Theorem 1.14 (Polynomial Freiman-Ruzsa, due to Gowers–Green–Manners–Tao 2024). Assuming that:

Then there exists a subspace

K \leq 𝔽_{p}^{n}

of size at most

C_{1} (K) | A |

such that for some

x \in 𝔽_{p}^{n}

| A \cap (x + K) | \geq \frac{| A |}{C_{2} (K)},

where $C_{1} (K)$ and $C_{2} (K)$ are polynomial in $K$ .