The density-probability principle is true!

I never properly wrote about this question here -- only obliquely referred to it here with the text "proving it seems annoying" -- but now it has an answer! This is a question about automatic sets, so I should probably explain what those are, since a lot of you won't know. A b-automatic set is a set of whole numbers which is given by, you have some DFA on the alphabet {0,...,b-1}, you write…