On Erd\H{o}s Chains in the Plane

TL;DR

This paper establishes lower bounds on the number of distinct distance chains in a finite point set in the plane, using advanced combinatorial and geometric techniques, and extends results to graph-distance sets with Hamiltonian paths.

Contribution

It introduces new lower bounds for Erd ext{o}s distance chains and graph-distance sets, utilizing energy methods and recent bounds on incidences of lines, advancing understanding of geometric configurations.

Findings

Lower bound for distance n-chains: |\u0394_n(P)|   rac{|P|^n}{ ext{log}^{rac{13}{2}(n-1)}|P|}

Lower bound for graph-distance sets with Hamiltonian path: |_G(P)|   rac{|P|^{m-1}}{ ext{polylog}|P|}

Uses energy construction and Rudnev's line incidence bounds to achieve results.

Abstract

Let $P$ be a finite point set in $\mathbb{R}^2$ with the set of distance $n$-chains defined as $$ \Delta_n(P)=\{(|p_1-p_2|,|p_2-p_3|,\ldots,|p_n-p_{n+1}|):p_i \in P\}.$$ We show that for $2\leq n=O_{|P|}(1)$ we have $$|\Delta_n(P)|\gtrsim \frac{|P|^{n}}{\log^{\frac{13}{2}(n-1)}|P|}.$$ Our argument uses the energy construction of Elekes and a general version of Rudnev's rich-line bound implicit in Rudnev's recent hinge paper which allows one to iterate efficiently on highly intersecting nested subsets of Guth-Katz lines. Let $G$ is a simple connected graph on $m=O(1)$ vertices with $m\geq 2$. Define the graph-distance set $\Delta_G(P)$ as $$ \Delta_G(P) = \{ (|p_{i}-p_{j}|)_{\{i,j\}\in E(G)} : p_i,p_j \in P\}.$$ Combining with results of Guth and Katz and Rudnev with the above, if $G$ has a Hamiltonian path we have $$ |\Delta_G(P)| \gtrsim \frac{|P|^{m-1}}{\text{polylog}|P|}. $$ \end{abstract}