1. FormalConjectures
  2. GreensOpenProblems
  3. «3»
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Ben Green's Open Problem 3

Reference: [Ben Green's Open Problem 3](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.3 Problem 3)

open Set MeasureTheory namespace Green3

Suppose that $A \subset [0,1]$ is open and has measure greater than $\frac{1}{3}$. Is there a solution to $xy = z$ with $x, y, z \in A$?

@[category research open, AMS 11] theorem declaration uses 'sorry'green_3 : answer(sorry) ↔ ∀ 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