  Let $t \geq 2$ be an integer and $C(b,D)$ be a generalized Cantor set with $b
\geq 3$. We study how close can numbers in $C(b,D)$ be approximated by rational
numbers with denominators $t^n$. For any function $\psi : \mathbb{N} \to
(0,\infty)$, let $W_{t} (\psi)$ be the set of numbers in $[0,1]$ such that $|x
- p/t^n| < \psi(n)$ for infinitely many $(p,n) \in \mathbb{N}^2$. We correct an
error in a result of Levesley, Salp and Velani (Math. Ann., 338:97-118, 2007)
on the Hausdorff measure of $W_{t} (\psi) \cap C(b,D)$ when $b=t$, and also
prove a generalization when $b$ and $t$ are multiplicatively dependent. If $b$
and $t$ are multiplicatively independent but have the same prime divisors, we
obtain a partial result on the Hausdorff measure and bounds for the Hausdorff
dimension, which are close to the multiplicatively dependent case. Based on
these results, several conjectures are proposed.
