Consider the real number $0.123456789101112\dots$, where you concatenate the digits of the natural numbers.
Certain natural numbers are "early", meaning, they appear earlier as a substring of digits than they are supposed to.
For example: $12$, $23$ and $101$ are early natural numbers.
I have a few questions:
Is the set of early numbers infinite? What about the set of non-early numbers?
Does the set of early numbers have an asymptotic density? If so, is there a closed form expression for that real number?
Same as previous question, but regarding the set of non-early numbers.