Part c gives $x+B(\Gamma,\alpha)\subseteq A+A+A$ with $d=|\Gamma|\leq2\alpha^{-3}$. By the <arithmetic progression in a cyclic Bohr set>, $B(\Gamma,\alpha)$ contains a centered progression of length at least
$$
\frac18\alpha N^{1/d}
\geq\frac18\alpha N^{\alpha^3/2}.
$$
Its translate by $x$ is the required <arithmetic progression> in $A+A+A$.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-129/2/e/solution