TITLE: Countable shifts of closed positive sets
QUESTION [9 upvotes]: 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?

REPLY [3 votes]: Easier (well, I mean: requiring less deep knowledge) reason than in the answers by Goldstern and Will Brian.
Your set $C$ may be approximated by a disjoint union of small dyadic intervals (simply disjoint intervals also work) with accuracy 1 percent. Thus by pigeonhole principle one of them consists of points of $C$ by at least 99 percents. Its disjoint (or almost disjoint) rational shifts, which form a partition of the whole segment $[0,1]$, show that $[0,1]$ consists of points of $C+\mathbb{Q}$ by at least 99 percents.