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Distinct Distances on Algebraic Curves in the Plane

Published online by Cambridge University Press:  15 July 2016

JÁNOS PACH  and
FRANK DE ZEEUW
JÁNOS PACH
Affiliation:
Department of Mathematics, EPFL, Lausanne, Switzerland and Rényi Institute, Budapest, Hungary (e-mail: pach@cims.nyu.edu)
FRANK DE ZEEUW
Affiliation:
Department of Mathematics, EPFL, Lausanne, Switzerland (e-mail: fdezeeuw@gmail.com)
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Abstract

Let S be a set of n points in ${\mathbb R}^{2}$ contained in an algebraic curve C of degree d. We prove that the number of distinct distances determined by S is at least cdn 4/3, unless C contains a line or a circle.

We also prove the lower bound cd ′ min{m 2/3 n 2/3, m 2, n 2} for the number of distinct distances between m points on one irreducible plane algebraic curve and n points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer and Solymosi in [19].

Keywords

Primary 52C10Secondary 52C45

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 1 , January 2017 , pp. 99 - 117
Copyright
Copyright © Cambridge University Press 2016 

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