Is there an irrational number that the digits never repeat anywhere and have all 10 digits appear everywhere?

Question

Is there an irrational number that the digits never repeat anywhere and have all 10 digits appear everywhere?

let's look at one that doesn't work like $$\pi=3.141592653589793238462643383...$$ starting at the 23rd digit you get 33 so it fails another example of one that fails is $0.10102101023135791...$ even tho no digit ever repeats twice a pair of digits do $10,10$ and and here 5 digits in a row do $10102,10102$.

my question is there an irrational number such that all digits are used equally and no sequence of the digits repeat twice like this. $123547123547,8989,0909,182182,99,...$

Answer 1

The sequence of digits you want is an infinite square-free word on the alphabet 0123456789.

EDIT: Consider an infinite square-free word on the alphabet 012, which we know exists by Thule's construction. Let the positions of $2$ in this word be $i_1, i_2, \ldots$: there must be infinitely many, otherwise after some point we would have an infinite square-free word on alphabet 01, which is impossible. For each $k$, change the letter in position $i_k$ to $3$, $4$, \ldots $9$ or leave it as $2$ if $k \equiv 0, 1, \ldots, 7 \mod 8$ respectively. The resulting infinite word is still square-free, and now has infinitely many of each of $0,1, \ldots, 9$.