Borel set as union of $G_\delta$ and countable set

Question

What is an example of a Borel set of $\mathbb{R}$ which cannot be written as a union of a $G_\delta$ and a countable set?

Answer 1

Let $K\subset [0,1]$ be a Cantor set. Put $A=\mathbb Q+K$. Clearly, that the set $A$ is Borel. Assume that there exists a countable set $C\subset A$ such that a set $A\setminus C$ is $G_\delta$. Since the set $A\setminus C$ is dense in $\mathbb R$, the set $\mathbb R\setminus (A\setminus C)$ is meager (that is a countable union of nowhere dense sets). Then the set $\mathbb R=A\cup \mathbb R\setminus (A\setminus C)=\bigcup_{q\in\mathbb Q} (q+K)\cup\mathbb R\setminus (A\setminus C)$ is meager too, a contradiction with Baire theorem.