Kakeya set

A bounded subset of ${\mathbb{R}}^n$ is a Kakeya set if it contains a unit line segment in every direction. Segment positions may vary with direction. Such a set need not have positive Lebesgue measure; the dimension problem concerns its Minkowski dimension or other notions of dimension.

A Besicovitch set is a planar Kakeya set of Lebesgue measure zero containing a unit line segment in every direction.

Choose a compact $K\subset [0,1]\times \mathbb{R}$ with full horizontal projection and zero-length projections $y+tx$ for $t\in [0,1]$. Then the displayed set is compact, has zero area by the Fubini theorem, and contains a segment of every slope in $[0,1]$. Such parameter sets exist by the Baire category theorem in the Hausdorff metric: fine parallel segments approximate any parameter set while making a fixed band of projections uniformly small. Finite grids and a countable intersection make every required projection null. Four rotated copies supply every planar direction.

A Perron tree is a finite triangular splitting-and-sliding arrangement retaining a prescribed angular range while overlapping its pieces strongly. The compression parameter and the number of generations can make union area an arbitrarily small fraction of total piece area. Suitable longitudinal extensions separate the pieces into disjoint rectangles. This stronger finite geometry is used by the ball-multiplier counterexample; a compact measure-zero Besicovitch set alone does not assert that arbitrary segment translates are disjoint.

The dimension assertion asks that every bounded Kakeya set in ${\mathbb{R}}^n$ have full Minkowski dimension. The Kakeya maximal conjecture implies the stronger lower-dimension conclusion: its application to $E_{\delta }$ gives $|E_{\delta }|{\gtrsim }_{n,\epsilon }{\delta }^{n\epsilon }$, hence $N_{\delta }(E){\gtrsim }_{n,\epsilon }{\delta }^{-n+n\epsilon }$ for every $\epsilon >0$.

A Kakeya tube is a neighborhood of transverse radius $\delta$ of a line segment of fixed unit length. Round cross-sections and comparable rectangular cross-sections are interchangeable up to fixed constants. The tube has Lebesgue measure comparable to ${\delta }^{n-1}$ in ${\mathbb{R}}^n$.