Degree reduction and graininess for Kakeya-type sets in ${\mathbb{R}}^3$

Larry Guth

Let $\mathfrak{T}$ be a set of cylindrical tubes in ${\mathbb{R}}^3$ of length $N$ and radius 1. If the union of the tubes has volume $N^{3-\sigma }$, and each point in the union lies in tube…