# Revista Matemática Iberoamericana

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**Volume 32, Issue 2, 2016, pp. 447–494**

**DOI: 10.4171/RMI/891**

Published online: 2016-06-08

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

Larry Guth^{[1]}(1) Massachusetts Institute of Technology, Cambridge, USA

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 $N^{3 - \sigma}$, 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.

*Keywords: *Kakeya set, incidence geometry, polynomial method

Guth Larry: Degree reduction and graininess for Kakeya-type sets in $\mathbb R^3$. *Rev. Mat. Iberoamericana* 32 (2016), 447-494. doi: 10.4171/RMI/891