The Kakeya set conjecture in four dimensions (Hausdorff dimension)

Theorem 1.1: if $K\subset\mathbb R^4$ contains, for every $e\in S^3$, a segment $\{a_e+te:0\le t\le1\}$, then $\dim_H K=4$. No measurability, compactness, stickiness or packing-dimension hypothesis is imposed. The paper also derives packing and Minkowski-dimension consequences, a projection lower bound in higher dimensions, and Nikodym and curved-Kakeya applications via cited transfers. It does NOT prove the four-dimensional Kakeya maximal conjecture, and does not settle the conjecture in dimension five or higher.

Theorem 1.1: if $K\subset\mathbb R^4$ contains, for every $e\in S^3$, a segment $\{a_e+te:0\le t\le1\}$, then $\dim_H K=4$. No measurability, compactness, stickiness or packing-dimension hypothesis is imposed. The paper also derives packing and Minkowski-dimension consequences, a projection lower bound in higher dimensions, and Nikodym and curved-Kakeya applications via cited transfers. It does NOT prove the four-dimensional Kakeya maximal conjecture, and does not settle the conjecture in dimension five or higher.