Distinct Distance Estimates and Low Degree Polynomial Partitioning
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454_2014_Article_9648.pdf
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221.4 KB
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Checksum (MD5)
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Author(s)
Guth, Lawrence
Date Issued
December 2014
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Guth, Larry. “Distinct Distance Estimates and Low Degree Polynomial Partitioning.” Discrete & Computational Geometry 53.2 (2015): 428–444.
Version
Author's final manuscript
Abstract
We give a shorter proof of a slightly weaker version of a theorem from Guth and Katz (Ann Math 181:155–190, 2015): we prove that if L is a set of L lines in R[superscript 3] with at most L[superscript 1/2] lines in any low degree algebraic surface, then the number of r-rich points of is L is ≲ L[superscript (3/2) + ε] r[superscript -2]. This result is one of the main ingredients in the proof of the distinct distance estimate in Guth and Katz (2015). With our slightly weaker theorem, we get a slightly weaker distinct distance estimate: any set of N points in R[superscript 2] c[subscript ε]N[superscript 1-ε] distinct distances.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00454-014-9648-8
