{L}_n$.\n\nExample: the lattices $\\Lambda_t = \\Z (t, 0) \\oplus \\Z (0, 1/t)$ in $\\R^2$ have covolume $1$.\nTheir shortest nonzero vector has length $t$ for $0 \u003c t \\le 1$.\nAs $t \\to 0$ condition 2 fails, and indeed $\\Lambda_t$ leaves every compact subset of $\\mathcal{L}_2$.\n\nBoth conditions are needed.\nThe lattices $t\\Z^2$ with $t \\to \\infty$ satisfy condition 2 but not condition 1, and they also leave every compact set.\n\n## Explanations\n\n### L1 human\n\nA lattice in the plane is a regular grid of points.\nThe theorem says when a family of grids stays within a bounded region of the space of all grids.\nThe grids must not have too large a cell, and they must not have two points too close together.\nA grid that degenerates, by squashing in one direction, always produces short vectors.\n\n### L2 human\n\nFor unimodular lattices, condition 1 is automatic, so a set of unimodular lattices is relatively compact exactly when the lattices avoid a fixed ball around $0$.\nThis gives the basic criterion for divergence in the homogeneous space $\\mathrm{SL}_n(\\R)/\\mathrm{SL}_n(\\Z)$.\nIt is used in homogeneous dynamics, for example in the non-divergence results of Margulis and Dani for unipotent flows, and in Diophantine approximation through Dani's correspondence.\n",
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