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  "body": "\n## Statement\n\nLet $\\alpha$ be a real [[algebraic number|0MNF0BYFQC]] that is [[irrational|0DZXAXEJQ9]], and let $\\varepsilon \u003e 0$.\nThen there are only finitely many [[rational numbers|0NCJ5WXN6Z]] $p/q$, with $p, q \\in \\Z$ and $q \\ge 1$, such that\n$$\\abs{\\alpha - \\frac{p}{q}} \u003c \\frac{1}{q^{2 + \\varepsilon}}.$$\n\n## Remarks\n\nThe proof builds an auxiliary polynomial in many variables with integer coefficients that vanishes to high order at $(\\alpha, \\dots, \\alpha)$.\nSuppose there were many very good approximations with fast growing denominators.\nThen this polynomial would have to vanish to high order at a rational point too, and a nonvanishing lemma (Roth's lemma) rules this out.\nThe proof is not effective: it gives a bound for the number of exceptional fractions, but not for their size.\nThe exponent $2$ cannot be lowered, because [[Dirichlet's approximation theorem|04JAY9DGB7]] gives infinitely many $p/q$ with $\\abs{\\alpha - p/q} \u003c 1/q^2$.\nThe hypothesis that $\\alpha$ is algebraic is necessary: a [[Liouville number|0PYEZM6CN9]] has, for each exponent $\\mu$, infinitely many approximations with error less than $1/q^{\\mu}$.\nFor example, $\\alpha = \\sqrt[3]{2}$ has only finitely many approximations with error less than $1/q^{2.01}$.\nThe theorem sharpens [[Liouville's theorem on Diophantine approximation|0N3WFZEFPM]], where the exponent is the degree of $\\alpha$, and the later results of Thue and Siegel.\n\n## Explanations\n\n### L1 human\n\nAn algebraic irrational number, such as $\\sqrt{2}$ or $\\sqrt[3]{2}$, cannot be approximated by fractions much better than any irrational can.\nEvery irrational has infinitely many fractions $p/q$ within $1/q^2$, but only finitely many within $1/q^{2.001}$ when the number is algebraic.\nNumbers that are approximated extremely well, like Liouville numbers, are therefore transcendental.\n\n### L2 human\n\nRoth's theorem implies that a [[Thue equation|3Q6CFPQZDF]] $F(x, y) = m$, with $F$ an irreducible binary form of degree at least $3$, has only finitely many integer solutions.\nIts higher-dimensional generalization is the [[Schmidt subspace theorem|9TPS3W497D]].\nThe theorem says that the irrationality exponent of each algebraic irrational is $2$, the same as for almost all real numbers.\n",
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