We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine whether a number $a \in [0,1]$ is in the Cantor set by looking at its ternary expansion. If there is a $1$ in the ternary expansion, then the number is not in the Cantor set.
Now what if I make some modification on the Cantor set. At each iteration, instead of removing the middle third, I remove an open interval of length $r \cdot l$ in the middle of each line segment, where $l$ is the length of the original line segment, and $0 < r < 1$. The resulting set is expected to have the same topological structure as the Cantor set, but the members of the sets are different (unless $r=1/3$, of course). Does anyone have any idea on how to determine whether a given number $a \in [0,1]$ is in the resulting set?
