Let $E\subset S^1$ have positive Lebesgue measure. Do there exist finitely many rotations $r_1, r_2, \dots ,r_n$ such that $r_1E\cup r_2E\cup \dots\cup r_nE$ has measure $2\pi$? Or is there a counterexample?
The answer is no. Take a compact set $E$ with positive measure buth empty interior and assume that $K=r_1 E\cup r_2 E \cdots \cup r_n E$ has measure $2\pi$. Then $K$ would be dense in $S^1$, hence equal to $S^1$, since it is closed. But this is impossible, since $K$ has empty interior, too.