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A Density Condition on Point Sets with Slowly-Scaling Distinct Dot Products

Anshula Gandhi

TL;DR

This work tackles the slowly-scaling distinct dot products problem in the plane by proving a density-type theorem: if $|D(P_n)| \ll n^{3/4}$ for a sequence of point sets, then for every $b \in (0,1)$ there exists a subsequence in which a line through the origin contains a $b$-dense, point-rich subset with size at least $|L| \gtrsim n^{1/2}$. The authors develop a framework around line and circle configurations, showing that optimal line and circle arrangements yield only linear growth of dot products, and that cross-interactions between a popular line and a popular circle can force $|D(P_n)|$ up to $\gtrsim n^{3/4}$. They introduce and leverage the concepts of supporting lines/circles, popular lines/circles, and a density condition on line-point distributions to connect geometric structure with dot-product counts. The findings provide a concrete structural barrier to sublinear or sub-$n^{3/4}$ scaling, guiding future work toward either constructing such slowly-scaling sets or proving their nonexistence, and thereby narrowing the gap between lower bounds and known constructions.

Abstract

The distinct dot products problem, a variation on the Erdős distinct distance problem, asks "Given a set $P_n$ of $n$ points in $\mathbb{R}^2$, what is the minimum number $|D(P_n)|$ of distinct dot products formed between them, asymptotically?" The best proven lower-bound is $|D(P_n)| \gtrsim n^{2/3+7/1425}$, due to work by Hanson$\unicode{x2013}$Roche-Newton$\unicode{x2013}$Senger, and a recent improvement by Kokkinos. However, the slowest-scaling known constructions have $|D(P_n)|\sim n$, leaving quite a large gap in the bound. Finding a sublinearly-scaling construction, or disproving its existence, would narrow this gap. We provide a condition that a sequence of point configurations $(P_n)_{n \in \mathbb{N}}$ must satisfy in order for $|D(P_n)|$ to scale 'slowly' i.e. $|D(P_n)| \ll n^{3/4}$. Namely, we prove that any such configuration must contain a point-rich line that gets arbitrarily 'dense' as the sequence progresses.

A Density Condition on Point Sets with Slowly-Scaling Distinct Dot Products

TL;DR

This work tackles the slowly-scaling distinct dot products problem in the plane by proving a density-type theorem: if for a sequence of point sets, then for every there exists a subsequence in which a line through the origin contains a -dense, point-rich subset with size at least . The authors develop a framework around line and circle configurations, showing that optimal line and circle arrangements yield only linear growth of dot products, and that cross-interactions between a popular line and a popular circle can force up to . They introduce and leverage the concepts of supporting lines/circles, popular lines/circles, and a density condition on line-point distributions to connect geometric structure with dot-product counts. The findings provide a concrete structural barrier to sublinear or sub- scaling, guiding future work toward either constructing such slowly-scaling sets or proving their nonexistence, and thereby narrowing the gap between lower bounds and known constructions.

Abstract

The distinct dot products problem, a variation on the Erdős distinct distance problem, asks "Given a set of points in , what is the minimum number of distinct dot products formed between them, asymptotically?" The best proven lower-bound is , due to work by HansonRoche-NewtonSenger, and a recent improvement by Kokkinos. However, the slowest-scaling known constructions have , leaving quite a large gap in the bound. Finding a sublinearly-scaling construction, or disproving its existence, would narrow this gap. We provide a condition that a sequence of point configurations must satisfy in order for to scale 'slowly' i.e. . Namely, we prove that any such configuration must contain a point-rich line that gets arbitrarily 'dense' as the sequence progresses.
Paper Structure (18 sections, 16 theorems, 14 equations, 16 figures)

This paper contains 18 sections, 16 theorems, 14 equations, 16 figures.

Table of Contents

  1. Introduction
  2. The main result
  3. Line configurations
  4. Dot products among points on a line scale at least linearly
  5. Bounding the number of supporting lines
  6. Existence of a popular line
  7. Circle configurations
  8. Dot products among points on a circle scale at least linearly
  9. Bounding the number of supporting circles
  10. Existence of a popular circle
  11. Line-and-circle configurations
  12. Combining an optimal line and optimal circle configuration
  13. Combining a line and optimal circle configuration
  14. Combining a line and circle configuration
  15. Line-and-circle configurations in slowly-scaling sets
  16. ...and 3 more sections

Key Result

Theorem 1

(A density condition for slow scaling). Suppose that $\mathcal{P} = (P_n)_{n\in \mathbb{N}}$ is a sequence of point configurations, where each $P_n$ is a set of $n$ distinct points in $\mathbb{R}^2$. And suppose $|D(P_n)| \ll n^{3/4}$. Then, for all $b \in (0,1)$, there exists a subsequence $(P_{k_1

Figures (16)

  • Figure 1: Any $n$ points in geometric progression form $\sim n$ dot products.
  • Figure 2: Any $n$ points on a line form $\gtrsim n$ dot products.
  • Figure 3: A set of points and their supporting lines.
  • Figure 4: If the number of dot products scales $\ll n^{\alpha}$, there exists some popular line with $\gg n^{2-2\alpha}$ points..
  • Figure 5: Any $n$ points equally spaced on a circle form only $\sim n$ dot products.
  • ...and 11 more figures

Theorems & Definitions (38)

  • Conjecture
  • Theorem 1: \ref{['densitycond']}
  • Lemma 3.1: Points in geometric progression along a line
  • proof
  • Lemma 3.2: Bounding dot products on a line
  • proof
  • Definition 3.3: Supporting lines
  • Lemma 3.4: Lower bound on # supporting lines
  • proof
  • Lemma 3.5: Upper bound on # supporting lines
  • ...and 28 more