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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 $\epsilon >0$.

For a unit direction $e$, let $T_{\delta }(a,e)$ be a length-one Kakeya tube with transverse radius $\delta$, centered at $a$, and define the Kakeya maximal function byUsing normalized surface measure on the direction sphere, the Kakeya maximal conjecture is the following family of estimates, in the formulation relevant to this paper:The constant is independent of $\delta$ and $f$. Replacing round Kakeya tubes by comparable rectangular tubes changes only dimensional constants.

A bounded Kakeya set contains a unit line segment in every direction. The Kakeya Minkowski dimension conjecture says that every such set has full Minkowski dimension $n$. The maximal estimate in fact gives full lower as well as upper Minkowski dimension.

To prove that implication, let $E_{\delta }=\{x:\mathrm{dist}(x,E)<\delta \}$. Each unit segment in $E$ has a thinner tube contained in $E_{\delta }$, so ${\mathcal{K}}_{c\delta }{1}_{E_{\delta }}(e)\ge 1$ for every $e$, with a fixed dimensional $c>0$. Apply the maximal estimate to this indicator function:If $N_{\delta }(E)$ is the smallest number of radius-$\delta$ balls covering $E$, that cover, enlarged by a fixed factor, covers $E_{\delta }$. Thus $|E_{\delta }|{\lesssim }_n{\delta }^n{N}_{\delta }(E)$ andTaking the lower limit of $\log N_{\delta }(E)/\log (1/\delta )$ and then letting $\epsilon \downarrow 0$ gives lower Minkowski dimension at least $n$. Bounded subsets of ${\mathbb{R}}^n$ have upper Minkowski dimension at most $n$. Consequently both dimensions equal $n$, which proves the requested implication.

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.