I'm interested in the set given in this question. For clarity, set
$$S_{\epsilon} = \cup_{j = 1}^{\infty} \left( q_j + \frac{\epsilon}{2^j}, q_j - \frac{\epsilon}{2^j}\right)$$
where $\\{q_j\\}$ is an enumeration of the rationals. Consider
$$S = \cap_{k = 1}^{\infty} (S_{1/k} \backslash \\{q_k\\}) $$
Since each $S_{1/k} \backslash \\{q_k\\} $ is open and dense, the Baire category theorem gives that $S$ must be baire generic in $\mathbb{R}$, and hence dense. However, $S$ cannot contain any rational number - so what is this set? Moreover, does it depend on the enumeration of $\mathbb{Q}$?
I have some feeling that this is related to Liouville numbers, but lack any references.