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

## Examples

Consider the golden ratio 𝜙:

${\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}.}$

Then the nth convergent ${\displaystyle {\frac {p_{n}}{q_{n}}}}$ can be found via the recurrence relation:[1]

${\displaystyle {\begin{cases}p_{n}=p_{n-1}+p_{n-2}&{\text{ with }}p_{0}=1,p_{1}=2,\\q_{n}=q_{n-1}+q_{n-2}&{\text{ with }}q_{0}=q_{1}=1.\end{cases}}}$

It is easy to see that ${\displaystyle q_{n+1} for ${\displaystyle n\geq 2}$, as a result

${\displaystyle {\frac {\log {q_{n+1}}}{q_{n}}}<{\frac {2\log {q_{n}}}{q_{n}}}{\text{ for }}n\geq 2}$

and since it can be proven that ${\displaystyle \sum _{n=0}^{\infty }{\frac {\log q_{n}}{q_{n}}}<\infty }$ 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.[2]

By contrast, consider the constant ${\displaystyle \alpha =[a_{0},a_{1},a_{2},\ldots ]}$ with ${\displaystyle (a_{n})}$ defined as

${\displaystyle a_{n}={\begin{cases}10&{\text{ if }}n=0,1,\\q_{n}^{q_{n-1}}&{\text{ if }}n\geq 2.\end{cases}}}$

Then ${\displaystyle q_{n+1}>q_{n}^{\frac {2q_{n}}{q_{n-1}}}}$, so we have by the ratio test that ${\displaystyle \sum _{n=0}^{\infty }{\frac {\log q_{n+1}}{q_{n}}}}$ diverges. ${\displaystyle \alpha }$ is therefore not a Brjuno number.[3]

## 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

### Brjuno sum

The Brjuno sum or Brjuno function ${\displaystyle B}$ is

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

where:

• ${\displaystyle q_{n}}$ is the denominator of the nth convergent ${\displaystyle {\tfrac {p_{n}}{q_{n}}}}$ of the continued fraction expansion of ${\displaystyle \alpha }$.

### Real variant

The real Brjuno function ${\displaystyle B(\alpha )}$ is defined for irrational numbers ${\displaystyle \alpha }$ [4]

${\displaystyle B:\mathbb {R} \setminus \mathbb {Q} \to \mathbb {R} \cup \{+\infty \}}$

and satisfies

{\displaystyle {\begin{aligned}B(\alpha )&=B(\alpha +1)\\B(\alpha )&=-\log \alpha +\alpha B(1/\alpha )\end{aligned}}}

for all irrational ${\displaystyle \alpha }$ between 0 and 1.

### Yoccoz's variant

Yoccoz's variant of the Brjuno sum defined as follows:[5]

${\displaystyle Y(\alpha )=\sum _{n=0}^{\infty }\alpha _{0}\cdots \alpha _{n-1}\log {\frac {1}{\alpha _{n}}},}$

where:

• ${\displaystyle \alpha }$ is irrational real number: ${\displaystyle \alpha \in \mathbb {R} \setminus \mathbb {Q} }$
• ${\displaystyle \alpha _{0}}$ is the fractional part of ${\displaystyle \alpha }$
• ${\displaystyle \alpha _{n+1}}$ is the fractional part of ${\displaystyle {\frac {1}{\alpha _{n}}}}$

This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.