In my self-study, I came across the following two interesting, related results:
Let $E$ be Lebesgue measurable, with $\mu(E)>0$ (here $\mu$ denotes the Lebesgue measure). Then:
for any $0<\rho<1$, there exists an open interval $I$ such that $\mu(E \cap I)> \rho \cdot \mu(I)$. In turn we should be able to use (1) with $\rho > 3/4$ to prove that
the set $E-E = \{x-y : x, y \in E\}$ contains an (open) interval centered at $0$ (in particular, if $\rho > 3/4$, the text I am using suggests that $(-\frac{1}{2} \mu(I), \frac{1}{2}\mu(I)) \subseteq E-E$).
I would like to see if anyone visiting today would be up for proving (1) or (2) (inclusive-or). I have tried to work out (1) a few times, but none of my attempts have been satisfactory.