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

When $N$ is prime, simultaneous approximation of the frequencies shows that $B(\Gamma,\rho)\subseteq\mathbb Z/N\mathbb Z$ contains a centered <arithmetic progression> of length at least
$$
\frac18\rho N^{1/|\Gamma|}.
$$

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 Incoming links: Arithmetic progression in a cyclic Bohr set