Micha Sharir, Adam Sheffer, József Solymosi
Published 2013-02-13, updated 2013-06-02Version 2
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.
On lattices, distinct distances, and the Elekes-Sharir framework
On Distinct Distances Between a Variety and a Point Set
Distinct distances for points lying on curves in $\mathbb{R}^d$ -- the bipartite case
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