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arXiv · 1302.3081

Distinct distances on two lines

Micha Sharir·Adam Sheffer·József Solymosi
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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 Ω(\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 Ω(m^{4/3}), improving upon the previous bound Ω(m^{5/4}) of Elekes.

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Micha Sharir, Adam Sheffer, József Solymosi. 2013-06-02. Distinct distances on two lines. https://arxiv.org/abs/1302.3081

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