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  "body": "\n## Statement\n\nLet $n \\ge 2$, and let $L_1, \\dots, L_n$ be [[linearly independent|0CRMGPAH83]] [[linear forms|07YQ1C78G0]] in $n$ variables whose coefficients are real [[algebraic numbers|0MNF0BYFQC]].\nLet $\\varepsilon \u003e 0$.\nFor $x = (x_1, \\dots, x_n) \\in \\Z^n$ write $\\norm{x} = \\max_i \\abs{x_i}$.\nThen the set of nonzero $x \\in \\Z^n$ with\n$$\\abs{L_1(x) \\, L_2(x) \\cdots L_n(x)} \u003c \\norm{x}^{-\\varepsilon}$$\nis contained in the union of finitely many proper linear subspaces of $\\Q^n$.\n\n## Remarks\n\nFor $n = 2$, take $L_1(x, y) = \\alpha y - x$ and $L_2(x, y) = y$ with $\\alpha$ a real algebraic irrational.\nFor $x/y$ close to $\\alpha$, the norm $\\norm{(x, y)}$ is at most a constant times $\\abs{y}$.\nUp to this constant, the inequality then reads $\\abs{\\alpha - x/y} \u003c \\abs{y}^{-2-\\varepsilon}$, and the theorem gives the [[Thue–Siegel–Roth theorem|R9JZA3G2NX]].\nThe proof extends Roth's method with an auxiliary polynomial and uses the geometry of numbers.\nThe conclusion speaks of finitely many subspaces, not finitely many points: in the example above, all the points $(x, 0)$ satisfy the inequality, because $L_2$ vanishes on them.\nLike Roth's theorem, the result is not effective.\n\n## Explanations\n\n### L1 human\n\nRoth's theorem says that an algebraic number has few very good rational approximations.\nThe subspace theorem says the same for several algebraic linear forms at once.\nThe integer points where the product of the forms is very small do not spread out: they all lie on finitely many planes.\n\n### L2 human\n\nThe theorem is the main tool for finiteness results on Diophantine equations in several variables, such as $S$-unit equations $x_1 + \\dots + x_n = 1$.\nIt also gives transcendence results for numbers with special digit expansions.\nVersions with several places, the $p$-adic subspace theorem, are used for these applications.\n",
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