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0789dda | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | 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
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