Let P be a set of n points in the plane that determines at most n/5 distinct distances. We show that no line can contain more than O(n43/52polylog(n)) points of P. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span.
Bibliographical noteFunding Information:
Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation . Work by Micha Sharir was also supported by Grant 2012/229 from the US–Israel Binational Science Foundation , by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work by Oliver Roche-Newton was supported by the Austrian Science Fund (FWF), Project F5511-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation.
© 2015 Elsevier B.V.
- Distinct distances