FormalConjectures.GreensOpenProblems.«29»

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module

public import FormalConjecturesUtil

Ben Green's Open Problem 29

References:

Ben Green's Open Problem 29

[Gr12] Green, Ben. "What is... an approximate group." Notices Amer. Math. Soc 59.5 (2012): 655-656.

[Br13] Breuillard, Emmanuel, Ben Green, and Terence Tao. "Small doubling in groups." Erdős Centennial. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. 129-151.

[Sa10] Sanders, Tom. "On a nonabelian Balog–Szemerédi-type lemma." Journal of the Australian Mathematical Society 89.1 (2010): 127-132.

[CrSi10] Croot, Ernie, and Olof Sisask. "A probabilistic technique for finding almost-periods of convolutions." Geometric and functional analysis 20.6 (2010): 1367-1396.

@[expose] public sectionopen scoped Pointwisenamespace Green29

Suppose that $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \subset A$, $|S| \gg K^{-O(1)} |A|$, with $S^8 \subset A^4$?

The answer is negative. A counterexample is given, for arbitrarily large finite groups H, by A = ({-1} × H) ∪ ({1} × H) ∪ {(0, 1)} in Multiplicative ℤ × H.

@[category research solved, AMS 20,
  formal_proof using lean4 at
    "https://github.com/KitaKen1/green-29-counterexample/blob/1cbf80c3d5a333b2b79c203c0d60129924c8c712/lean/Green29FC.lean#L280-L287"]
theorem green_29 :
    answer(False) ↔
      ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
        ∀ {G : Type*} [Group G] [DecidableEq G] (K : ℝ) (A : Finset G),
          1 ≤ K → IsApproximateSubgroup K (A : Set G) →
            ∃ S ⊆ A, C * K ^ (-c) * (A.card : ℝ) ≤ (S.card : ℝ) ∧
            S ^ 8 ⊆ A ^ 4 := by⊢ False ↔
  ∃ C c,
    0 < C ∧
      0 < c ∧
        ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] (K : ℝ) (A : Finset G),
          1 ≤ K → IsApproximateSubgroup K ↑A → ∃ S ⊆ A, C * K ^ (-c) * ↑A.card ≤ ↑S.card ∧ S ^ 8 ⊆ A ^ 4
  sorryAll goals completed! 🐙

Such a conclusion is known with $|S| \gg_K |A|$ [Br13 Problem 6.5, CrSi10, Sa10].

@[category research solved, AMS 20]
theorem green_29.variant :
    ∀ K : ℝ, 1 ≤ K →
      ∃ c : ℝ, 0 < c ∧ -- Allow c to depend on K.
        ∀ {G : Type*} [Group G] [DecidableEq G] (A : Finset G),
          IsApproximateSubgroup K (A : Set G) →
            ∃ S : Finset G, -- No S ⊆ A requirement in this variant.
              c * (A.card : ℝ) ≤ (S.card : ℝ) ∧
              S ^ 8 ⊆ A ^ 4 := by⊢ ∀ (K : ℝ),
  1 ≤ K →
    ∃ c,
      0 < c ∧
        ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] (A : Finset G),
          IsApproximateSubgroup K ↑A → ∃ S, c * ↑A.card ≤ ↑S.card ∧ S ^ 8 ⊆ A ^ 4
  sorryAll goals completed! 🐙

end Green29