Is there infinite $1$ in the continued fraction of $\pi$?

Question

I asked this question just for curiosity! I guess that it's an unsolved problem, but I can't find any reference that mentions that.
You can see http://oeis.org/A001203 and How to find continued fraction of pi

Answer 1

I calculated the first $3.8\cdot 10^6$ entries of the continued fraction of $\pi$ (This requires about $2\cdot 10^6$ digits precision) and noticed the following :

If $d(n)$ denotes the ratio of the number of ones to the total number of entries, if we consider the first $n$ entries (The initial "$3$" counts as an entry as well), the following holds :

For $68692<n\le 3.8\cdot 10^6$ , we have $0.414\le d(n)\le 0.416$

Of course, there is no guarantee that this goes on forever, but it seems that roughly $d\approx 0.415$ for sufficient large $n$.

This is a good evidence (of course no proof) that infinite many ones appear in the continued fraction of $\pi$

Answer 2

As a rule, problems of this kind are extremely difficult. We know for any reasonable method of expressing real numbers as sequences of integers, for almost every real number we'll see a given digit (or more broadly any finite string of digits) with a particular frequency. For instance, if I'm looking at a number's decimal representation, then for almost every real number we'll see the digit string $17357$ with frequency $10^{-5}$. A similar rule holds for continued fractions, i.e. there exists a (in this case positive) constant $C > 0$ such that for almost every real number, the digit $1$ will occur with frequency $C$.

However, although we know almost every real number has this property, it's generally fairly difficult to construct examples of such numbers, and (as of this time) pretty much impossible to determine if a given number has this property. We can give very trivial answers to the negative (e.g. rationals are easy to check), but when it comes to determining whether a number encountered "in real life", say $\sqrt[7]{2}, e, \pi$ has this property, we really don't have any kind of machinery to deal with this. We have conjectures of what we think will probably happen, but no techniques have been discovered to test them. It's just an area of math where we have no clue how to approach the problem.

EDIT: I would bet money this problem is still open.