Powers-of-10-multiples of $\pi$ (or any irrational) are dense

Question

Very related, but not the same, to this question Multiples of an irrational number forming a dense subset, is the next one:

Is the sequence $(\{10^n\pi\})_{n=1}^\infty$ dense in the interval $[0,1]$? (where $\{x\}=x\ mod\,1=x-\lfloor x\rfloor$, is de decimal part of $x$)

I tried to extend the proof of HAskell in the comments of the prvious post, but I wasn't able.

EDIT: I change the question to any normal (in base 10) irrational

Answer 1

Not enough is known about the decimal expansion of $\pi$ to give a definitive answer to the question. For all anyone knows, there are no sevens in the decimal expansion of $\pi$ after some point, which would mean no terms in your sequence between .7 and .8 after some point.

It is widely believed that your sequence is not just dense but uniformly distributed but, as I said, no one can prove even far weaker results.