Can we show that the decimal expansion of $\pi$ doesn't occur in the decimal expansion of the Champernowne constant?

This question was inspired by an answer and some comments to this question.

Recall that the Champernowne constant is obtained by concatenating all natural numbers written in base 10 and then put $0.$ in front, that is, $$C_{10}=0.123456789101112131415161718192021222324\cdots$$

The question is does the decimal expansion of $\pi$ occur as a tail of this number? By which I mean is there some $n$ such that $$\pi=C_{10}\cdot10^{n+1}-10\cdot\lfloor C_{10}\cdot10^n\rfloor$$

Now it is obvious that any finite initial string of $\pi$ occurs in $C_{10}$. Not only that, it also occurs infinitely often (that's just because any finite string of numbers occurs infinitely often in $C_{10}$).

Hagen von Eitzen (who's answer inspired this question) also notes that proving that $\pi$ does occur as the tail of $C_{10}$ would imply that $\pi$ is base-$10$-normal which is an open question see e.g. here, so it is very unlikely we can prove that. He also notes "that if such a position exists [one where $C_{10}$ starts giving the digits of $\pi$] (somewhere in the middle of an $n$-digit integer, say) then the first $10^{n/2}n$ or so digits of $\pi$ turn out nearly regular and this should give rise to an unusually good rational approximation."

I tried to consider if there might be some relationship with relative algebraicity, but even in that direction little seems to be known since it is unknown even whether $\pi$ is algebraic over $e$.

In conclusion it seems intuitively extremely unlikely that $\pi$ would occur in $C_{10}$, but I can't think of a proof that it doesn't. A good answer would also be a reduction of this problem to some known open problem (note that we need to reduce that $\pi$ is not the tail of $C_{10}$).

PS If someone can come up with better tags please have a go at it.

• I do not understand: if we could prove that π occurs as the tail of Champernowne, we could prove that π is base-10-normal. Since the latter result is unknown, the answer to the question in your title is (trivially) "No we cannot". – Did Jun 6 '15 at 10:52
• @did Nope. That's what I address in the last paragraph. The answer to the question "Can we prove $\pi$ is the tail of $C_{10}$?" is no. But the question asks "Can we prove $\pi$ is not the tail of $C_{10}$?" and the answer to that might be yes. – DRF Jun 6 '15 at 11:03
• @Did Essentially this turns fully on what the correct answer is. If it is true that $\pi$ is the tail of $C_{10}$ than bad luck, since proving that would mean solving whether or not $\pi$ is base-$10$-normal, but if it's true (as I strongly believe) that $\pi$ is not the tail of $C_{10}$ there might be an easy proof of that since it doesn't imply anything hard as far as we know. – DRF Jun 6 '15 at 11:07
• @Did Why? The question as it stands currently asks exactly what it should and the things you added to the title are detailed in the post. I felt that title was way too long as is. – DRF Jun 6 '15 at 11:11
• @did sorry if I misphrased my comment. I thought you meant I should change the title to what you said I should rephrase the question as. I did not intend to come across hostile and I apologize if I did. – DRF Jun 6 '15 at 11:16

If there were such an equality, then the Champernowne number would be equally well (up to a scaling constant) approximated by rational numbers as $\pi$. In particular, they would have the same irrationality measure. However, the Champernowne constant is known to have irrationality measure $10$, whereas the irrationality measure of $\pi$ is expected to be $2$, and is known to be at most $7.6063\ldots$