Few distinct distances implies no heavy lines or circles

Adam Sheffer; Joshua Zahl; Frank de Zeeuw

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arXiv · 1308.5620

Few distinct distances implies no heavy lines or circles

Adam Sheffer·Joshua Zahl·Frank de Zeeuw

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Abstract

We 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 Ω(n^{7/8}) points of P and no circle contains Ω(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.

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Adam Sheffer, Joshua Zahl, Frank de Zeeuw. 2013-08-26. Few distinct distances implies no heavy lines or circles. https://arxiv.org/abs/1308.5620

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