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Generalised Erdős distance theory on graphs

Sean Dewar, Nora Frankl, Samuel Mansfield, Anthony Nixon, Jonathan Passant, Audie Warren

TL;DR

This work unifies generalized Erdős distance problems through g-rigidity, replacing geometric distance with polynomial measurements to study graph-based realisations of point sets. It develops an energy-based framework that translates bounds on the energy of point configurations under g-rigities into sharp lower bounds on the number of distinct realisations |f_{g,G}(P^V)|, applicable to Euclidean, pseudo-Euclidean, ℓ_p, dot-product, and tensor-completion settings. The authors obtain near-optimal results in 2D under rigidity assumptions, prove a second main theorem for low-degree algebraic curves, and extend methods to typical norms via graph coloring and pinning, while also exploring what happens when rigidity is dropped, including the rich behaviour of flexible graphs. Together, these results advance the understanding of how rigidity, algebraic constraints, and norm geometry govern the combinatorial complexity of point-set realisations, with implications for rigidity theory, matrix/tensor completion, and distance problems in varied geometries.

Abstract

The famous Erdős distinct distances problem asks the following: how many distinct distances must exist between a set of $n$ points in the plane? There are many generalisations of this question that ask one to consider different spaces and metrics, or larger structures of points. We bring these problems into a common framework using the concept of $g$-rigidity. Specifically, if $G=(V,E)$ is a (hyper)graph, $g$ is a map assigning polynomial measurements to the edges of $G$ and $f_{g,G}(P^V)$ gives the set of $g$-distinct realisations of the $g$-rigid graph $G$, where vertices must lie in a point set $P$, our main results describe sharp lower bounds for the size of $\big|f_{g,G}(P^V)\big|$. This allows us to obtain results for pseudo-Euclidean metrics, $\ell_p$ metrics, dot-product problems, matrix completion problems, and symmetric tensor completion problems. In addition, we use the recent work of Alon, Bucić and Sauermann along with a simple colouring argument to prove that the number of $\| \cdot\|$-distinct realisations of a graph $G=(V,E)$ within a $d$-dimensional point set $P$ is at least $Ω\left(\frac{|P|^{|V|-1}}{(\log |P|)^2} \right)$ for almost all $d$-norms. Our methods here also provide a short proof that the unit distance conjecture implies the pinned distance conjecture.

Generalised Erdős distance theory on graphs

TL;DR

This work unifies generalized Erdős distance problems through g-rigidity, replacing geometric distance with polynomial measurements to study graph-based realisations of point sets. It develops an energy-based framework that translates bounds on the energy of point configurations under g-rigities into sharp lower bounds on the number of distinct realisations |f_{g,G}(P^V)|, applicable to Euclidean, pseudo-Euclidean, ℓ_p, dot-product, and tensor-completion settings. The authors obtain near-optimal results in 2D under rigidity assumptions, prove a second main theorem for low-degree algebraic curves, and extend methods to typical norms via graph coloring and pinning, while also exploring what happens when rigidity is dropped, including the rich behaviour of flexible graphs. Together, these results advance the understanding of how rigidity, algebraic constraints, and norm geometry govern the combinatorial complexity of point-set realisations, with implications for rigidity theory, matrix/tensor completion, and distance problems in varied geometries.

Abstract

The famous Erdős distinct distances problem asks the following: how many distinct distances must exist between a set of points in the plane? There are many generalisations of this question that ask one to consider different spaces and metrics, or larger structures of points. We bring these problems into a common framework using the concept of -rigidity. Specifically, if is a (hyper)graph, is a map assigning polynomial measurements to the edges of and gives the set of -distinct realisations of the -rigid graph , where vertices must lie in a point set , our main results describe sharp lower bounds for the size of . This allows us to obtain results for pseudo-Euclidean metrics, metrics, dot-product problems, matrix completion problems, and symmetric tensor completion problems. In addition, we use the recent work of Alon, Bucić and Sauermann along with a simple colouring argument to prove that the number of -distinct realisations of a graph within a -dimensional point set is at least for almost all -norms. Our methods here also provide a short proof that the unit distance conjecture implies the pinned distance conjecture.
Paper Structure (35 sections, 64 theorems, 179 equations, 2 figures)

This paper contains 35 sections, 64 theorems, 179 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The graphical Erdős distance problem and rigidity
  3. Generalised Erdos problems via g-rigidity
  4. Results for specific g-maps
  5. Euclidean distance
  6. Pseudo-Euclidean metrics
  7. The lp metric
  8. The dot product and matrix completion
  9. Skew-symmetric bilinear forms
  10. Symmetric tensor completion
  11. Generalised Erdos problems via typical norms
  12. Layout of paper
  13. Notation and terminology
  14. Graphical Erdős problem with algebraic constraints
  15. g-rigidity preliminaries
  16. ...and 20 more sections

Key Result

Theorem 1.5

Let $G=(V,E)$ be a $g$-rigid $k$-uniform hypergraph with at least $d+1$ vertices and maximum degree $\Delta_G$, and let $P \subset \mathbb{R}^d$ be a finite set where for any hypersurface $C \subset \mathbb{R}^d$ with degree at most $\Delta_G \deg g$, the inequality $|P\cap C| \leq \frac{1}{100}|P|$

Figures (2)

  • Figure 1: Left: A possible realisation $p \in P^V$. Right: Two equivalent realisations (with edge colours red-blue and orange-blue respectively) that can be obtained via 0-extension.
  • Figure 2: Three 1 d.o.f. graphs with (from left to right) flex-genus 0, 1 and 5.

Theorems & Definitions (130)

  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.10
  • Corollary 1.11
  • Remark 1.12
  • Corollary 1.13
  • Corollary 1.14
  • ...and 120 more