Considering the Cantor set, which can be covered with a sequence of intervals defined as the following:
$$\{[a_{di}, b_{di}], d=1, 2, ..., \infty, i=1, 2, ..., n_d\}$$
where $d$ indexes the "depth" of each interval within the set, $i$ indexes the various intervals at depth $d$, and $n_d = 2^d$ represents the number of intervals present at depth $d$; that is, $[a_{11}, b_{11}] := [0, \frac{1}{3}], [a_{12}, b_{12}] := [\frac{2}{3}, 1]$, and so on, I am wondering the following:
Is every difference $(b_{di} - a_{di})$, that is the length of interval $i$ at depth $d$, also a member of the Cantor set? This is definitely true for the first few intervals, but I'm not sure how to verify if it's true for all of the intervals.
Every interval at depth $d$ has the same length - namely, $3^{-i}$. So you're just asking whether $3^{-i}$ is always an element of the Cantor set. And indeed it is - namely, it's the right endpoint of the leftmost interval at depth $i$.