Making a rational number from an irrational - Paul Erdos problem

Question

Given an irrational number $N$. You are able to insert at most $k$ digits between any two consecutive digits.

Does there exist a $k$, for which we are able to make a rational number? If yes, find the minimum value of $k$.

Answer 1

$k=1$ for some $N$. $k=9$ in general. In general you can't use k<9 because N could have all digits infinitely often and also have arbitrarily large runs of each individual digit that you have to break up using the other 9 digits. Each digit needs equal density in that case, or you'd need k bigger than 9 to break up the run of the lowest density digit. That means you need in the worst case to include the 9 other digits in each insertion. I show below that $k=9$ does work.

I can start with N as Liouville's transcendental that has 1 in the mth decimal place if m is in the range of the factorial function, and 0 otherwise. By inserting a 0, a 1, or nothing between any two digits, I can turn this into 0.101010....

Since this answer works in general for binary, I can do the same thing with $k=9$ for any N in base 10. Just create .01234567890123456789...