Distinct distances on two lines

TL;DR

This paper establishes a lower bound on the number of distinct distances between points on two non-parallel, non-orthogonal lines, improving previous bounds and contributing to combinatorial geometry.

Contribution

It proves a new lower bound of (min{|P_1|^{2/3}|P_2|^{2/3},|P_1|^2,|P_2|^2}) for distinct distances between points on two lines, surpassing earlier results.

Findings

Lower bound of (min{|P_1|^{2/3}|P_2|^{2/3},|P_1|^2,|P_2|^2}) for distinct distances

Special case: (m^{4/3}) distinct distances when |P_1|=|P_2|=m

Improves previous bound of (m^{5/4}) by Elekes

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.