feat(GreensOpenProblems): 52

Author: jeangud

FormalConjectures/GreensOpenProblems/52.lean

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+/-
+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+
+import FormalConjectures.Util.ProblemImports
+
+/-!
+# Green's Open Problem 52
+
+*Reference:* [Green's Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.52)
+
+-/
+
+open Filter Real
+open scoped Pointwise
+
+namespace Green52
+
+/-- The group $G = \mathbb{F}_2^n = (Z/2Z)^n$. -/
+abbrev 𝔽₂ (n : ℕ) := Fin n → ZMod 2
+
+/--
+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
+  sorry
+
+/--
+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 := by
+  sorry
+
+-- TODO(jgd): Implement variants from Green's comments [Gr24].
+
+end Green52