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Paper

A Kakeya maximal function estimate in four dimensions using planebrushes

Abstract

We obtain an improved Kakeya maximal function estimate and improved Kakeya Hausdorff dimension 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 3.049, and we prove that every Besicovitch set in must have Hausdorff dimension at least 3.059.