Computable Normal Numbers: Is There a Known Answer?

[Dragonfall]

Wikipedia is not very clear on this. Is there a known computable normal number?

I found this paper:

http://www.glyc.dc.uba.ar/santiago/papers/absnor.pdf

But I'm not sure if it's been peer reviewed.

 

[micromass]

What do you mean with "computable"?

Anyway, consider the Campernowne constant. It is just

$$0.123456789101112131415161718192021222324...$$

This is known to be normal (one of the very few explicit numbers known to be normal, although it is also known that "most" numbers are normal). And it will probably also satisfy your criterium of computability.

http://en.wikipedia.org/wiki/Champernowne_constant

Now, your paper (which certainly is peer-reviewed and correct!) shows the existence not only of a computable normal number, but of an computable absolutely normal number. This means that it is normal in any integer base $\geq 2$. Champernowne's constant is only known to be normal in base $10$. I don't think any other examples of absolutely normal computable numbers are known, but I'm not an expert.

 

[Dragonfall]

Yes, I meant "absolutely normal". Computable means digits are enumerable by a Turing machine or uniform family of circuits. That paper presents a super-exponential-time algorithm for computing Sierpinski's construction.

Is there a known polynomial-time computable normal number?

How many conjectures will the existence of such a number ruin?

 

[NateTG]

...

How many conjectures will the existence of such a number ruin?

Very few of significance, I expect. (In fact, it's conjectured that most of the computable mathematical constants we're familiar with pi or Euler's number are absolutely normal.)

 

[Dragonfall]

Good point.