| import Mathlib | |
| namespace OAI | |
| namespace Falconer | |
| open MeasureTheory | |
| open scoped ENNReal | |
| /-- A compact subset of Euclidean space with Hausdorff dimension greater than | |
| half the ambient dimension determines a distance set of positive Lebesgue measure. -/ | |
| theorem falconer_distance_conjecture : | |
| ∀ (d : ℕ), 2 ≤ d → ∀ E : Set (EuclideanSpace ℝ (Fin d)), IsCompact E → | |
| (d : ℝ≥0∞) / 2 < dimH E → | |
| 0 < volume {r : ℝ | ∃ x ∈ E, ∃ y ∈ E, dist x y = r} := by | |
| sorry | |
| end Falconer | |
| end OAI | |