Zero-full law for well approximable sets in missing digit sets

Let $b \geq 3$ be an integer and $C(b,D)$ be the set of real numbers in $[0,1]$ whose base $b$ expansion only consists of digits in a set $D \subseteq \{0,...,b-1\}$. We study how close can numbers in $C(b,D)$ be approximated by rational numbers with denominators being powers of some integer $t$ and obtain a zero-full law for its Hausdorff measure in several circumstances. When $b$ and $t$ are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann., 338:97-118, 2007) and generalize their theorem. When $b$ 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.