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Distinct Distances with $\ell_p$ Spaces

Moaaz AlQady, Riley Chabot, William Dudarov, Linus Ge, Mandar Juvekar, Srikanth Kundeti, Neloy Kundu, Kevin Lu, Yago Moreno, Sibo Peng, Samuel Speas, Julia Starzycka, Henry Steinthal, Anastasiia Vitko

Abstract

We study Erd\H os's distinct distances problem under $\ell_p$ metrics with integer $p$. We improve the current best bound for this problem from $Ω(n^{4/5})$ to $Ω(n^{6/7-ε})$, for any $ε>0$. We also characterize the sets that span an asymptotically minimal number of distinct distances under the $\ell_1$ and $\ell_\infty$ metrics.

Distinct Distances with $\ell_p$ Spaces

Abstract

We study Erd\H os's distinct distances problem under metrics with integer . We improve the current best bound for this problem from to , for any . We also characterize the sets that span an asymptotically minimal number of distinct distances under the and metrics.

Paper Structure

This paper contains 3 sections, 11 theorems, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Strictly convex $\ell_p$ metrics
  3. Structure in $\ell_1$ and $\ell_\infty$

Key Result

Theorem 1.1

Let $\mathcal{P}$ be a set of $n$ points in $\mathbb R^2$ and let $p> 1$. Then, under the $\ell_p$ metric, there exists a point $u\in \mathcal{P}$ such that

Figures (4)

  • Figure 1: In each of the nine regions, the bisector is a segment of an algebraic curve of degree at most $p$. Each inflection point is in a different region that contains a bounded segment of the bisector.
  • Figure 2: Case 1 when there are nine distinct rectangles.
  • Figure 3: Case 2 when there are nine distinct rectangles.
  • Figure 4: Placing disjoint arithmetic progressions with step $1/10n$ on horizontal lines.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Crossing lemma for multigraphs Szek97
  • Theorem 2.4
  • proof
  • ...and 5 more