Euclidean Kakeya Conjecture in Dimensions ≥4

Establish that every Kakeya set in Euclidean space ℝ^d—defined as a subset containing a unit line segment in every direction—has Hausdorff dimension exactly d for all dimensions d ≥ 4.

Background

The Euclidean Kakeya conjecture asserts that sets containing a unit line segment in every direction must have full Hausdorff dimension d. Despite the possibility of these sets having zero Lebesgue measure (as shown by Besicovitch), determining the exact fractal dimension is central to harmonic analysis and geometric measure theory.

The conjecture has been resolved in low dimensions: Davies proved the case d = 2, and Wang–Zahl recently settled d = 3. The paper under consideration focuses on finite-field analogues, but it explicitly notes that the Euclidean problem remains unresolved for d ≥ 4.