TY - JOUR

T1 - Distinct distances on two lines

AU - Sharir, Micha

AU - Sheffer, Adam

AU - Solymosi, József

N1 - Funding Information:
Work on this paper by the first two authors was partially supported by Grant 338/09 from the Israel Science Fund and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11) . Work by Micha Sharir was also supported by NSF Grant CCF-08-30272 and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University . Work by József Solymosi was supported by NSERC , ERC-AdG 321104 , and OTKA NK 104183 grants.

PY - 2013/9

Y1 - 2013/9

N2 - Let P1 and P2 be two finite sets of points in the plane, so that P1 is contained in a line ℓ1, P2 is contained in a line ℓ2, and ℓ1 and ℓ2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs of P1×P2 isΩ(min{|P1|2/3|P2|2/3,|P1|2,|P2|2}). In particular, if |P1|=|P2|=m, then the number of these distinct distances is Ω(m4/3), improving upon the previous bound Ω(m5/4) of Elekes (1999) [3].

AB - Let P1 and P2 be two finite sets of points in the plane, so that P1 is contained in a line ℓ1, P2 is contained in a line ℓ2, and ℓ1 and ℓ2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs of P1×P2 isΩ(min{|P1|2/3|P2|2/3,|P1|2,|P2|2}). In particular, if |P1|=|P2|=m, then the number of these distinct distances is Ω(m4/3), improving upon the previous bound Ω(m5/4) of Elekes (1999) [3].

KW - Combinatorial geometry

KW - Distinct distances

KW - Incidences

UR - http://www.scopus.com/inward/record.url?scp=84879760274&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2013.06.009

DO - 10.1016/j.jcta.2013.06.009

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AN - SCOPUS:84879760274

SN - 0097-3165

VL - 120

SP - 1732

EP - 1736

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

IS - 7

ER -