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Many unit distances requires many directions

Gabriel Currier, József Solymosi

TL;DR

The paper addresses the planar unit distance problem by showing that if a point set $P$ with $n$ points determines many unit distances from a restricted set of directions $D$ with $|D|=O(n^{1/3})$, then the number of such unit distances is $o(n^{4/3})$. The authors combine Guth–Katz polynomial partitioning with additive-combinatorics tools (Balog–Szemerédi–Gowers, Freiman–Ruzsa) and Chang’s bound on factorizations from generalized arithmetic progressions to deduce that many points must lie in a low-dimensional GAP, which then conflicts with circle-incidence bounds. The main contribution is a direction-size-agnostic obstruction: increasing the diversity of directions beyond $O(n^{1/3})$ is necessary to realize near-optimal unit-distance counts, establishing a strong structural constraint on point configurations with many unit distances. This advances understanding of the interplay between geometric incidence structure and arithmetic structure, with implications for the Erdős unit distance problem and related lattice-structure considerations.

Abstract

In this note, we show that in planar pointsets determining many unit distances, these unit distances must span many directions. Specifically, we show that a set of $n$ points can determine only $o(n^{4/3})$ unit distances from a set of at most $O(n^{1/3})$ directions.

Many unit distances requires many directions

TL;DR

The paper addresses the planar unit distance problem by showing that if a point set with points determines many unit distances from a restricted set of directions with , then the number of such unit distances is . The authors combine Guth–Katz polynomial partitioning with additive-combinatorics tools (Balog–Szemerédi–Gowers, Freiman–Ruzsa) and Chang’s bound on factorizations from generalized arithmetic progressions to deduce that many points must lie in a low-dimensional GAP, which then conflicts with circle-incidence bounds. The main contribution is a direction-size-agnostic obstruction: increasing the diversity of directions beyond is necessary to realize near-optimal unit-distance counts, establishing a strong structural constraint on point configurations with many unit distances. This advances understanding of the interplay between geometric incidence structure and arithmetic structure, with implications for the Erdős unit distance problem and related lattice-structure considerations.

Abstract

In this note, we show that in planar pointsets determining many unit distances, these unit distances must span many directions. Specifically, we show that a set of points can determine only unit distances from a set of at most directions.

Paper Structure

This paper contains 5 sections, 8 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem
  4. Concluding Remarks
  5. Acknowledgements

Key Result

Theorem 1.2

Let $P$ be a collection of $n$ points, and $D$ a collection of $O(n^{1/3})$ unit vectors. Then, $P$ can determine at most $o(n^{4/3})$ unit distances from vectors in $D$.

Figures (1)

  • Figure 1: A self-intersecting $P_3$ consisting of points $q_1,q_2,q_3,q_4$. Note that the vector difference $p_1-p_3$ is uniquely determined by the choice of arcs $a_1,a_2,a_3$.

Theorems & Definitions (18)

  • Conjecture 1.1: Erdős, 1946
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6: Balog-Szemerédi-Gowers BSG1BSG2
  • Definition 2.7
  • Theorem 2.8: Freiman-Ruzsa, FREIMANRUZSA
  • ...and 8 more