Let $\mu$ be the Lebesgue measure, and $+$ be addition modulo $1$ in the interval $[0,1)$.

Question1: Is there a closed set $C\subset [0,1)$ of positive measure such that for any countable set $D\subset [0,1)$, we have $\mu(C+D)<1$?

Question2: Is there a closed set $C\subset [0,1)$ of positive measure such that $\mu(\mathbb{Q}+C)<1$, where $\mathbb{Q}$ is the set of rational numbers?