Difference-set overlap bound on an interval
= Difference-set overlap bound on an interval
If a measurable set $E\subseteq[0,L]$ has measure $m(E)=L/2+\alpha$, then for $|t|<2\alpha$ the sets $E$ and $E+t$ lie in an interval of length $L+|t|$, so
$$
m(E\cap(E+t))\geq2m(E)-(L+|t|)>0.
$$
Thus $(-2\alpha,2\alpha)\subseteq E-E$. This quantitative overlap argument is a bounded form of the <Steinhaus theorem>.