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module
public import FormalConjecturesUtilBen Green's Open Problem 3
References:
[Ben Green's Open Problem 3](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.3 Problem 3)
[FGY26] Franchi, Leonardo and Gowers, W. T. and Yip, Fredy, Product-free subsets of $(0,1)$, arXiv:2607.06073, Theorem 1.1.
@[expose] public sectionopen Set MeasureTheorynamespace Green3Suppose that $A \subset [0,1]$ is open and has measure greater than $\frac{1}{3}$. Is there a solution to $xy = z$ with $x, y, z \in A$?
The answer is yes. [FGY26] prove that an open product-free subset of $(0,1)$ has measure strictly less than $\frac{1}{3}$.
@[category research solved, AMS 11]
theorem green_3 :
answer(True) ↔ ∀ A : Set ℝ,
IsOpen A → A ⊆ Icc 0 1 → volume A > 1/3 →
∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x * y = z := ⊢ True ↔ ∀ (A : Set ℝ), IsOpen A → A ⊆ Icc 0 1 → volume A > 1 / 3 → ∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x * y = z
All goals completed! 🐙end Green3