Let $A_1, A_2$ be bounded disjoint Lebesgue measurable subsets of $\mathbb R$, each of which has positive measure. Is there a translation $\tau:\mathbb R\to \mathbb R$ for which $\tau(A_1)\cap A_2$ has positive Lebesgue measure?
The question I'm actually interested in is the following: Let $A_1, A_2\subset \mathbb S^n$ be disjoint measurable subsets, each of which has positive measure. Is there a rotation $R$ of the sphere for which the measure of $R(A_1)\cap A_2$ is positive? I think I may be able to answer it if I can get some insight to the simpler question posed above.