for a given $n\in\mathbb{N}_{>0}$ $$A_{n,\epsilon}=\bigcup_{\frac{p}{q}\in\mathbb{Q}}(\frac{p}{q}-\frac{\epsilon}{q^n},\frac{p}{q}+\frac{\epsilon}{q^n}), A_n=\bigcap_{\epsilon>0}A_{n,\epsilon}$$
1.$\bigcap_{n}A_{n}\ne\mathbb{Q}$.
2.$\bigcap_{n}A_{n}$ is an uncountable set.
3.$\bigcap_{n}A_{n}-\mathbb{Q}$ is dense in $\mathbb{R}$
I think it's enough to consider $\widetilde{A}_{n,\epsilon}=A_{n,\epsilon}\bigcap[0,1],\widetilde{A}_{n}=\bigcap_{\epsilon>0}\widetilde{A}_{n,\epsilon}$ satisfying those properites.
As we change the position of $\bigcup$ and $\bigcap$, we get A exactly the $\mathbb{Q}$, so I'm interested in this one.