Degree reduction and graininess for Kakeya-type sets in R[superscript 3]
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1402.0518.pdf
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457.65 KB
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Author(s)
Guth, Lawrence
Date Issued
June 2016
Journal
Revista Matemática Iberoamericana
Publisher
European Mathematical Publishing House
Citation
Guth, Larry. “Degree Reduction and Graininess for Kakeya-Type Sets in R[superscript 3].” Revista Matemática Iberoamericana, vol. 32, no. 2, 2016, pp. 447–94.
Version
Original manuscript
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
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.4171/RMI/891
