Distinct distances on two lines
Combinatorics
2013-06-04 v2 Computational Geometry
Abstract
Let P_1 and P_2 be two sets of points in the plane, so that P_1 is contained in a line L_1, P_2 is contained in a line L_2, and L_1 and L_2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs of P_1xP_2 is \Omega(\min{|P_1|^{2/3}|P_2|^{2/3},|P_1|^2, |P_2|^2}). In particular, if |P_1|=|P_2|=m, then the number of these distinct distances is \Omega(m^{4/3}), improving upon the previous bound \Omega(m^{5/4}) of Elekes.
Cite
@article{arxiv.1302.3081,
title = {Distinct distances on two lines},
author = {Micha Sharir and Adam Sheffer and József Solymosi},
journal= {arXiv preprint arXiv:1302.3081},
year = {2013}
}