Kakeya set conjecture in dimensions n ≥ 3

Prove that every Kakeya set in R^n has Minkowski dimension n and Hausdorff dimension n for all n ≥ 3. A Kakeya set is a compact subset of R^n that contains a unit line segment pointing in every direction; the conjecture is resolved in the plane but remains unresolved in higher dimensions.

Background

The classical Kakeya set conjecture posits full Hausdorff and Minkowski dimension for sets that contain a unit line segment in every direction. While the conjecture is settled in two dimensions, the higher-dimensional case is a central open problem in geometric measure theory and harmonic analysis.

This paper proves weaker, yet significant, structural results: every Kakeya set in R^3 has Assouad dimension 3, and under a stability condition on the equality of Hausdorff and packing dimensions, Kakeya sets in R^3 have full Hausdorff and packing dimension. These results do not resolve the classical conjecture but advance understanding of the multi-scale structure of Kakeya sets.