- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:48
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
32
2016
2
Degree reduction and graininess for Kakeya-type sets in $\mathbb R^3$
Larry
Guth
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Kakeya set, incidence geometry, polynomial method
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.
Fourier analysis
447
494
10.4171/RMI/891
http://www.ems-ph.org/doi/10.4171/RMI/891