Problem in the proof of the dimension of the Cantor set

Question

From the proof of the Hausdorff dimension of the middle third Cantor set. I cannot understand the last sentence in this proof. I cannot see how we have counted $2^j \leq \sum_i 2^j3^s|U_i|^s$

Answer 1

We have covering intervals $\{U_i\}$. For every generation $j$, the intervals $\{U_i\}$ must cover all $2^j$ basic intervals of length $3^{-j}$. For each $i$ we will count the number of $j$-generation intervals that $U_i$ can intersect. The sum of these counts must be at least $2^j$, else we would have a $j$-generation interval that is disjoint from all $U_i$.

The author chooses $j$ such that $3^{-j-1}\le \min_i |U_i|$ and observes that $U_i$ can intersect at most $2^j 3^s |U_i|^s$ intervals of generation $j$. Therefore, $$\sum_i 2^j 3^s |U_i|^s \ge 2^j$$