Let $q,p \in \mathbb{Q}^+$ with $0<1-q \leq \frac{1}{p}$. Does it always exist a $c:\mathbb{N} \to \{0,1\}$ such that $$\sum_{k=0}^\infty c(k) \, q^k = p$$
If yes, is $c$ unique or are there many $c$'s for a given $(q,p)$? If no, what additional conditions need to be imposed on $(q,p)$ to guarantee the existence of a $c$ that satisfies the equality above?