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 Template:Harvtxt.

Formal definition

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

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

converges to a finite number.

Here:

• $q_n$ is the denominator of the Template:Mvarth convergent $\frac{p_n}{q_n}$ of the continued fraction expansion of $\alpha$.
• $B$ is a Brjuno function

Examples

Consider the golden ratio Template:Phi:

$\varphi =\frac{1+\sqrt{5}}{2}=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\ddots }}}}.$

Then the nth convergent $\frac{p_n}{q_n}$ can be found via the recurrence relation:Template:Sfn

$\{\begin{array}{cc}{p}_n=p_{n-1}+p_{n-2}& \text{ with }{p}_0=1,p_1=2,\\ q_n=q_{n-1}+q_{n-2}& \text{ with }{q}_0=q_1=1.\end{array}$

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 $\sum _{n=0}^{\mathrm{\infty }}\frac{\log q_n}{q_n}<\mathrm{\infty }$ for any irrational number, Template:Phi 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.Template:Sfn

By contrast, consider the constant $\alpha =[a_0,a_1,a_2,\dots ]$ with $(a_n)$ defined as

$a_n=\{\begin{array}{cc}10& \text{ if }n=0,1,\\ q_n^{q_{n-1}}& \text{ if }n\ge 2.\end{array}$

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

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. Template:Harvs 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 (Template:Math)th convergent is exponentially larger than that of the Template:Mvarth 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

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

where:

• $q_n$ is the denominator of the Template:Mvarth convergent $\frac{p_n}{q_n}$ of the continued fraction expansion of $\alpha$.

Real variant

Brjuno function

The real Brjuno function $B(\alpha )$ is defined for irrational numbers $\alpha$

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

and satisfies

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

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

Yoccoz's variant

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

$Y(\alpha )=\sum _{n=0}^{\mathrm{\infty }}{\alpha }_0\cdots {\alpha }_{n-1}\log \frac{1}{{\alpha }_n},$

where:

• $\alpha$ is irrational real number: $\alpha \in ℝ\setminus \mathbb{Q}$
• ${\alpha }_0$ 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.

See also

• Irrationality measure
• Markov constant

References

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