Let $E\subseteq \Bbb R$ be a set of Lebesgue measure zero. Show that there exists $a \in \Bbb R$ such that the set $$E+a :=\{x+a:x\in E\}$$ contain no rational numbers.
I tried to use there is a $G_{\delta}$ set equal measure zero set with measure zero, and I also tried to use contradiction. But I am getting nowhere. I have no idea how to deal with the containing no rational numbers.
Could someone provide a hint?
Suppose for a contradiction that $E+a$ always contains a rational. Then $$ \bigcup_{q\in \mathbb Q}(E-q) $$ would contain every $a\in \mathbb R$, but would also be a countable union of measure-zero sets.