Would you please tell me how to prove such statement?
For every $n \in \mathbb{Z}$ let $C_n$ be a closed set with empty interior contained in the interval $[n, n+1]$. Let $D=\bigcup_{x \in \mathbb{Z} }C_n$
Show that there exists $t\in \mathbb{R}$ such that $t+D \subset \mathbb{R-Q}$. Notation: $t+D=\{ x\in \mathbb{R}: x=t+d, d\in D \}$.
I suppose we can use somehow the Baire's theorem. I know it in such a version:
A countable union $\bigcup F$ ,where each $F$ is a closed subset with empty interior of a complete metric space, has empty interior.
Consider $E=\bigcup_{q\in\Bbb Q} (D+q)$, this is the same as $$\bigcup_{q\in\Bbb Q}\bigcup_{n\in\Bbb Z} (C_n+q),$$ so it is a countable union of closed sets with empty interior, hence $E$ has empty interior by the Baire category theorem. Let $x_0\in\Bbb R\setminus E$ and consider $D+x_0$.