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import FormalConjecturesUtil
namespace Green12
open Finset
Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15} N^{10}$ tuples $(x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10}$ such that $x_i + y_j \in A$ whenever $j \in {i, i+1, i+2}$?
Note: We interpret indices modulo 5.
@[category research open, AMS 5 11]
theorem green_12 : answer(sorry) ↔
∀ {G : Type*} [AddCommGroup G] [Fintype G] [DecidableEq G],
∀ (A : Finset G),
let N := Fintype.card G
let α := (A.card : ℝ) / N
let valid_tuples : Finset ((Fin 5 → G) × (Fin 5 → G)) := Finset.univ.filter (fun t =>
∀ i : Fin 5, ∀ j ∈ ({i, i + 1, i + 2} : Finset (Fin 5)), t.1 i + t.2 j ∈ A)
(valid_tuples.card : ℝ) ≥ α ^ 15 * (N : ℝ) ^ 10 := ⊢ True ↔
∀ {G : Type u_1} [inst : AddCommGroup G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (A : Finset G),
let N := Fintype.card G;
let α := ↑(#A) / ↑N;
let valid_tuples := {t | ∀ (i j : Fin 5), j ∈ {i, i + 1, i + 2} → t.1 i + t.2 j ∈ A};
↑(#valid_tuples) ≥ α ^ 15 * ↑N ^ 10
All goals completed! 🐙
end Green12