Suppose $X \subseteq \mathbb{R}$ has measure zero. Can we find an uncountable $A \subseteq \mathbb{R}$ such that $X + A = \bigcup \{a + x: a \in A, x \in X\}$ does not have measure zero?

Clearly, the answer is yes under Martin's axiom and $\mathfrak{c} > \aleph_1$. But can we do this without additional assumptions.