Question

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: http://mathoverflow.net/questions/26942/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 $C\sqrt{n}$?

Answer 1

I believe that Gerry's answer to the earlier question on MO is exhaustive enough for this question as well. The normality of $\pi$ (which would imply the irrationality measure 2) is not known yet, and even the questions like "all decimal digits from some point on are 0s and 1s" cannot be shown unconditionally.

The best known estimate for the irrationality measure of $\pi$, due to V. Salikhov [Russ. Math. Surv. 63:3 (2008), 570--572] (see also MR2483171 (2010b:11082)) reads $|\pi-p/q| > q^{-7.6063\dots}$ for all integers $p$ and $q>q_0$. The improvement of Hata's record is achieved by constructing rational approximations to $\pi$ alone (Hata's result is in fact about $\mathbb Q$-linear independence of $1$, $\log 2$ and $\pi$).

An interesting approach of attacking the normality of numbers like $\pi$, as well as problems like 0s--1s, is related to the so-called BBP formulas. However all such formulas for $\pi$ correspond to its binary (also hexadecimal) expansions. The last piece of news is a base 10 formula for $1/\pi^2$ due to G. Almkvist and J. Guillera (arXiv:1009.5202 [math.NT]), $$ \sum_{n=0}^\infty\frac{(6n)!}{n!^6} (532n^2+126n+9)\frac1{10^{6n}}=\frac{375}{4\pi^2}, $$ which is not of BBP type but close to it in a certain sense.

A nice exposition of The Life of $\pi$ is J. Borwein's public lecture given at the 2010 Australian Math Society meeting.

Answer 2

I think the closest we know is Furstenberg's result that $10^m3^n\pi$ is dense modulo $1$ (say). On the other hand, this is true for any irrational in place of $\pi$. A simple proof appears here. See also this generalization.