Ben Green's Open Problem 50

Suppose that $A \subset \mathbb{F}_2^n$ is a set of density $\alpha$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\log(1/\alpha))$?

Here $kA$ denotes the $k$-fold iterated sumset, i.e., the set of all sums of $k$ elements from $A$ (with repetition allowed). In Mathlib, this is denoted k • A using pointwise scalar multiplication on sets.

Reference: [Ben Green's Open Problem 50](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.6 Problem 50)

theorem Green50.green_50 : True ↔ ∃ C > 0, ∀ (n : ℕ) (A : Finset (Fin n → ZMod 2)), A.Nonempty → have α := ↑A.dens; ∃ (W : Submodule (ZMod 2) (Fin n → ZMod 2)) (v : Fin n → ZMod 2), v +ᵥ ↑W ⊆ ↑(10 • A) ∧ ↑n - C * Real.logb 2 (1 / α) ≤ ↑(Module.finrank (ZMod 2) ↥W)

Let $A \subset \mathbb{F}_2^n$ be a set of density $\alpha > 0$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\log(1/\alpha))$?

More precisely: does there exist an absolute constant $C > 0$ such that for all $n \geq 1$ and all nonempty $A \subseteq \mathbb{F}_2^n$ with density $\alpha > 0$, the sumset $10A$ contains a coset of some subspace of dimension at least $n - C \log_2(1/\alpha)$?

The sumset $10A$ is defined as $\{a_1 + a_2 + \cdots + a_{10} : a_i \in A\}$, using the pointwise scalar multiplication notation 10 • A where • denotes the iterated addition of a set.

Note: We model $\mathbb{F}_2^n$ as Fin n → ZMod 2, which is an $n$-dimensional vector space over $\mathbb{F}_2$.