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import FormalConjecturesUtilBen Green's Open Problem 18
[Gr26] Ben Green's Open Problem 18
[Au16] Austin, Tim. "Ajtai–Szemerédi theorems over quasirandom groups." Recent trends in combinatorics. Cham: Springer International Publishing, 2016. 453-484.
[So13] Solymosi, Jozsef. "Roth-type theorems in finite groups." European Journal of Combinatorics 34.8 (2013): 1454-1458.
[Go01] Gowers, William T. "A new proof of Szemerédi's theorem." Geometric & Functional Analysis GAFA 11.3 (2001): 465-588.
open Finset
namespace Green18
The number of triples $(x, y, g)$ in $G^3$ such that $g \neq e$, and $(x, y), (gx, y), (x, gy)$ are all in $A$. These are called "naive corners" by [Au16].
Note: the shortened formulation from [Gr26] does not mention $g \neq e$, but this is the original statement from [Au16], which ensure non-trivial corners. Note however that [Au16] use more generally compact groups and not just finite discrete groups.
def numNaiveCorners {G : Type*} [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)) : ℕ :=
( (univ : Finset (G × G × G)).filter
fun ⟨x, y, g⟩ => g ≠ 1 ∧ (x, y) ∈ A ∧ (g * x, y) ∈ A ∧ (x, g * y) ∈ A
).card
Suppose that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Is it true that there are $\gg_\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$?
Note: A is taken as $\alpha$-dense, i.e. $|A| \ge \alpha |G|^2$ [Au16, Question 2]
@[category research open, AMS 5 11 20]
theorem green_18 : answer(sorry) ↔
∀ α > 0, ∃ c > 0, ∃ m₀ : ℕ,
∀ (G : Type*) [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)),
Fintype.card G ≥ m₀ →
(A.card : ℝ) ≥ α * (Fintype.card G) ^ 2 →
(numNaiveCorners A : ℝ) ≥ c * (Fintype.card G) ^ 3 := ⊢ True ↔
∀ α > 0,
∃ c > 0,
∃ m₀,
∀ (G : Type u_1) [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (A : Finset (G × G)),
Fintype.card G ≥ m₀ → ↑(#A) ≥ α * ↑(Fintype.card G) ^ 2 → ↑(numNaiveCorners A) ≥ c * ↑(Fintype.card G) ^ 3
All goals completed! 🐙
The number of triples $(x, y, g)$ in $G^3$ such that $g \neq e$, and $(x, y), (xg, y), (x, gy)$ are all in $A$. These are called "BMZ corners" by [Au16].
def numBmzCorners {G : Type*} [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)) : ℕ :=
(
(univ : Finset (G × G × G)).filter
fun ⟨x, y, g⟩ => g ≠ 1 ∧ (x, y) ∈ A ∧ (x * g, y) ∈ A ∧ (x, g * y) ∈ A
).card
[So13] proved this is true for "BMZ corners". Follows from the proof of Theorem 2.1, p.1456-1457.
@[category research solved, AMS 5 11 20]
theorem green_18.bmz_corners : ∀ α > 0, ∃ c > 0, ∃ m₀ : ℕ,
∀ (G : Type*) [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)),
Fintype.card G ≥ m₀ →
(A.card : ℝ) ≥ α * (Fintype.card G) ^ 2 →
(numBmzCorners A : ℝ) ≥ c * (Fintype.card G) ^ 3 := ⊢ ∀ α > 0,
∃ c > 0,
∃ m₀,
∀ (G : Type u_1) [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (A : Finset (G × G)),
Fintype.card G ≥ m₀ → ↑(#A) ≥ α * ↑(Fintype.card G) ^ 2 → ↑(numBmzCorners A) ≥ c * ↑(Fintype.card G) ^ 3
All goals completed! 🐙
end Green18