A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a *[Brjuno number](https://en.wikipedia.org/wiki/Brjuno_number)* if $$\sum\_{i=1}^\infty\frac{\log q\_{i+1}}{q\_i} < \infty$$ where $q\_i>0$ is the denominator of the $i$th [convergent](https://en.wikipedia.org/wiki/Continued_fraction#Infinite_continued_fractions_and_convergents) $\frac{p\_i}{q\_i}$ of the [continued fraction](https://en.wikipedia.org/wiki/Continued_fraction) expansion of $\alpha$.


Is the set $B$ of Brjuno numbers measurable, and if yes, what is the measure of $B\cap[0,1]$?



Measurability of Brjuno numbers
I replace $\alpha$ by $x$ below.


Restricting ourselves to positive numbers is useless, since the property considered does not depend on $a\_0(x) = \lfloor x \rfloor$. Therefore, $B$ is the union of all images of $B \cap [0,1[$ by integer translations.


Now, call $G$ the Gauss-Kuzmin map from $[0,1[ \setminus \mathbb{Q}$ to itself: $G(x) = 1/x-\lfloor 1/x \rfloor$.
This map preserves the probability measure $\mu$ with density $x \mapsto (\ln 2)^{-1}(1+x)^{-1}$ on $[0,1[ \setminus \mathbb{Q}$ and is ergodic.


The partial quotients $(a\_n(x))\_{n \ge 1}$ of $\alpha$ are the integer parts of the iterates $(G^n(x))\_{n \ge 1}$. Hence $(a\_n)\_{n \ge 1}$ and $(q\_n)\_{n \ge 1}$ are Borel functions, so $B$ is a Borel set.


One checks that $\ln(a\_1+1)$ is integrable with regard to $\mu$. Birkhoff theorem applies, so when $n \to +\infty$
$$\frac{\ln q\_n}{n} \le \frac1n\sum\_{k=1}^n \ln(a\_k+1) \to C:= \int\ln(a\_1+1)\mathrm{d}\mu \quad \mu \text{-a.s..}$$


On the other hand, $(q\_n)\_{n \ge 1}$ is bounded below by Fibonacci sequence. Therefore, by comparison, the series $\sum\_n \frac{\ln q\_{n+1}}{q\_n} $ converges $\mu$-almost surely, and also almost surely for the Lebesgue measure.


