JournalsrmiVol. 37, No. 1pp. 317–359

A Kakeya maximal function estimate in four dimensions using planebrushes

  • Nets Hawk Katz

    California Institute of Technology, Pasadena, USA
  • Joshua Zahl

    The University of British Columbia, Vancouver, Canada
A Kakeya maximal function estimate in four dimensions using planebrushes cover

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Abstract

We obtain an improved Kakeya maximal function estimate in R4\mathbb R^4 using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff’s hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth–Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in R4\mathbb R^4 at dimension 3.059. As a consequence, every Besicovitch set in R4\mathbb R^4 must have Hausdorff dimension at least 3.059.

Cite this article

Nets Hawk Katz, Joshua Zahl, A Kakeya maximal function estimate in four dimensions using planebrushes. Rev. Mat. Iberoam. 37 (2021), no. 1, pp. 317–359

DOI 10.4171/RMI/1219