Kakeya inequality

Kakeya inequalities bound overlaps of long thin tubes with controlled directions.

These tube estimates are used to study Kakeya sets and their dimension.

For a direction $e$, the Kakeya maximal function is the supremum over translations of normalized averages of $|f|$ over unit-length Kakeya tubes in direction $e$. It measures how large a function can be on at least one thin tube in each direction. Normalizing by tube volume is essential when comparing different thicknesses.

The maximal estimate used in the classical Kakeya set problem asks, for every $\epsilon >0$, for a bound from $L^n({\mathbb{R}}^n)$ to $L^n(S^{n-1})$ with loss at most ${\delta }^{-\epsilon }$. Constants may depend on dimension and $\epsilon$, but not tube thickness or the function. Applied to an indicator function of a neighborhood of a Kakeya set, it gives the Kakeya Minkowski dimension conjecture.

For a separated angular net, two translated unit rectangles of width $\delta$ intersect in area at most a constant times ${\delta }^2/(\delta +\alpha )$, where $\alpha$ is their unoriented angular distance. Each row of their intersection matrix therefore has sum at most $C\delta \log (2/\delta )$. Applying the Schur test to the adjoint averaging operator gives the displayed bound. A wider-tube comparison extends it from the net to all directions. Since $\sqrt{\log (2/\delta )}{\lesssim }_{\epsilon }{\delta }^{-\epsilon }$, this proves the planar version of the Kakeya maximal conjecture.

Choose $M\asymp {\delta }^{-1}$ angularly separated unit segments in a bounded Kakeya set, and let $F$ sum the indicators of their Kakeya tubes. Then $\int F\asymp 1$. Two tubes at angular separation $\alpha$ intersect in area at most $C{\delta }^2/(\delta +\alpha )$, so summing each harmonic angular row gives $\int F^2\lesssim \log (2/\delta )$. The Cauchy-Schwarz inequality on their union proves the displayed neighborhood lower bound. Since $|E_{\delta }|\lesssim {\delta }^2{N}_{\delta }(E)$, both planar Minkowski dimensions equal two.

For three transverse families of $R^{1/2}\times R^{1/2}\times R$ tubes in ${\mathbb{R}}^3$,