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

Brjuno function

The real Brjuno function ${\displaystyle B(\alpha )}$ is defined for irrational numbers ${\displaystyle \alpha }$ ^[1]

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

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:^[2]

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

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.