In Diophantine approximation, for a given positive real number $\alpha$ let $[a_0, a_1, \cdots]$ denote its continued fraction expansion and let $p_n/q_n = [a_1, \cdots, a_n]$. Then it is known that $q_n$ grows at least exponentially. In 1935, Paul Levy proved that in fact for almost all real $\alpha$, we have $\displaystyle \lim_{n \rightarrow \infty} q_n^{1/n} = \exp(\pi^2/12 \log 2)$. The result was proved using Ergodic theory. Now my question is, does there exist a single known example of a real number $\beta$ such that $\beta = [b_0, b_1, \cdots]$, $p_n/q_n = [b_0, \cdots, b_n]$, and $\displaystyle \lim_{n \rightarrow \infty} q_n^{1/n} = \exp(\pi^2/12 \log 2)$?
For example, one might expect $\beta = \exp(\pi^2/12 \log 2)$ to do the trick…
Best Answer
Yes. There is a classical construction in number theory due to Champernowne of a number that has the right frequency of each block in its decimal expansion. The number is just 0.12345678910111213141516171819202122$\ldots$. You have to do a certain amount of work to check this property. Such a number is called a normal number base 10.
Adler, Keane and Smorodinsky in 1981 constructed a "continued fraction normal number" analogous to the Champernowne number for the continued fraction transformation in a reasonably explicit way - they gave an essentially explicit description of the $b_n$'s that appear. This continued fraction normality is (much) stronger than the condition that you are asking for: it implies that the denominators grow at the correct rate and also any other average quantity belonging to a very wide class defined on the basis of the underlying dynamical system takes the same value for this number as it does for a set of Lebesgue measure 1 in [0,1].
Incidentally, experimentally $\pi$ has the correct denominator growth, whereas $e$ has an anomalous denominator growth rate (provably).