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  "body": "\n## Statement\n\nLet $\\xi$ be an [[irrational number|0DZXAXEJQ9]].\nThen there are infinitely many [[rational numbers|0NCJ5WXN6Z]] $p/q$, with $p, q \\in \\Z$ and $q \\ge 1$, such that\n$$\\abs{\\xi - \\frac{p}{q}} \u003c \\frac{1}{\\sqrt{5}\\, q^2}.$$\nThe constant $\\sqrt{5}$ is best possible.\nThat is, for each real $A \u003e \\sqrt{5}$ there is an irrational $\\xi$ with only finitely many rationals $p/q$ that satisfy $\\abs{\\xi - p/q} \u003c 1/(A q^2)$.\n\n## Remarks\n\nThe proof uses the [[continued fraction expansion|04VA28F3HB]] of $\\xi$.\nAmong any three consecutive convergents $p_n/q_n$ of $\\xi$, at least one satisfies the inequality, and there are infinitely many convergents.\nFor the second part, take the [[golden ratio|0SC8T7548D]] $\\varphi = (1 + \\sqrt{5})/2$, whose continued fraction is $[1; 1, 1, 1, \\dots]$.\nIts convergents are quotients of consecutive Fibonacci numbers, and $q^2 \\abs{\\varphi - p/q}$ tends to $1/\\sqrt{5}$ along them.\nThe theorem improves the constant $1$ of [[Dirichlet's approximation theorem|04JAY9DGB7]].\nIf one removes $\\varphi$ and all numbers equivalent to it under the action of $\\mathrm{GL}_2(\\Z)$, the best constant becomes $\\sqrt{8}$.\n\n## Explanations\n\n### L1 human\n\nEach irrational number can be approached by fractions $p/q$ with an error much smaller than $1/q^2$.\nHurwitz's theorem gives the exact best factor: the error can be made smaller than $1/(\\sqrt{5} q^2)$ infinitely often.\nThe golden ratio is the hardest number to approximate, and it shows that $\\sqrt{5}$ cannot be improved.\n\n### L2 human\n\nThe theorem belongs to the theory of the Lagrange spectrum, which records the best constants for each irrational number.\nContinued fractions are the main tool: the convergents give the best approximations.\nNumbers with bounded partial quotients, such as quadratic irrationals, are badly approximable, and the golden ratio is the extreme case.\n",
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