A subscription is required to access this article.
We obtain an improved Kakeya maximal function estimate in 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 at dimension 3.059. As a consequence, every Besicovitch set in 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. 317–359DOI 10.4171/RMI/1219