"Explicit" examples of Irrational numbers very well approximated by rational numbers

Question

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.

Answer 1

Your question seems somewhat ill-posed. What type of construction exactly do you allow? I suppose that you might ask for the number to be obtained by basic algebraic operations and perhaps some elementary functions from rational numbers, and perhaps some standard transcendental constants such as $\pi$ and $e$. However, to make it a precise question would require some work, and in some sense whatever you come up with would likely be arbitrary.

I strongly believe that no such number is known, though I do not have a definitive reference stating this, and there are people who know more about this. However, it seems that, where the continued fraction expansions are explicitly known (such as for $e$), the growth of the expansion tends to be at most linear, a far cry from what you would need to violate Bryuno's condition.

The next number you might try is probably $\pi$. However, as far as I understand, it hasn't even been proved that its continued fraction expansion is unbounded! Based on experimental data of the initial part of this expansion, it certainly doesn't seem likely that $\pi$ violates Bryuno condition.

Indeed, given that almost every rational number is Bryuno, it seems rather unlikely that any specific one that we might try will fail the condition, unless there is a specific reason for it.

Moreover, imagine yourself undertaking some work in an unrelated field of mathematics, and you come across a new significant constant $\alpha$. Unbeknownst to you, $\alpha$ is Bryuno, which means it likely looks an awful lot like a rational number $\rho$. Unless you know for some other reason that $\alpha$ must be irrational (which would be rather lucky indeed), how likely are you to be able to tell that $\alpha\neq\rho$?

However, most importantly in my view, it is not clear that you are asking the right question. Suppose that by some miracle, you knew that $x:=\sqrt[5]{e+\pi+\zeta(5)}$ is a non-Bryuno number, how would that help you? (I suppose it would say interesting things about the number from a number-theoretic point of view, but I am assuming that is not really what you are after.)

It seems rather difficult to argue that the number $x$ is much more natural than any other (let's say efficiently) computable real number. And the number in Anthony's comment will certainly be very efficiently computable.