Theorem 7.1 (Green, Manners, Tao, Gowers). There is a polynomial $p$ with the following property:

n \in ℕ

and

A \subset 𝔽_{2}^{n}

is such that

| A + A | \leq C | A |

, then there is a subspace

H \subset 𝔽_{2}^{n}

of size at most

| A |

such that

A

is contained in the union of at most

p (C)

translates of

H

. (Equivalently, there exists

K \subset 𝔽_{2}^{n}

| K | \leq p (C)

such that

A \subset K + H