This question relates to this one and that one.

**Some background**

In the setting of discrete holomorphic dynamics (say, Julia sets) an irrational $\lambda$ is said to be well approximated by rational numbers when \begin{eqnarray*} \sum_{n=0}^{+\infty}\frac{\ln q_{n+1}}{q_{n}} & = & +\infty\, \end{eqnarray*} where $\left(\frac{p_{n}}{q_{n}}\right)_{n\in\mathbb{N}}$ is its sequence of convergents (see continued fractions). This "arithmetic" condition was an improvement of Cremer condition (replace $\Sigma$ by $\limsup$) by Brjuno to tackle the (difficult) problem of linearization of germs of a biholomorphism \begin{eqnarray*} \Delta\left(z\right) & = & e^{2\mathtt{i}\pi\lambda}z+o\left(z\right). \end{eqnarray*} The theorem of Siegel-Brjuno asserts that if the condition does not hold then any such germ $\Delta$ is locally conformally conjugate to its linear part (the irrational rotation). The converse is a (very clever, needless to say) construction produced by Yoccoz: if the above condition holds then there exists some germs $\Delta$ which are not locally linearizable. His construction boils down to building a $\Delta$ with periodic orbits accumulating on $0$. To this day the moduli space of conjugacy of such germs is not known, and describing it remains an important open question in the field. Somehow all this is also connected to the recent (beautiful) work of Buff and Chéritat, where they build a Julia set of full Lebesgue measure.

It also relates to more "conventional" dynamics, as Perez-Marco exhibited other "arithmetic" conditions involved in the problem of classification of circle diffeomorphisms.

It is well known that irrational numbers well approximated by rational numbers, in the above sense, is a $PSL_{2}\left(\mathbb{Z}\right)$-invariant set (through the action by homographies on the real line) with zero Lebesgue measure. The fact that such numbers are so rare is related to the fact that, for almost every irrational number, the geometric mean of the integers $(a_{n})_{n\in\mathbb{N}}$ appearing in the continued fraction expansion converges to the Khinchin's constant $K_{0}\simeq2,68$.

**My question**

It is easy to produce theoretical examples of such numbers, just start from a sequence $\left(q_{n}\right)_{n\in\mathbb{N}}$ satisfying the condition and find an adequate sequence $\left(p_{n}\right)_{n\in\mathbb{N}}$, for instance by following a walk in the Stern-Brocot tree, such that $\lim\frac{p_{n}}{q_{n}}\notin\mathbb{Q}$.

Now, does anyone know about an "explicit" (for a reasonable notion of explicit) number which is well approximated by rational numbers?

To the best of my knowledge this question should be answered as "no", but the limits of my knowledge are not that far away, even from my point of view ;) Thank you in advance for any comment (better: answer!) to this wishful question.

2more comments