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

Examples

Consider the golden ratio 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:^[1]

{\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.^[2]

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.^[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 $e^{2\pi i\alpha }$ are linearizable if $\alpha$ 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 $n$ th 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 $n$ 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 $\alpha$ ^[4]

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:^[5]

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