claimroutineinferred_dep

Logarithmic volume bound for neighborhoods

Kakeya sets' δ\delta-neighborhoods have volume ≳L(1/δ)−2(n−1)\gtrsim L(1/\delta)^{-2(n-1)}
volume lower bound leading to dimension
Claim·§9.3 The maximal estimate and Hausdorff dimension·inside corQuantitative extension and Kakeya bounds

Statement (paper)

If F⊂RnF\subset\RR^n contains a unit line segment in every direction, then its open δ\delta-neighborhood FδF_\delta satisfies ∣Fδ∣≳L(1/δ)−2An=L(1/δ)−2(n−1)|F_\delta|\gtrsim L(1/\delta)^{-2An}=L(1/\delta)^{-2(n-1)} with A=2/pA=2/p, for 0<δ≤10<\delta\le 1.

In plain English

If F⊂RnF\subset\RR^n contains a unit line segment in every direction, then its open δ\delta-neighborhood satisfies ∣Fδ∣≳L(1/δ)−2An=L(1/δ)−2(n−1)|F_\delta|\gtrsim L(1/\delta)^{-2An}=L(1/\delta)^{-2(n-1)} with A=2/pA=2/p, for 0<δ≤10<\delta\le1.

Why it's here

This turns the Kakeya maximal estimate into a lower bound on volume: testing the maximal inequality on the indicator function of FδF_\delta shows that FδF_\delta cannot be too small, losing only a power of log⁡(1/δ)\log(1/\delta). This logarithmic bound is the quantitative input for the Hausdorff dimension nn conclusion in the corollary.»}
Classified routine:
Standard argument (Bourgain/Córdoba) of testing the Kakeya maximal inequality on the indicator of FδF_\delta: Kδ1Fδ≳1K_\delta 1_{F_\delta}\gtrsim1 on the sphere, which gives a volume lower bound; it follows directly from the preceding claim.

Uses

claimKakeya maximal estimate by duality

Used by

Nothing in the paper

Sources

PaperOpen
Section
§9.3 The maximal estimate and Hausdorff dimension
Page
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sections/kakeya.tex:203
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Friction

  • info
    inferred_dep
    outline: "Logarithmic volume bound for neighborhoods" rests on claim-cor-quant-kakeya-1