Champernowne Constant is Normal

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Theorem

The Champernowne constant:

$0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$

is normal with respect to base $10$.


Proof


Historical Note

The Champernowne constant was constructed by David Gawen Champernowne as an example of a real number which was demonstrably normal.

He did this in the paper of 1933: The Construction of Decimals Normal in the Scale of Ten (J. London Math. Soc. Vol. 8: 254 – 260), while still an undergraduate.


There is confusion in the literature as to whether he proved that the Champernowne constant is in fact absolutely normal.

It would be apparent from the title of the above paper that he did in fact prove normal only with respect to base $10$.

Research is ongoing.


Sources