Distinct distances between a collinear set and an arbitrary set of\n points

2016/12/15 by Ariel Bruner, Micha Sharir, Bruner, Ariel +1
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1612.04940

openalex publication_date 2016/12/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We consider the number of distinct distances between two finite sets of\npoints in bf R^k, for any constant dimension k\≥ 2, where one set P₁\nconsists of n points on a line l, and the other set P₂ consists of m\narbitrary points, such that no hyperplane orthogonal to l and no\nhypercylinder having l as its axis contains more than O(1) points of P₂.\nThe number of distinct distances between P₁ and P₂ is then \n
Omega
left(
min
left
n^2/3m^2/3,
;\n
fracn^10/11m^4/11
log^2/11m,
; n²,
; m²
right

right) .\nWithout the assumption on P₂, there exist sets P₁, P₂ as above, with\nonly O(m+n) distinct distances between them.\n

Citations

• Bounds of incidences between points and algebraic curves
• Polynomials vanishing on grids: The Elekes-Rónyai problem revisited

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