$A$ be a subset of $[0,1]$ with non-empty interior ; then is it true that $\mathbb Q+A=\mathbb R$? Let $A$ be  a subset of $[0,1]$ with non-empty interior ; then is it true that $\mathbb Q+A=\mathbb R$ ?
Sure. If $A$ has non-empty interior, that means it contains a ball around some point $x \in [0,1]$ of radius $2 r$. Since there's always a rational $q$ less than $r$ away from $y-x$ for any real number $y$, $y$ will be in the ball centered at $q+x$ of radius $2r$ in $q + A$, which is contained in $\mathbb{Q}+A$.
ℝ=ℚ +ℚ̅(irrational numbers),what ever we choose A other than irrational numbers,it is not possible to cover all ℝ,
so ℚ+A will be the proper subset of ℝ.