🔢 mathematics

An incomplete attack on the upper bound of the unit distance problem

This paper presents an incomplete attempt to demonstrate that the established upper bound of approximately $n^{4/3}$ for the number of unit distances determined by $n$ points in the plane is not sharp, while also offering insights into configurations of points and lines that achieve the sharp Szemerédi-Trotter incidence bound.

Original authors: Steven Senger

Published 2026-05-27

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Steven Senger

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Party Distance" Problem

Imagine you throw a massive party with $n$ guests standing anywhere on a large dance floor. You want to know: What is the maximum number of pairs of people who can stand exactly one meter apart from each other?

In 1946, a famous mathematician named Erdős guessed that this number couldn't be too huge. He thought it would be roughly $n$ (the number of guests) plus a tiny bit of extra.

However, the best mathematical proof we have right now (from the 1980s) says the number could be as high as $n^{4/3}$ . To visualize this: if you have 1,000,000 guests, the "safe" limit is about 10,000,000 pairs standing one meter apart.

The Goal of This Paper:
The author, Steven Senger, is trying to prove that this limit ( $n^{4/3}$ ) is too high. He wants to show that you can't actually pack that many people one meter apart without breaking the laws of geometry. He calls this an "incomplete attack" because he got stuck and couldn't finish the proof, but he wants to share his map in case someone else can finish the journey.

The Strategy: The "Crossing Roads" Analogy

To prove his point, Senger uses a clever trick involving traffic jams (mathematicians call this the "Crossing Number Lemma").

The Map: Imagine drawing a map where every guest is a dot. If two guests are exactly one meter apart, you draw a curved line (an arc) connecting them.
The Traffic: When you draw all these lines, they will inevitably cross each other. The "Crossing Number" is just a count of how many times these lines intersect.
The Rule: There is a known mathematical rule that says: If you have a lot of lines (edges) and not enough dots (vertices), the lines are forced to cross each other a huge number of times.

Senger's Logic:
He assumes the worst-case scenario: that the $n^{4/3}$ limit is actually true. If this were true, he argues, the "traffic jam" of crossing lines would have to be incredibly specific and uniform.

The Analogy: Imagine a highway where every single car is involved in exactly the same number of near-misses with other cars.
The Claim: Senger shows that for the math to work, almost every "one-meter connection" must have about $n^{2/3}$ other lines crossing it. It's like saying every single person at the party must be standing in a spot where exactly 1,000 other people's "one-meter zones" overlap them.

The "Lune" and the "Strip"

To make this concrete, Senger breaks the dance floor down into smaller, manageable pieces.

The Lunes (Crescent Shapes):
If you have two people, Alice and Bob, who are less than a meter apart, their "one-meter circles" overlap in a shape that looks like a crescent moon (mathematicians call this a lune).
- Senger argues that if the $n^{4/3}$ limit is true, there must be huge crowds of people packed into these crescent shapes.
- He identifies "Typical Points": These are guests who are surrounded by so many other people that they are part of many of these crowded crescents.
The Two Squares:
He zooms in on just two small squares on the dance floor that contain the most "one-meter pairs." He then prunes the crowd, removing people who are too close together or too far apart, leaving a "clean" group of people.
- The Result: He ends up with a group of people packed into a thin horizontal strip. In this strip, the people are arranged in a very rigid, grid-like pattern where the gaps between them are all roughly the same size.

The "Incomplete" Part: The Trap

This is where the paper stops working. Senger sets up a trap for the geometry:

He finds the two people in this tight group who are closest to each other along a curved path. Let's call the distance between them $\epsilon$ (a tiny number).
He argues that because the group is so packed, there must be another pair of people even closer together than $\epsilon$ .
The Hope: If you keep finding pairs that are closer and closer, eventually the distance would have to become zero (two people occupying the same spot), which is impossible. This would prove the original assumption ( $n^{4/3}$ ) was wrong.

Why it failed:
Senger admits he got stuck. He couldn't mathematically prove that the "closer pair" actually existed in a way that created a contradiction. He mentions that other mathematicians (Katz and Silier) have since found stronger results that might have solved the problem, so he abandoned his specific approach.

The Side Note: Points and Lines

The paper also briefly discusses a related problem: Points and Lines.

Imagine you have $n$ dots and $n$ lines. How many times can a dot sit exactly on a line?
The math says the maximum is also around $n^{4/3}$ .
Senger shows that if you hit this maximum, the dots and lines must be arranged in a very specific, "sharp" pattern, similar to the crowded dance floor. He uses a "mirror" technique (called duality) to show that the rules for crowded dots are the same as the rules for crowded lines.

Summary

The Problem: Can we prove that you can't have as many "one-meter pairs" as the current math allows ( $n^{4/3}$ )?
The Method: Assume you can have that many. This forces the points to be arranged in a very specific, crowded way where "traffic" (crossing lines) is perfectly uniform.
The Attempt: Senger tried to show that this perfect uniformity leads to a geometric impossibility (people getting infinitely close).
The Outcome: He got stuck. He couldn't finish the proof, but he laid out the "roadmap" of how the points would have to behave if the limit were true. He hopes someone else can pick up the map and finish the trip.

Note: The paper explicitly states this is an incomplete attempt. It does not claim to have solved the problem, nor does it offer new applications for AI or other fields. It is purely a mathematical exploration of a specific geometric puzzle.

Technical Summary: An Incomplete Attack on the Upper Bound of the Unit Distance Problem

Problem Statement
The paper addresses the unit distance problem in the plane, which seeks an upper bound on the number of pairs of points separated by a unit distance in a set of $n$ points. The established upper bound, proven by Spencer, Szemerédi, and Trotter, is $O(n^{4/3})$ . While Erdős conjectured a bound of $n^{1+\epsilon}$ , recent results (including those disproving the conjecture via AI and explicit constructions by Sawin) have narrowed the gap but not closed it. The author investigates whether the $n^{4/3}$ bound is sharp. Specifically, the paper attempts to derive a contradiction by assuming the bound is tight and analyzing the geometric structure of point sets that would achieve this maximum.

Methodology
The author employs a proof by contradiction, assuming the existence of a set $P$ of $n$ points determining $\Theta(n^{4/3})$ unit distances. The methodology proceeds through several stages of structural analysis and geometric reduction:

Graph-Theoretic Framework: The proof utilizes the crossing number lemma (Lemma 1.2), which relates the number of edges $|E|$ and vertices $|V|$ in a graph to its crossing number $cr(G)$. By constructing a graph where vertices are points and edges are arcs of unit circles connecting consecutive points, the author links the number of unit distances to the number of circle intersections.
Average Crossing Analysis: The author establishes that if the $n^{4/3}$ bound is tight, the associated graph must have "typical" edges with approximately $n^{2/3}$ crossings. This leads to the existence of "typical" points (Proposition 1.5) that lie on $\approx n^{1/3}$ circles, where each circle contains $\approx n^{1/3}$ points, and the regions (lunes) between consecutive points on these circles contain $\approx n^{2/3}$ points.
Geometric Reduction (Pigeonholing): The plane is decomposed into squares, and the analysis is restricted to a pair of squares containing the densest concentration of unit distances. The point set is further refined by:
- Selecting a horizontal strip $H_\delta$ to maximize unit distances between two subsets $A$ and $B$ .
- Pruning points to ensure a minimum arc length $\epsilon$ between consecutive points on any unit circle, resulting in a subset $E$ with $\gtrsim n$ points and $\gtrsim n^{4/3}$ unit distances.
Trigonometric Estimation: The author derives estimates for distances between points on intersecting circles using Taylor expansions of sine and cosine. Lemma 3.2 and Corollary 3.3 provide approximations for the distance $|a-c|$ based on arc lengths $\alpha, \beta$ and the angle $\theta$ between tangents. The core idea is that if points are tightly clustered, a second pair of points should be found even closer than the initial minimum separation $\epsilon$ , potentially leading to a contradiction.
Connection to Szemerédi-Trotter Incidence Bound: The paper draws a parallel to the Szemerédi-Trotter theorem regarding point-line incidences. It analyzes "ST-sharp pairs" (sets of $n$ points and $n$ lines with $\Theta(n^{4/3})$ incidences) to demonstrate similar structural properties, such as the distribution of points on lines and the crossing behavior of dual graphs.

Key Contributions and Results

Structural Characterization: The paper provides a detailed characterization of the geometric constraints any point set must satisfy if it were to attain the $n^{4/3}$ upper bound. This includes the necessity of "typical" points, the specific distribution of points within lunes, and the minimum separation $\epsilon$ between points on unit circles.
Refined Geometric Lemmas: The author presents specific trigonometric estimates (Proposition 3.1, Lemma 3.2, Corollary 3.3) to analyze the behavior of points in the limit of small arc lengths and angles, distinguishing between cases where distances behave like arc differences versus arc-angle products.
Duality Analysis: The paper extends the crossing number argument to the dual setting of point-line incidences, showing that ST-sharp pairs exhibit analogous structural regularities (Propositions 4.2–4.4).

Significance and Limitations
The paper explicitly frames itself as an "incomplete attempt." The author states that the work was conducted between 2015 and 2021 but was abandoned because the author encountered a "stuck" point in the derivation of the final contradiction. Furthermore, the author notes that subsequent work by Katz and Silier (referenced as [4]) appears to have produced significantly stronger results, rendering this specific attack obsolete.

The significance of the document lies not in a solved theorem, but in the description of the approach. The author hopes the detailed breakdown of the pigeonholing scheme, the pruning algorithm, and the trigonometric estimates may prove useful to others investigating the sharpness of the unit distance bound or the structure of Szemerédi-Trotter extremal sets. The paper serves as a repository of a specific line of reasoning that, while failing to produce a contradiction in this instance, offers a rigorous exploration of the geometric landscape of potential counterexamples to the $n^{4/3}$ bound.