Let $F$ be the set of of numbers $x\in [0,1]$ with base 3 expansions $0.a_1a_2...$ for which there exists an integer k such that $a_i\neq 1$ for all $i\geq k$. Find the Hausdorff dimension of $F$.
Note that $F$ is the countable union of $F_k$ where $F_k$ is the points $0.a_1a_2...a_k..$ such that for all $i\geq k$ we have $a_i\neq 1$.
So it suffices to find that $supdim_HF_k$. Note that $F_1$ is the cantor set, with Hausdorff dimension $log2/log3$.
I'm not sure how to approach this. Im supposed to use self similarity, but I'm having a hard time understanding which sets are self similar to others in this case, $I$ suspect each of the $F_ks$ are self similar to $F$ , but im not sure how or why.
May someone elaborate?
Look at $F_2$ this way: divide the unit interval in three intervals of length $1/3$ according to the first digit of the base 3 expanaion. In each of them, you look at the points without a 1 in the rest of their expansion. This translates to seeing a cantor set in each of the three intervals. So $F_2$ is essentially three scaled down copies of $F_1$. Can you finish it from that?