Is the Minkowski sum of the rational numbers and a nonmeasurable set again nonmeasurable?

Question

Let $\mathbb Q$ be the set of all rational numbers and $X\subset \mathbb R$ nonmeasurable (in the sense of Lebesgue). What we can say about the Minkowski sum $X+\mathbb Q$, is it measurable or not?

Answer 1

The sum $X+\mathbb Q$ may be measurable or nonmeasurable, depending on the choice of the nonmeasurable set $X.$

If $X$ is a Vitali set, then $X+\mathbb Q=\mathbb R$ is measurable.

A Bernstein set is a set $X\subset\mathbb R$ such that both $F\cap X$ and $F\setminus X$ are nonempty for every uncountable closed set $F.$ A Bernstein set is nonmeasurable; moreover, the usual construction of a Bernstein set (by transfinite induction) can easily be modified so as to guarantee that $X+\mathbb Q=X.$