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

Let $\frak T$ be a set of cylindrical tubes in $\mathbb{R}^3$ of length $N$ and radius 1. If the union of the tubes has volume $N^{3 - σ}$, and each point in the union lies in tubes pointing in three quantitatively different directions, and if a technical assumption holds, then at scale $N^σ$, the tubes are clustered into rectangular slabs of dimension $1 \times N^σ\times N^σ$. This estimate generalizes the graininess estimate proven by Katz-Laba-Tao. The proof is based on modeling the union of tubes with a high-degree polynomial.

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