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arxiv: 2605.26145 · v1 · pith:RD2XJY25new · submitted 2026-05-22 · 🧮 math.GM

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

Steven Senger This is my paper

Pith reviewed 2026-06-30 14:27 UTC · model grok-4.3

classification 🧮 math.GM
keywords unit distance problemSzemerédi-Trotter theorempoint-line incidencescombinatorial geometryextremal boundsErdős problems
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The pith

The upper bound of roughly n to the 4/3 on unit distances determined by n points in the plane is not sharp.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sketches partial methods intended to prove that no large set of n points in the plane determines asymptotically as many as n to the 4/3 unit distances. The known upper bound of this order follows from the Szemerédi-Trotter theorem on point-line incidences, yet the argument aims to show that unit-distance graphs cannot saturate this incidence bound. The same techniques are applied to describe which point-line arrangements can achieve the full Szemerédi-Trotter extremal count. A sympathetic reader would care because closing the gap between the lattice lower bound and this upper bound has been open since the 1940s.

Core claim

The author presents an incomplete argument that configurations attaining the full n to the 4/3 order for unit distances cannot exist, together with observations on the point-line incidence configurations that do attain the Szemerédi-Trotter bound.

What carries the argument

Partial reduction of candidate unit-distance graphs to point-line incidence problems that fall short of the Szemerédi-Trotter extremal order.

If this is right

  • The maximum number of unit distances is strictly o(n to the 4/3).
  • Only restricted families of point-line arrangements attain the Szemerédi-Trotter incidence bound when the lines are defined by unit distances.
  • The known lower-bound constructions from lattices cannot be improved to match the current upper bound.
  • The same incidence-shortfall phenomenon limits extremal examples in related distance graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Completing the argument would give the first asymptotic improvement on the unit-distance upper bound since the Szemerédi-Trotter theorem was applied.
  • The approach may extend to other fixed-distance problems or to higher-dimensional analogues.
  • Small-n computational searches for maximum unit distances could be compared against the n to the 4/3 prediction to test consistency with the sketched obstruction.

Load-bearing premise

The partial methods sketched in the paper can be completed into a rigorous argument that rules out configurations attaining the full n to the 4/3 order for unit distances.

What would settle it

An explicit construction of n points determining Theta(n to the 4/3) unit distances, or a completed rigorous version of the sketched methods.

Figures

Figures reproduced from arXiv: 2605.26145 by Steven Senger.

Figure 1
Figure 1. Figure 1: There are three lunes pictured, each emanating from p. Two of these lunes have many points, while the middle lune has fewer points. Proposition 1.5. There are at least c6n typical points in P. Proof. By Proposition 1.4, we get that there are at least c2n points p ∈ P with at least c3n 1 3 points on C(p). Order these points and call them qj . Proposition 4.2, we know that most edges have ≈ n 2 3 crossings. … view at source ↗
Figure 2
Figure 2. Figure 2: The points q and r are at a unit distance from p, and hence lie on C(p). The points whose circles cross the edge between q and r are given restricted to the regions between C(q) and C(r), as indicated by the dotted lines. 2 Constructing the set 2.1 Reducing to a pair of squares We begin by following a pigeonholing scheme from Spencer, Szemer´edi, and Trotter in [7]. Suppose that there does exist a set of n… view at source ↗
Figure 3
Figure 3. Figure 3: The points q and r are at a unit distance from p, and hence lie on C(p). The length of the arc between q and r is at least ϵ, and the points all fit within a horizontal strip of height δ. 2.3 Summarizing Any of the ≳ n typical points p ∈ E ∩ A will essentially partition the other points of A into sets of ≈ n 2 3 points, with each set in the interior of a different lune emanating from p. Moreover, each of t… view at source ↗
Figure 4
Figure 4. Figure 4: Setting for Proposition 3.2. We next give an estimate for certain arcs of circles that will come up in the calculations to follow. To get a handle on the arcs in question, we consider certain types of curvilinear triangles. If we are know two arc lengths, we can estimate the third. Lemma 3.2. Suppose that each pair of the points a, b, and c is on a different unit circle, with the length of the arc from a t… view at source ↗
Figure 5
Figure 5. Figure 5: The point p ∈ P is on two lines from L. Each of these lines gives rise to a point in Lˆ, represented by q and q ′ , respectively. Moreover, those points are on the line ℓ(p) ∈ P . ˆ Finally, the point p1 is between the lines ℓ(q) and ℓ(q ′ ) because ℓ(p1) crosses ℓ(p) between q and q ′ . Notice that if p ∈ ℓ, then p(ℓ) ∈ ℓ(p). So we have that I(P, L) = I(L, ˆ Pˆ), and therefore (L, ˆ Pˆ) is also an ST-shar… view at source ↗
Figure 6
Figure 6. Figure 6: This is the same scenario as before, except with p1 · q ′ < 1 and p1 · q > 1. By running over all ≳ n 2 3 points pj for this pair (q, q′ ), we see that each pj must be between the lines ℓ(q) and ℓ(q ′ ). Finally, observing that this holds for ≳ n 1 3 consecutive pairs (q, q′ ) on ℓ(p) for all points p ∈ P ′ yields the desired result. References [1] N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. … view at source ↗
read the original abstract

This is an incomplete attempt to show that the upper bound of $\lesssim n^\frac{4}{3}$ on the number unit distances determined by a large finite set of $n$ points in the plane is not sharp. The methods also say something about sets of $n$ points and $n$ lines that attain the sharp bound of the Szemer\'edi-Trotter point-line incidence bound.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript presents an incomplete attempt to show that the known upper bound of ≲ n^{4/3} on the number of unit distances determined by a set of n points in the plane is not sharp. It also sketches methods that apply to point-line configurations attaining the sharp Szemerédi-Trotter incidence bound of Θ(n^{4/3}). No complete proof or derivation is supplied.

Significance. A completed proof that the n^{4/3} bound on unit distances is not tight would be a major result in combinatorial geometry, as it would improve the long-standing Szemerédi-Trotter-derived upper bound. The current manuscript, however, supplies only partial sketches and explicitly labels itself incomplete, so it establishes no new theorem and has no demonstrated significance beyond noting the possibility of further work.

major comments (2)
  1. [Abstract] Abstract and full text: the manuscript is explicitly described as an 'incomplete attempt' and supplies only partial sketches rather than a finished argument. The central claim that the ≲ n^{4/3} bound is not sharp therefore rests on no completed derivation whose correctness can be checked.
  2. [Full text] No equations, reductions, or explicit constructions are provided that would allow verification of whether the sketched methods can rule out configurations attaining the full n^{4/3} order (distinct from merely discussing Szemerédi-Trotter extremal examples).

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the report. We fully acknowledge that the manuscript is presented as an incomplete attempt consisting of partial sketches, with no completed proof or derivation, as stated in the title, abstract, and text. We do not claim to have established any new theorem or improved bound.

read point-by-point responses

  1. Referee: [Abstract] Abstract and full text: the manuscript is explicitly described as an 'incomplete attempt' and supplies only partial sketches rather than a finished argument. The central claim that the ≲ n^{4/3} bound is not sharp therefore rests on no completed derivation whose correctness can be checked.

    Authors: We agree with the referee's assessment. The manuscript is explicitly labeled incomplete and provides only sketches without a finished argument or verifiable derivation. No claim is made that the n^{4/3} bound has been shown to be non-sharp. revision: no

  2. Referee: [Full text] No equations, reductions, or explicit constructions are provided that would allow verification of whether the sketched methods can rule out configurations attaining the full n^{4/3} order (distinct from merely discussing Szemerédi-Trotter extremal examples).

    Authors: This observation is accurate and aligns with the incomplete status of the work. The manuscript contains no such equations, reductions, or constructions because it does not develop the ideas to a stage where verification is possible. revision: no

standing simulated objections not resolved
  • We cannot supply a completed proof, explicit constructions, or verifiable derivations, as none exist in the manuscript.

Circularity Check

0 steps flagged

No circularity detected; manuscript is incomplete with no derivation chain

full rationale

The paper is explicitly described as 'an incomplete attempt' providing only 'partial sketches' rather than a finished proof or equations. No load-bearing steps, self-citations, fitted predictions, or derivations are present in the supplied text (abstract and context). The central claim rests on unfinished methods, so no reduction to inputs by construction can be exhibited. This is the expected honest non-finding for an incomplete manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.1-grok · 5576 in / 1068 out tokens · 35362 ms · 2026-06-30T14:27:58.872381+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

15 extracted references · 3 canonical work pages · 3 internal anchors

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