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29 March 2024
 
  » arxiv » 1308.5620

 Article overview


Few distinct distances implies no heavy lines or circles
Adam Sheffer ; Joshua Zahl ; Frank de Zeeuw ;
Date 26 Aug 2013
AbstractWe study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distances, then no line contains Omega(n^{7/8}) points of P and no circle contains Omega(n^{5/6}) points of P.
We rely on the bipartite and partial variant of the Elekes-Sharir framework that was presented by Sharir, Sheffer, and Solymosi in cite{SSS13}. For the case of lines we combine this framework with a theorem from additive combinatorics, and for the case of circles we combine it with some basic algebraic geometry and a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang cite{WYZ13}. A significant difference between our approach and that of cite{SSS13} (and other recent extensions) is that, instead of dealing with distances between two point sets that are restricted to one-dimensional curves, we consider distances between one set that is restricted to a curve and one set with no restrictions on it.
Source arXiv, 1308.5620
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