Few distinct distances implies no heavy lines or circles
Author(s)Sheffer, Adam; Zahl, Joshua; de Zeeuw, Frank
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We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set PP of n points determines o(n) distinct distances, then no line contains Ω(n[superscript 7/8]) points of PP and no circle contains Ω(n[superscript 5/6]) points of PP . We rely on the partial variant of the Elekes-Sharir framework that was introduced by Sharir, Sheffer, and Solymosi in  for bipartite distinct distance problems. To prove our bound for the case of lines we combine this framework with a theorem from additive combinatorics, and for our bound 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 . A significant difference between our approach and that of  (and of other related results) 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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Springer Berlin Heidelberg
Sheffer, Adam, Joshua Zahl, and Frank de Zeeuw. “Few Distinct Distances Implies No Heavy Lines or Circles.” Combinatorica 36.3 (2016): 349–364.
Author's final manuscript