Degree reduction and graininess for Kakeya-type sets in R[superscript 3]

Abstract

Let T be a set of cylindrical tubes in ℝ[superscrpit 3] of length N and radius 1. If the union of the tubes has volume N[superscript 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[superscript σ ], the tubes are clustered into rectangular slabs of dimension 1 x N[superscript σ] x N[superscript σ]. 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