Almost sharp bounds on the number of discrete chains in the plane

2019/11/30 by Nóra Frankl, Frankl, Nora, Andrey Kupavskii +1
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · doi:10.48550/arxiv.1912.00224

openalex publication_date 2019/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. Given k positive real numbers δ₁,…,δ_k, a (k+1)-tuple (p₁,…,p_k+1) in ℝ^d is called a (δ,k)-chain if ‖p_j-p_j+1‖ = δ_j for every 1≤ j ≤ k. What is the maximum number C_k^d(n) of (k,δ)-chains in a set of n points in ℝ^d, where the maximum is taken over all δ? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all k in the planar case. error term It is only for k≡ 1 (mod) 3 that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in 3 dimension.

Related

• On the Number of Discrete Chains
• Unit Distance Problems
• Dot product chains
• Unit and distinct distances in typical norms
• On Erdős Chains in the Plane