Distinct distances on two lines

Let P 1 and P 2 be two finite 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 1 × P 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 (1999) [3].