A Kakeya maximal function estimate in four dimensions using planebrushe

• Nets Hawk Katz ^[1] ; Joshua Zahl ^[2]
  1. [1] California Institute of Technology

     California Institute of Technology

     Estados Unidos

     
  2. [2] University of British Columbia

     University of British Columbia

     Canadá

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 37, Nº 1, 2021, págs. 317-359
• Idioma: inglés
• DOI: 10.4171/rmi/1219
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• Resumen

  • We obtain an improved Kakeya maximal function estimate in R4 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 at dimension 3.059. As a consequence, every Besicovitch set in R4 must have Hausdorff dimension at least 3.059.