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  "body": "\n## Definition\n\nLet $\\alpha$ be an [[irrational number|0DZXAXEJQ9]].\nLet $p_n / q_n$, for $n \\ge 0$, be the [[convergents|0D474C7ES1]] of the [[continued fraction|04P2RJ4WPS]] expansion of $\\alpha$, with $q_n \\ge 1$.\nThe number $\\alpha$ is a *Brjuno number* if\n$$\\sum_{n = 0}^{\\infty} \\frac{\\log q_{n+1}}{q_n} \u003c \\infty.$$\n\n## Remarks\n\nExample: for the golden mean $\\alpha = (\\sqrt{5} - 1)/2$, the $q_n$ are Fibonacci numbers.\nThey satisfy $q_{n+1} \\le 2 q_n$ and $q_n \\ge (3/2)^{n-1}$ for $n \\ge 1$.\nSo the terms are at most $\\log(2 q_n) / q_n$, which decay geometrically, and the series converges.\nNon-example: choose partial quotients $a_n \\ge 1$ inductively with $a_{n+1} \\ge e^{q_n}$.\nThen $q_{n+1} \\ge a_{n+1} q_n \\ge e^{q_n}$, so each term is at least $1$ and the series diverges.\nEvery Diophantine number, and hence almost every real number, is a Brjuno number.\nBrjuno proved that if $\\alpha$ is a Brjuno number, every germ $f(z) = e^{2\\pi i \\alpha} z + O(z^2)$ is linearizable at $0$.\nYoccoz proved that for the quadratic germ $e^{2\\pi i \\alpha} z + z^2$ the condition is also necessary.\nSo $e^{2\\pi i \\alpha} z + z^2$ has a [[Siegel disc|KNDYA9T8ZK]] around $0$ exactly when $\\alpha$ is a Brjuno number.\n",
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