Brjuno number

In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in.

Formal definition
An irrational number $$\alpha$$ is called a Brjuno number when the infinite sum
 * $$B(\alpha) = \sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n}$$

converges to a finite number.

Here:
 * $$ q_n $$ is the denominator of the $n$th convergent $$\tfrac{p_n}{q_n}$$ of the continued fraction expansion of $$\alpha$$.
 * $$B$$ is a Brjuno function

Examples
Consider the golden ratio $\phi$:
 * $$\phi = \frac{1+\sqrt{5}}{2} = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}}.$$

Then the nth convergent $$\frac{p_n}{q_n}$$ can be found via the recurrence relation:
 * $$\begin{cases}

p_n = p_{n-1} + p_{n-2} & \text{ with } p_0=1,p_1=2, \\ q_n = q_{n-1} + q_{n-2} & \text{ with } q_0=q_1=1. \end{cases}$$ It is easy to see that $$q_{n+1}<q_n^2$$ for $$n \ge 2$$, as a result
 * $$\frac{\log{q_{n+1}}}{q_n} < \frac{2\log{q_{n}}}{q_n} \text{ for } n \ge 2$$

and since it can be proven that $$\sum_{n=0}^\infty \frac{\log q_n}{q_n} < \infty$$ for any irrational number, 𝜙 is a Brjuno number. Moreover, a similar method can be used to prove that any irrational number whose continued fraction expansion ends with a string of 1's is a Brjuno number.

By contrast, consider the constant $$\alpha = [a_0,a_1,a_2,\ldots]$$ with $$(a_n)$$ defined as
 * $$a_n = \begin{cases}

10           & \text{ if } n  =  0,1, \\ q_n^{q_{n-1}} & \text{ if } n \ge 2. \end{cases}$$ Then $$q_{n+1}>q_n^\frac{2q_n}{q_{n-1}}$$, so we have by the ratio test that $$\sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n}$$ diverges. $$\alpha$$ is therefore not a Brjuno number.

Importance
The Brjuno numbers are important in the one–dimensional analytic small divisors problems. Bruno improved the diophantine condition in Siegel's Theorem, showed that germs of holomorphic functions with linear part $$e^{2\pi i \alpha}$$ are linearizable if $$\alpha$$ is a Brjuno number. showed in 1987 that this condition is also necessary, and for quadratic polynomials is necessary and sufficient.

Properties
Intuitively, these numbers do not have many large "jumps" in the sequence of convergents, in which the denominator of the ($n + 1$)th convergent is exponentially larger than that of the $n$th convergent. Thus, in contrast to the Liouville numbers, they do not have unusually accurate diophantine approximations by rational numbers.

Brjuno sum
The Brjuno sum or Brjuno function $$B$$ is


 * $$B(\alpha) = \sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n}$$

where:
 * $$ q_n $$ is the denominator of the $n$th convergent $$\tfrac{p_n}{q_n}$$ of the continued fraction expansion of $$\alpha$$.

Real variant
The real Brjuno function $$B(\alpha)$$ is defined for irrational numbers $$\alpha$$
 * $$ B : \R \setminus \Q \to \R \cup \{ +\infty \} $$

and satisfies


 * $$\begin{align}

B(\alpha) &= B(\alpha+1) \\ B(\alpha) &= - \log \alpha + \alpha B(1/\alpha) \end{align}$$

for all irrational $$\alpha$$ between 0 and 1.

Yoccoz's variant
Yoccoz's variant of the Brjuno sum defined as follows:


 * $$Y(\alpha)=\sum_{n=0}^{\infty} \alpha_0\cdots \alpha_{n-1} \log \frac{1}{\alpha_n},$$

where: This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.
 * $$\alpha$$ is irrational real number: $$\alpha\in \R \setminus \Q $$
 * $$\alpha_0$$ is the fractional part of $$\alpha$$
 * $$\alpha_{n+1}$$ is the fractional part of $$\frac{1}{\alpha_n}$$