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