I have two questions:

1. This is a more general question, how to find the periodicity, say of $1/252$ base 3?
2. The question is this: Find all $x$ such that both $x$ and $2x$ belong to the cantor set(CS).

Attempted Solution:

1. So, my idea is this: Let $x=.a\_1a\_2a\_3...$ be its ternary expansion, then if $x$ belongs to the CS, $a\_i\in\{0,2\}.$ However, since all the $a\_i's$ in the ternary expansion of $x$ that are $2's$ will be $1's$ in the ternary expansion of $2x$, and since we want $2x$ to belong to the Cantor Set, then I think we are looking for all those numbers $x$ that have two different ternary expansions.

I think that all the numbers that are endpoints of the extracted intervals (i.e. $1/3, 1/9, 7/9$); that is all numbers of the form $m/3^k$ have two different ternary expansions. But I am not sure how to prove this last part??