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-/
import FormalConjecturesUtilBen Green's Open Problem 19
[Gr26] Ben Green's Open Problems
[FSS20] Fox, Jacob, et al. "Triforce and corners." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 169. No. 1. Cambridge University Press, 2020.
[Ma21] Mandache, Matei. "A variant of the Corners theorem." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 171. No. 3. Cambridge University Press, 2021.
[Ch11] Chu, Qing. "Multiple recurrence for two commuting transformations." Ergodic Theory and Dynamical Systems 31.3 (2011): 771-792.
open Finset Real Classical
namespace Green19
section GroupDefs
-- Abstract representation to ease notation [FSS20].
variable {G : Type*} [AddCommGroup G] [Fintype G] [DecidableEq G]A corner in $A$ with common difference $d$ [FSS20].
def IsCorner (A : Finset (G × G)) (x y d : G) : Prop :=
(x, y) ∈ A ∧ (x + d, y) ∈ A ∧ (x, y + d) ∈ A
From [FSS20]: given $A \subseteq G \times G$ and $d \in G$, let $$S_d(A) = \lbrace (x, y) \in G \times G : (x, y), (x + d, y), (x, y + d) \in A \rbrace$$
noncomputable def S (d : G) (A : Finset (G × G)) : Finset (G × G) :=
univ.filter (fun p => IsCorner A p.1 p.2 d)
end GroupDefs
True if the given exponent satisfies Green's conditions [Gr26].
def ValidExponent (c : ℝ) : Prop :=
∃ K > 0,
∀ α, 0 < α → α < 1 →
∀ᶠ n in Filter.atTop,
∀ A : Finset (𝔽₂ n × 𝔽₂ n),
let N : ℝ := (Fintype.card (𝔽₂ n) : ℝ)
(A.card : ℝ) ≥ α * N^2 →
∃ d : 𝔽₂ n, d ≠ 0 ∧ ((S d A).card : ℝ) ≥ K * α^c * N^2The infimum of all valid exponents [Gr26].
noncomputable def C : ℝ := sInf {c | ValidExponent c}
What is $C$, the infimum of all exponents $c$ for which the following is true, uniformly for $0 < \alpha < 1$? Suppose that $A \subset \mathbb{F}_2^n \times \mathbb{F}_2^n$ is a set of density $\alpha$. Write $N := 2^n$. Then there is some $d \neq 0$ such that $A$ contains $\gg \alpha^c N^2$ corners $(x,y), (x,y+d), (x+d,y)$.
This question has been resolved by [FSS20], showing that $C = 4$.
@[category research solved, AMS 5 11]
theorem green_19 : C = 4 := ⊢ C = 4
All goals completed! 🐙[Ma21] showed that $3.13 \leq C$.
@[category research open, AMS 5 11]
theorem green_19.lower : C >= 3.13 := ⊢ C ≥ 3.13
All goals completed! 🐙[Ma21] showed that $C \leq 4$.
@[category research open, AMS 5 11]
theorem green_19.upper : C <= 4 := ⊢ C ≤ 4
All goals completed! 🐙
/- TODO(jeangud): in [FSS20] they mention that the corresponding question for squares
$(x, y), (x, y + d), (x + d, y), (x + d, y + d)$ is wide open (and here it is not even clear that
$C$ exists). -/
end Green19