claimroutineinferred_dep

Kakeya sets have infinite gauge Hausdorff measure

Kakeya sets have infinite Hausdorff measure for the log gauge hγh_\gamma
gives Hausdorff dimension nn
Claim·§9.3 The maximal estimate and Hausdorff dimension·inside corQuantitative extension and Kakeya bounds

Statement (paper)

Let F⊂RnF\subset\RR^n contain a unit line segment in every direction. Then for every γ>n−1\gamma>n-1 and the gauge hγ(r)=rn[log⁡(1/r)]2An[log⁡log⁡(1/r)]γh_\gamma(r)=r^n[\log(1/r)]^{2An}[\log\log(1/r)]^\gamma (A=2/pA=2/p, rr small), the Hausdorff measure of FF with respect to hγh_\gamma is infinite. Consequently, since hγ(r)=o(rs)h_\gamma(r)=o(r^s) for every s<ns<n, dim⁡HF=n\dim_{\mathrm H}F=n.

In plain English

If F⊂RnF\subset\mathbb R^n contains a unit line segment in every direction, then for every γ>n−1\gamma>n-1 its Hausdorff measure with gauge hγ(r)=rn[log⁡(1/r)]2An[log⁡log⁡(1/r)]γh_\gamma(r)=r^n[\log(1/r)]^{2An}[\log\log(1/r)]^\gamma (with A=2/pA=2/p) is infinite. Hence FF has Hausdorff dimension nn.

Why it's here

This converts the logarithmically lossy Kakeya maximal bound from the previous claim into a lower bound on covering sums, using Keich's grouping of covers by dyadic scale. Since hγ(r)=o(rs)h_\gamma(r)=o(r^s) for every s<ns<n, infinite hγh_\gamma-measure forces infinite ss-dimensional measure for all s<ns<n, which gives the Hausdorff dimension statement in the corollary.
Classified routine:
Standard Bourgain/Keich argument that converts a Kakeya maximal estimate with logarithmic loss into a lower bound for gauge Hausdorff measure, grouping the cover by dyadic scale (Keich 1999). Here it is applied to the paper's maximal bound from the previous claim.

Uses

claimKakeya maximal estimate by duality

Used by

Nothing in the paper

Sources

PaperOpen
Section
§9.3 The maximal estimate and Hausdorff dimension
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sections/kakeya.tex:211
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Friction

  • info
    inferred_dep
    outline: "Infinite gauge Hausdorff measure via Keich grouping" rests on claim-cor-quant-kakeya-1