Is this the middle fourth cantor set?

Question

Let $ D $ be the set of all $ x ∈ [0, 1] $ having a representation in the form $$ \sum_{i=1}^{\infty} {a_i}/{4^i} $$

where each $ a_i $ is either 0 or 3.

Does this represent the middle fourth cantor set? That is, $ D_1 $ is $ [0,1] - (0.375, 0.625) $? If so, is it calculated the same way as the Middle Third Cantor Set? If not, any guidance on how to give a geometric description of this set would be appreciated.

Answer 1

It's more of a middle-halves Cantor set, since at each level you're keeping only the outer fourths at each end, that is, removing the middle half of the interval.

In most respects this behaves just like the usual middle-thirds Cantor set.

Answer 2

This is the middle-half Cantor set: the first interval removed is $(1/4,3/4)$. Its Hausdorff dimension can be computed in the usual way: it's the number $s$ such that $2\lambda^s=1$ where $\lambda$ is the scaling factor. Here $\lambda=1/4$ and so the dimension is $1/2$.