Famously, it is not known whether $\pi$ is a normal number. Indeed, there are far weaker statements that are not known, such as the statement that there are infinitely many 7s in the decimal expansion of $\pi$. I'd like to have some idea of where the boundary lies between what we know and what we do not know. For example, I would guess that it is not even known not to be the case that all decimal digits from some point on are 0s and 1s. Am I right about this?

A partial answer to this question was given by Timothy Chow in a discussion of another question: Is pi a good random number generator?. He pointed out that some very weak facts can be deduced from known results about how well $\pi$ can be approximated by rationals. I suppose I could ask whether that is essentially the only technique we have. Could it be, for instance (as far as what is proved is concerned -- obviously it isn't actually the case) that the digits of $\pi$ are all 0s and 1s from some point on and that there is a constant $C$ such that the number of 1s in the first $n$ digits is never more than $C\sqrt{n}$?

`$$ f(n) = \begin{cases} 1 & \text{if there is a a sequence of }n\text{ consecutive 7s in the decimal expansion of }\pi, \\ 0 & \text{otherwise}, \end{cases} $$`

I was slightly startled to read in a book by Hartley Rogers that this function is computable. The proof that there is an algorithm that, when given $n$, returns $f(n)$, is short, simple, and somewhat thought-provoking. $\endgroup$ – Michael Hardy Jan 12 '11 at 20:21