# Brjuno number

In mathematics, a Brjuno number is a special type of irrational number.

## Formal definition

An irrational number ${\displaystyle \alpha }$ is called a Brjuno number when the infinite sum

${\displaystyle B(\alpha )=\sum _{n=0}^{\infty }{\frac {\log q_{n+1}}{q_{n}}}}$

converges to a finite number

Here:

• ${\displaystyle q_{n}}$ is the denominator of the nth convergent ${\displaystyle {\frac {p_{n}}{q_{n}}}}$ of the continued fraction expansion of ${\displaystyle \alpha }$.
• ${\displaystyle B}$ is a Brjuno function

## Name

The Brjuno numbers are named after Alexander Bruno, who introduced them in Brjuno (1971); they are also occasionally spelled Bruno numbers or Bryuno numbers.

## 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 ${\displaystyle e^{2\pi i\alpha }}$ are linearizable if ${\displaystyle \alpha }$ is a Brjuno number. Jean-Christophe Yoccoz (1995) 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 nth convergent. Thus, in contrast to the Liouville numbers, they do not have unusually accurate diophantine approximations by rational numbers.

## Brjuno function

The real Brjuno function ${\displaystyle B(x)}$ is defined for irrational x and satisfies

${\displaystyle B(x)=B(x+1)}$
${\displaystyle B(x)=-\log x+xB(1/x)}$ for all irrational x between 0 and 1.