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import FormalConjecturesUtilErdős Problem 120
St20 Steinhaus, Hugo, Sur les distances des points dans les ensembles de measure positive. Fund. Math. (1920), 93-104.
open Set MeasureTheory
namespace Erdos120
There exists a set $E \subseteq \mathbb{R}$, dependent on set $A \subseteq \mathbb{R}$, of positive measure which does not contain any set of the shape $a * A + b$ for some $a,b \in \mathbb{R}$ and $a \neq 0$?
def Erdos120For (A : Set ℝ) : Prop := ∃ E : Set ℝ,
MeasurableSet E ∧ 0 < volume E ∧ ∀ a b : ℝ, a ≠ 0 → ¬ .image (fun x => a * x + b) A ⊆ E
Let $A \subseteq \mathbb{R}$ be an infinite set. Must there be a set $E \subseteq \mathbb{R}$ of positive measure which does not contain any set of the shape $a * A + b$ for some $a,b \in \mathbb{R}$ and $a \neq 0$?
@[category research open, AMS 5 28]
theorem erdos_120 : answer(sorry) ↔ ∀ A : Set ℝ, A.Infinite → Erdos120For A := ⊢ True ↔ ∀ (A : Set ℝ), A.Infinite → Erdos120For A
All goals completed! 🐙
Steinhaus [St20] has proved Erdős 120 to be false whenever $A$ is a finite set.
@[category research solved, AMS 5 28]
theorem erdos_120.variants.finite_set {A : Set ℝ} (h : A.Finite) : ¬ Erdos120For A := A:Set ℝh:A.Finite⊢ ¬Erdos120For A
All goals completed! 🐙
end Erdos120