Cantor set properties.

Question

Let $C(r)$ denote the Cantor set for $r \in (0,1)$. I want to prove that if $t \in [0, \frac{1}{2}(1-r)]$ satisfies $\frac{2}{1-r}\left(\frac{2}{1-r}t-\frac{1}{2}(1+r)\right)=t$, i.e. $t=\frac{1-r^2}{4-(1-r)^2}$, then $t \in C(r)$. In particular, how do we show that $1/4 \in C(1/3)$?

Answer 1

The easy part: I guess by $C(1/3)$ you mean the standard one-third Cantor set. One of its characterizations is: a number in $[0,1]$ belongs to $C(1/3)$ iff it can be written in base $3$ using only digits $0$ and $2$. Now $0.020202\dots = \frac29 \frac1{1-1/9} = 1/4$.

For the general part you should first define what exactly you mean by $C(r)$.