Resumen de Degree reduction and graininess for Kakeya-type sets in R3

Larry Guth

• Let T be a set of cylindrical tubes in R3 of length N and radius 1. If the union of the tubes has volume N3−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 Nσ, the tubes are clustered into rectangular slabs of dimension 1×Nσ×Nσ. This estimate generalizes the graininess estimate in [7]. The proof is based on modeling the union of tubes with a high-degree polynomial.