Degree reduction and graininess for Kakeya-type sets in
![Degree reduction and graininess for Kakeya-type sets in $\mathbb R^3$ cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-rmi-volume-32-issue-2.png&w=3840&q=90)
Let be a set of cylindrical tubes in of length and radius 1. If the union of the tubes has volume , and each point in the union lies in tubes pointing in three quantitatively different directions, and if a technical assumption holds, then at scale , the tubes are clustered into rectangular slabs of dimension . This estimate generalizes the graininess estimate in [7]. The proof is based on modeling the union of tubes with a high-degree polynomial.
Larry Guth, Degree reduction and graininess for Kakeya-type sets in . Rev. Mat. Iberoam. 32 (2016), no. 2, pp. 447–494
DOI 10.4171/RMI/891