Kakeya set conjecture in dimensions n ≥ 4

A Kakeya set, also called a Besicovitch set, is a compact set $E\subseteq\mathbb R^n$ that contains a unit line segment in every direction. That is, for every $e\in S^{n-1}$ there is an $a\in\mathbb R^n$ with

$$\{a+te:0\le t\le1\}\subseteq E .$$

Conjecture. For every integer $n\ge 4$, every Kakeya set $E\subseteq\mathbb R^n$ has Hausdorff dimension $n$:

$$\dim_H E=n .$$

The same assertion for $n=2$ and $n=3$ is a theorem (see below). The weaker statement with Minkowski (box-counting) dimension in place of Hausdorff dimension is also open for $n\ge4$, since $\dim_H E\le\underline{\dim}_M E$. A stronger conjecture concerns the Kakeya maximal function: for every $\varepsilon>0$,

$$\Bigl\|\sup_{T\ni x}\tfrac{1}{|T|}\int_T|f|\Bigr\|_{L^n(\mathbb R^n)}\le C_\varepsilon\,\delta^{-\varepsilon}\|f\|_{L^n(\mathbb R^n)},$$

where the supremum runs over $1\times\delta$ tubes $T$ containing $x$. This maximal estimate implies the set conjecture. [1, 2]

Background

Besicovitch showed in 1919–1928 that Kakeya sets can have Lebesgue measure zero. In the plane, a needle can be turned around inside a set of arbitrarily small area. The conjecture says that such sets must still be as large as possible in the sense of dimension. Interest grew after Fefferman's 1971 disproof of the ball multiplier conjecture, which used Besicovitch sets. Kakeya-type estimates were then recognized as obstructions underlying Fourier restriction, Bochner–Riesz summation and local smoothing. The restriction conjecture implies the Kakeya conjecture. Analogues in finite fields and other settings, together with connections to additive combinatorics, made the problem a meeting point of harmonic analysis, geometric measure theory and combinatorics. [1, 2]