Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 26J / f / 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,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