Can we show that the decimal expansion of $\pi$ doesn't occur in the decimal expansion of the Champernowne constant?
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 $7.6063\ldots$