Cantor Ternary Set Problem, Ternary Expansions

Question

I'm reading Lebesgue Integration on Euclidean Space by Frank Jones and I'm stuck on this problem, of page 41, problem 15.

Let $C$ the Cantor Ternary Set.

Let $G_{1}=(\frac{1}{3},\frac{2}{3})$, $G_{2}=(\frac{1}{9},\frac{2}{9})\cup(\frac{7}{9},\frac{8}{9})$ then $C=[0,1]-\bigcup_{k=1}^{\infty}{G_{k}}$

Now let $x\in C$. Prove that $x$ is an end point of some extracted interval belonging to some $G_k\Leftrightarrow$ $x$ has two different ternary expansions.

Answer 1

HINT: If a number in $[0,1)$ has two different ternary expansions, they are of the form $0.d_1d_2\ldots d_n02222\ldots$ and $0.d_1d_2\ldots d_n10000\ldots$. Show that these numbers are the rational numbers whose denominators in lowest terms are powers of $3$. Note that this shows that only one of the two implications is actually true.