We obtain an improved Kakeya maximal function estimate in $\mathbb R^4$ 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 $\mathbb R^4$ at dimension 3.059. As a consequence, every Besicovitch set in $\mathbb R^4$ must have Hausdorff dimension at least 3.059.

中文翻译：

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使用平面刷在四个维度上进行Kakeya最大函数估计

我们使用称为几何画笔的新几何参数在$ \ mathbb R ^ 4 $中获得改进的Kakeya最大函数估计。平面画笔是Wolff画笔的更高维度的类似物，当通过典型点的线集中到平面中时，它可以有效控制Besicovitch集的大小。当Besicovitch集不具有此属性时，可以使用现有的Guth–Zahl三线性估计来约束Besicovitch集的大小。特别地，我们在维度3.059在$ \ mathbb R ^ 4 $中建立最大函数估计。结果，$ \ mathbb R ^ 4 $中设置的每个Besicovitch必须具有至少3.059的Hausdorff维数。