Solution
= Solution
Let $|t|<2\alpha$. Both $E$ and its translate $E+t$ lie in an interval of length $1+|t|$. By finite additivity and translation invariance of <Lebesgue measure>,
$$
m(E\cap(E+t))
=m(E)+m(E+t)-m(E\cup(E+t))
\geq2\left(\frac12+\alpha\right)-(1+|t|)
=2\alpha-|t|>0.
$$
Choose $x\in E\cap(E+t)$. Then $x\in E$ and $x-t\in E$, so $t=x-(x-t)\in E-E$. Therefore the <difference-set overlap bound on an interval> gives
$$
\boxed{(-2\alpha,2\alpha)\subseteq E-E.}
$$