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

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 tubes pointing in three quantitatively different directions, and if a technical assumption holds, then at scale $N^{\sigma }$, the tubes are clustered into rectangular slabs of dimension $1\times N^{\sigma }\times N^{\sigma }$. This estimate generalizes the graininess estimate in [7]. The proof is based on modeling the union of tubes with a high-degree polynomial.