Difference set for E measurable

Question

Let $E\subset \mathbb {R^n}$ measurable (Lebesgue) such that $\mu(E)>0$. Prove that $D(E)=\{x-y:x,y\in E\}$ contains a ball centered in $0$.

Any hint, please? Thanks!

Answer 1

Hints: (i) $x\in D(E)$ if and only if $E\cap(E+x)$ is non-empty; (ii) $g: x\mapsto\mu(E\cap(E+x))$ is continuous, and (iii) $g(0)=\mu(E)>0$.