# May Champernowne constants $C_m$ be related to other numbers than $m$?

[This question is related to another question concerning normal numbers I asked at Math SE.]

Has it ever been found worth to ask the question if the Champernowne constants $C_m$, especially $C_2$ might be related to other (known or nameable, algebraic or transcendental, probably normal) numbers like $\sqrt{2}$, $\pi$ or $e$? Is there any chance, or are there good arguments that $C_2$ probably won't be related to e.g. $\pi$, i.e. there's no closed formula $\phi(\cdot)$ saying $C_2 = \phi(\pi)$.

An argument might go like this:

The value of $C_2$ depends on the arbitrary base 2 while the value of $\pi$ doesn't.

But maybe 2 isn't so arbitrary? (To say the very least: 2 is a prime number. And there's a deep connection between prime numbers and $\pi$.)

As infinite series they are defined by

and take values like

$C_2 = 0.862240125868054571557790283249394578565764742768299094516\dots$

$C_3 = 0.598958167538433992500172217929436590978208768676105936754\dots$

We obviously don't see at a glance that $C_2 = \frac{\pi}{3} = 1.047\dots$ or $C_2 = \frac{\pi}{4} = 0.785\dots$ or $C_2 = \frac{\pi^3}{2^5} = 0.968\dots$. ( Endless combinations abound.)

But who assures us - just for example - that no enumerable sequence $(\alpha_k)$ with $\alpha_k \in \mathbb{Q}$ exists with $C_2 = \sum \alpha_k \pi^k$?

Note that many infinite series resulting in a closed expression over $\pi$ – first of all Leibniz' formula for $\pi$ – came somehow as a surprise. Can this be turned around?

• It's trivial to construct lots of infinite sequences $\alpha_k\in\mathbb Q$ such that $C_2$ (or any real number you like) equals $\sum \alpha_k \pi^k$ (or also, if that's what you meant, $\sum \alpha_k \pi^{-k}$)—just by the density of $\mathbb Q$ in $\mathbb R$. Do you mean something else by "enumerable"? – Greg Martin Sep 14 at 18:34

One argument against $\pi$ being related to $C_2$: while $C_2$ is normal, its normality is really terrible in the sense that the convergence of its digit-sequence frequencies to uniform is extremely slow. Almost all real numbers, and (presumably) all algebraic expressions in $\pi$, should be not only normal but should approach the limiting frequencies about as fast as sequences of coin tosses would.
• What can be learned from the fact, that it's not known whether the decimal expansion of $\sqrt{2}$ contains infinitely many zeros, while it is obvious, that its binary expansion must contain infinitely many zeros (and ones). The first base where it's unclear would be 3. – Hans Stricker Sep 18 at 9:15