Rational Points in $m$-adic Cantor Sets

For any natural numbers $m\geq 3$ and $s$, $0<s<m-1$ it is defined Cantor $m$-adic sets $C(m,s)$, the set of real numbers in segment [0, 1] having an expansion on base $m$ without the cipher $s$. It is proved that for any prime number $p>m^2$ the set of simplified fractions of the form $\tfrac{s}{p^t}$ where $s$ and $t$ are and integer is finite (possibly empty).