= Planar Kakeya neighborhood lower bound
{title2=$|E_\delta|\gtrsim1/\log(2/\delta)$}
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 $\int F\asymp1$. 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 $|E_\delta|\lesssim\delta^2N_\delta(E)$, both planar <Minkowski dimensions> equal two.
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