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?



Acting with a finite number of rotations on a set of positive measure can you fill almost the whole circle?
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.


