FormalConjectures.GreensOpenProblems.«36»

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import FormalConjecturesUtil

Green's Open Problem 36

References:

[Gr24] Green's Open Problems #36

[CKS05] Cohn, H., Kleinberg, R., Szegedy, B., and Umans, C. "Group-theoretic Algorithms for Matrix Multiplication" (Problem 4.7)


open Classical Filter

open scoped Pointwise


namespace Green36

The simultaneous double product property [CKS05, 4.1].

def SimultaneousDoubleProduct {ι H : Type*} [AddCommGroup H]
    (A B : ι → Finset H) : Prop :=
  (∀ i, (A i + B i).card = (A i).card * (B i).card) ∧
  (∀ i j k, i ≠ k → Disjoint (A i + B j) (A j + B k))

A variant of the simultaneous double product property, as stated in [Gr24, Problem 36].

def Green36Property {ι H : Type*} [AddCommGroup H]
    (A B : ι → Finset H) : Prop :=
  (∀ i, (A i + B i).card = (A i).card * (B i).card) ∧
  (∀ i j k, j ≠ k → Disjoint (A i + B i) (A j + B k))

Do the following exist, for arbitrarily large $n$? An abelian group $H$ with $|H| = n^{2+o(1)}$, together with subsets $A_1, ..., A_n, B_1, ..., B_n$ satisfying $|A_i||B_i| \ge n^{2-o(1)}$ and $|A_i + B_i| = |A_i||B_i|$, such that the sets $A_i + B_i$ are disjoint from the sets $A_j + B_k$ ($j \neq k$)?

NOTE: according to [CKS05, 4.1], the conditions should be $A_i + B_j$ disjoint from $A_j + B_k$ for $i \neq k$. See green_36.variants.cks05.

@[category research open, AMS 5 20]
theorem declaration uses 'sorry'green_36 :
    answer(sorry) ↔
      ∀ ε > (0 : ℝ), ∃ᶠ n in atTop,
        ∃ (H : Type) (_ : AddCommGroup H) (_ : Finite H) (A B : Fin n → Finset H),
          (n : ℝ) ^ (2 - ε) ≤ Nat.card H ∧ Nat.card H ≤ (n : ℝ) ^ (2 + ε) ∧
          (∀ i, (n : ℝ) ^ (2 - ε) ≤ (A i).card * (B i).card) ∧
          Green36Property A B := by⊢ True ↔
  ∀ ε > 0,
    ∃ᶠ (n : ℕ) in atTop,
      ∃ H x,
        ∃ (_ : Finite H),
          ∃ A B,
            ↑n ^ (2 - ε) ≤ ↑(Nat.card H) ∧
              ↑(Nat.card H) ≤ ↑n ^ (2 + ε) ∧
                (∀ (i : Fin n), ↑n ^ (2 - ε) ≤ ↑(A i).card * ↑(B i).card) ∧ Green36Property A B
  sorryAll goals completed! 🐙

Variant using the exact simultaneous double product property from [CKS05, 4.1].

@[category research open, AMS 5 20]
theorem declaration uses 'sorry'green_36.variants.cks05 :
    answer(sorry) ↔
      ∀ ε > (0 : ℝ), ∃ᶠ n in atTop,
        ∃ (H : Type) (_ : AddCommGroup H) (_ : Finite H) (A B : Fin n → Finset H),
          (n : ℝ) ^ (2 - ε) ≤ Nat.card H ∧ Nat.card H ≤ (n : ℝ) ^ (2 + ε) ∧
          (∀ i, (n : ℝ) ^ (2 - ε) ≤ (A i).card * (B i).card) ∧
          SimultaneousDoubleProduct A B := by⊢ True ↔
  ∀ ε > 0,
    ∃ᶠ (n : ℕ) in atTop,
      ∃ H x,
        ∃ (_ : Finite H),
          ∃ A B,
            ↑n ^ (2 - ε) ≤ ↑(Nat.card H) ∧
              ↑(Nat.card H) ≤ ↑n ^ (2 + ε) ∧
                (∀ (i : Fin n), ↑n ^ (2 - ε) ≤ ↑(A i).card * ↑(B i).card) ∧ SimultaneousDoubleProduct A B
  sorryAll goals completed! 🐙


-- TODO(jeangud) Add variants mentioned in [Gr24, Problem 36] comments.

end Green36