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import FormalConjecturesUtilGreen's Open Problem 52
open Filter Realopen scoped Pointwise
namespace Green52
Suppose that $A \subset \mathbb{F}_2^n$ is a set with an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$?
@[category research open, AMS 5 11]
theorem green_52 :
answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)),
S.card = K → A + (S : Set (𝔽₂ n)) = Set.univ →
∃ (V : AffineSubspace (ZMod 2) (𝔽₂ n)), (V : Set (𝔽₂ n)) ⊆ A + A ∧
n ≤ Module.finrank (ZMod 2) V.direction + c K := ⊢ True ↔
∃ c,
∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)),
S.card = K → A + ↑S = Set.univ → ∃ V, ↑V ⊆ A + A ∧ n ≤ Module.finrank (ZMod 2) ↥V.direction + c K
All goals completed! 🐙
Could $2A$ even contain a coset of codimension $O(\log K)$?
@[category research open, AMS 5 11]
theorem green_52_log :
answer(sorry) ↔ ∃ (C D : ℝ), ∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)),
0 < K → S.card = K → A + (S : Set (𝔽₂ n)) = Set.univ →
∃ (V : AffineSubspace (ZMod 2) (𝔽₂ n)), (V : Set (𝔽₂ n)) ⊆ A + A ∧
(n : ℝ) ≤ (Module.finrank (ZMod 2) V.direction : ℝ) + C * log (K : ℝ) + D := ⊢ True ↔
∃ C D,
∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)),
0 < K →
S.card = K → A + ↑S = Set.univ → ∃ V, ↑V ⊆ A + A ∧ ↑n ≤ ↑(Module.finrank (ZMod 2) ↥V.direction) + C * log ↑K + D
All goals completed! 🐙
-- TODO(jgd): Implement variants from Green's comments [Gr24].
end Green52