Distinct distances on two lines

作者：Sharir Micha^*; Sheffer Adam; Solymosi Jozsef

来源：Journal of Combinatorial Theory - Series A, 2013, 120(7): 1732-1736.

DOI：10.1016/j.jcta.2013.06.009

• Distinct distances
• Combinatorial geometry
• Incidences

Let P-1 and P-2 be two finite sets of points in the plane, so that PI 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 X P-2 is
Omega (min{vertical bar P-1 vertical bar(2/3)vertical bar P-2 vertical bar(2/3), vertical bar P-1 vertical bar(2), vertical bar P-2 vertical bar(2)}).
In particular, if vertical bar P-1 vertical bar = vertical bar P-2 vertical bar = m, then the number of these distinct distances is Omega(m(4/3)), improving upon the previous bound Omega(m(5/4)) of Elekes (1999) [3].

• 出版日期2013-9