sequences based upon the cantor space

Considering the cantor space i.e. $2^\mathbb{N}$, what would be examples elements $x$ in the cantor space such that $\lim \frac{|\{m < n:x(m) = 1\}|}{n}$ does not converge. If $x$ has a finite amount of $1s$ or $0s$, then it is clear that it converges(applying monotone converge theorem). Another example would be $1010101010\ldots$ which converges to $1/2$.