Brjuno number

Remarks

Example: for the golden mean $\alpha = (\sqrt{5} - 1)/2$, the $q_n$ are Fibonacci numbers. They satisfy $q_{n+1} \le 2 q_n$ and $q_n \ge (3/2)^{n-1}$ for $n \ge 1$. So the terms are at most $\log(2 q_n) / q_n$, which decay geometrically, and the series converges. Non-example: choose partial quotients $a_n \ge 1$ inductively with $a_{n+1} \ge e^{q_n}$. Then $q_{n+1} \ge a_{n+1} q_n \ge e^{q_n}$, so each term is at least $1$ and the series diverges. Every Diophantine number, and hence almost every real number, is a Brjuno number. Brjuno proved that if $\alpha$ is a Brjuno number, every germ $f(z) = e^{2\pi i \alpha} z + O(z^2)$ is linearizable at $0$. Yoccoz proved that for the quadratic germ $e^{2\pi i \alpha} z + z^2$ the condition is also necessary. So $e^{2\pi i \alpha} z + z^2$ has a Siegel disc around $0$ exactly when $\alpha$ is a Brjuno number.