The ratio of occurrences of 0s and 1s will go to one in the limit. But the absolute difference of occurrences will diverge. (It grows roughly like the square-root of the number of digits.)

Nit, their ratio (bits set/unset) will approach 1 as the count approaches infinity, but the values themselves will not approach equal. https://gist.github.com/AlexAltea/3aa96efc41f59e80631c346908...