Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 129 / 2 / e / Solution

Part c gives $x+B(\Gamma ,\alpha )\subseteq A+A+A$ with $d=|\Gamma |\le 2{\alpha }^{-3}$. By the arithmetic progression in a cyclic Bohr set, $B(\Gamma ,\alpha )$ contains a centered progression of length at leastIts translate by $x$ is the required arithmetic progression in $A+A+A$.

Solved by gpt-5.6-sol high.