Bull. Korean Math. Soc. 2021; 58(5): 1279-1300
Online first article July 7, 2021 Printed September 30, 2021
https://doi.org/10.4134/BKMS.b200937
Copyright © The Korean Mathematical Society.
Jonathan Passant
University of Rochester
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 \cite{R19}, which allows one to iterate efficiently on 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 \cite{GK15} and Rudnev \cite{R19} with the above, if $G$ has a Hamiltonian path we have $$ |\Delta_G(P)| \gtrsim \frac{|P|^{m-1}}{\text{polylog}|P|}. $$
Keywords: Incidence geometry, distinct distances
MSC numbers: 52C10, 51A20
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd