# Questions regarding early natural numbers.

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:

1. Is the set of early numbers infinite? What about the set of non-early numbers?

2. Does the set of early numbers have an asymptotic density? If so, is there a closed form expression for that real number?

3. Same as previous question, but regarding the set of non-early numbers.

• The set of early numbers is definitely infinite, since $12,123,1234,12345, \ldots$ are early. – Stefan Mesken Jun 6 '14 at 6:50

The set of early numbers is infinite, as it contains at least the following set: $$\{12,123,1234,12345,123456,1234567,\dots\}$$
As for the set of non-early numbers, I think you can show that the number $1111\dots 1$ is not early for any number of ones it contains, meaning the set of non-early numbers is also infinite.