Degree reduction and graininess for Kakeya-type sets in

  • Larry Guth

    Massachusetts Institute of Technology, Cambridge, USA

Abstract

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.

Cite this article

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