Suppose we have an irrational number represented in base $3$ such that there can only be a maximum of $n$ consecutive $1$'s or $2$'s in the ternary expansion. Furthermore, suppose the only digit immediately before or after a $1$ is a $0$ or $1$, and likewise, the only digit immediately before or after a $2$ is a $0$ or $2$. Does this imply there are arbitrarily long sequences of $0$'s in the expansion? Or does such a number even exist?
For example, numbers of this form would look something like: