# 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 Brjuno (1971).

## 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 nth convergent ${\tfrac {p_{n}}{q_{n}}}$ of the continued fraction expansion of $\alpha$.
* $B$ is a Brjuno function

## Examples

Consider the golden ratio 𝜙:
    $\phi ={\frac {1+{\sqrt {5}}}{2}}=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}.$
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 $n\geq 2$, as a result
    ${\frac {\log {q_{n+1}}}{q_{n}}}<{\frac {2\log {q_{n}}}{q_{n}}}{\text{ for }}n\geq 2$
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
    $a_{n}={\begin{cases}10&{\text{ if }}n=0,1,\\q_{n}^{q_{n-1}}&{\text{ if }}n\geq 2.\end{cases}}$
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 by showing that germs of holomorphic functions with linear part $e^{2\pi i\alpha }$ are linearizable if $\alpha$ is a Brjuno number. Jean-Christophe Yoccoz (1995) showed in 1987 that Brjuno's condition is sharp; more precisely, he proved that for quadratic polynomials, this condition is not only sufficient but also necessary for linearizability.

## 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 nth convergent. Thus, in contrast to the Liouville numbers, they do not have unusually accurate diophantine approximations by rational numbers.

## Brjuno function
### 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 nth 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 $\alpha$
    $B:\mathbb {R} \setminus \mathbb {Q} \to \mathbb {R} \cup \{+\infty \}$
and satisfies
    ${\begin{aligned}B(\alpha )&=B(\alpha +1)\\B(\alpha )&=-\log \alpha +\alpha B(1/\alpha )\end{aligned}}$
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:
* $\alpha$ is irrational real number: $\alpha \in \mathbb {R} \setminus \mathbb {Q}$
* $\alpha _{0}$ is the fractional part of $\alpha$
* $\alpha _{n+1}$ is the fractional part of ${\frac {1}{\alpha _{n}}}$
This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.