Does $G_{\delta}+q$ sets cover $\Bbb{R}$ a.e

Question

Let $G_{\delta}$ be countable intersections of given open sets with positive Lebesgue measure on $[a,b]$. My question is that if $G_{\delta}+q$ covers $\Bbb{R}$ a.e, i.e. is $$ \bigcup_{q \in \mathbb{Q}}(q+G_{\delta})=\Bbb{R}-N $$ true? ($N$ is of Lebesgue measure zero). $G_{\delta}$ must be uncountable for it has positive Lebesgue measure. But it may has empty interior. I need help on this question.

Answer 1

Claim: Let $G$ be any Borel set of reals of positive measure. Then $H = G + \mathbb{Q} = \{g + q : g \in G, q \in \mathbb{Q}\}$ is conull.

Verbose proof: Suppose not. Using Lebesgue density theorem, pick two intervals $I, J$ of same length such that $H$, $\mathbb{R} \setminus H$ have more than $99$% measure in $I, J$ respectively. Now choose a rational $r$ such that $I + r$ and $J$ overlap by more than $99$%. But now both $H + r = H$ and its complement have more than $60$% measure in $J$: Contradiction.