FormalConjectures.GreensOpenProblems.«52»

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import FormalConjecturesUtil

Green's Open Problem 52

Reference: Green's Open Problems

open Filter Realopen scoped Pointwisenamespace 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 := by⊢ 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
  sorryAll goals completed! 🐙

Could $2A$ even contain a coset of codimension $O(\log K)$?

From Green's 2025 update:

Update 2025. Kaave Hosseini and Ryan Alweiss have independently pointed out that the second question is far too optimistic. To see this, let $A$ be a Hamming ball of radius $n/2 - \sqrt{n}$. There is a set $S$ of size $O(n)$ such that $A + S = \mathbb{F}_2^n$; a random choice of $S$ will work. However, every subspace contained in $A - A$ has codimension $\gg \sqrt{n}$.

@[category research solved, AMS 5 11,
  formal_proof using lean4 at
    "https://github.com/KitaKen1/green-52-log-counterexample/blob/16cb5d0/lean/Green52LogCounterexampleFC.lean#L661-L668"]
theorem green_52_log :
    answer(False) ↔ ∃ (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 := by⊢ False ↔
  ∃ 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
  sorryAll goals completed! 🐙

-- TODO(jgd): Implement variants from Green's comments [Gr24].

end Green52