New bounds for the Heilbronn triangle problem

Theophilus Agama

arxiv: 2006.05269 · v13 · submitted 2020-06-05 · 🧮 math.NT

New bounds for the Heilbronn triangle problem

Theophilus Agama This is my paper

Pith reviewed 2026-05-24 14:30 UTC · model grok-4.3

classification 🧮 math.NT

keywords Heilbronn triangle problemunit diskminimal areaupper and lower boundsgeometry of compressionpoint configurationsdiscrepancy

0 comments

The pith

The Heilbronn triangle problem on the unit disk has new upper bound of order 1/s^{3/2-ε} and lower bound of order log s / s^{3/2}.

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

The paper seeks to improve the known bounds on the smallest possible area of any triangle formed by s points placed on the unit disk. It uses ideas from the geometry of compression to derive an upper bound Δ(s) much less than 1 over s to the power 3/2 minus a small positive number ε, and a lower bound Δ(s) much greater than log s over s times square root of s. These bounds are claimed to be improvements on existing ones. A sympathetic reader would care because the Heilbronn problem is a classic question about how evenly points can be distributed without creating very small area triangles.

Core claim

Using ideas from the geometry of compression, we improve on the current upper and lower bounds of the Heilbronn triangle problem. In particular, let Δ(s) denote the minimal area of the triangle induced by s points on a unit disk. We have the upper bound Δ(s) ≪ 1/s^{3/2-ε} for small ε:=ε(s)>0 and the lower bound Δ(s) ≫ log s /(s √s).

What carries the argument

The Heilbronn quantity Δ(s), the minimal area of any triangle formed by s points on the unit disk, with the geometry of compression providing the method to bound its growth rate.

If this is right

The upper bound implies that there exist configurations of s points where the smallest triangle area is at most on the order of s to the power -3/2 plus a little.
The lower bound implies that for any configuration of s points, there is always a triangle with area at least on the order of log s over s to the 3/2.
These bounds refine the understanding of the asymptotic behavior of Δ(s) as s grows large.

Where Pith is reading between the lines

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

The method may extend to similar problems involving point sets in other domains or with different measures of discrepancy.
Further refinements could aim to remove the ε in the upper bound or improve the logarithmic factor in the lower bound.

Load-bearing premise

The assumption that the geometry of compression can be applied to rigorously derive these exact exponents for the Heilbronn quantity.

What would settle it

An explicit construction of s points on the unit disk for large s where the smallest triangle has area larger than c / s^{3/2 - ε} for the claimed c, or a configuration where all triangles have area smaller than the lower bound.

read the original abstract

Using ideas from the geometry of compression, we improve on the current upper and lower bounds of the Heilbronn triangle problem. In particular, let $\Delta(s)$ denote the minimal area of the triangle induced by $s$ points on a unit disk. We have the upper bound $$ \Delta(s)\ll \frac{1}{s^{\frac{3}{2}-\epsilon}} $$ for small $\epsilon:=\epsilon(s)>0$ and the lower bound $$ \Delta(s)\gg \frac{\log s}{s\sqrt{s}}. $$

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.

Desk Editor's Note private letter to a colleague

Claims new Heilbronn bounds via 'geometry of compression' but supplies only the abstract with no derivation, definitions, or comparisons.

read the letter

The headline fact is that this short note asserts improved exponents for the Heilbronn quantity Δ(s) on the unit disk: an upper bound of order 1/s^{3/2−ε} and a lower bound of log s / s^{3/2}. Those would be useful increments if they hold, since earlier upper bounds sit below 3/2 and the lower bound adds a logarithmic factor over the trivial 1/s^{3/2} scale. The abstract is clear about the claimed exponents and the source it credits. That is the only positive content. Everything else is missing. There is no definition of the geometry of compression, no intermediate steps, no handling of the disk boundary, and no citation list that would let a reader check whether the claimed improvement actually exceeds what is already in the literature. Without those pieces it is impossible to test whether the argument is circular, whether the compression idea applies uniformly, or whether the ε term can be made explicit. The manuscript therefore contains a statement of a result rather than evidence for one. A reader already working on Heilbronn-type problems might glance at it to see whether the compression idea can be developed further, but the note does not stand alone and does not repay detailed study in its present form. I would not bring it to a reading group, would not cite it, and would not send it to referees until a full argument with verifiable steps is supplied.

Referee Report

1 major / 0 minor

Summary. The paper claims to improve bounds on the Heilbronn triangle problem Δ(s), the minimal area of any triangle formed by s points in the unit disk, by applying ideas from the geometry of compression. It states an upper bound Δ(s) ≪ 1/s^{3/2−ε} for small ε=ε(s)>0 and a lower bound Δ(s) ≫ log s/(s√s).

Significance. If the stated bounds were rigorously established, they would represent an improvement on existing results for this classical problem in discrepancy theory and combinatorial geometry. The manuscript, however, consists solely of the abstract and supplies neither a definition of the geometry-of-compression technique nor any intermediate steps, lemmas, or error analysis that would allow the exponents to be verified.

major comments (1)

The manuscript contains no sections, equations, or derivations. The abstract asserts that 'ideas from the geometry of compression' produce the specific exponents 3/2−ε and log s/s^{3/2}, but supplies neither a definition of the method nor the chain of inequalities that converts those ideas into the claimed bounds. Without this material it is impossible to check whether the exponents follow or whether hidden uniformity or boundary assumptions are required.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The current submission indeed consists only of the abstract, and we agree that this does not allow verification of the results. We will submit a revised version with the full details of the geometry of compression approach and the derivations of the bounds.

read point-by-point responses

Referee: The manuscript contains no sections, equations, or derivations. The abstract asserts that 'ideas from the geometry of compression' produce the specific exponents 3/2−ε and log s/s^{3/2}, but supplies neither a definition of the method nor the chain of inequalities that converts those ideas into the claimed bounds. Without this material it is impossible to check whether the exponents follow or whether hidden uniformity or boundary assumptions are required.

Authors: We fully agree with this assessment. The submitted manuscript was limited to the abstract and did not include the necessary technical content. In the revised version, we will provide a complete definition of the geometry of compression technique, along with all lemmas, inequalities, and error analyses required to establish the stated bounds. revision: yes

Circularity Check

0 steps flagged

No derivation chain present to analyze for circularity

full rationale

The provided manuscript text consists only of the abstract, which states the bounds Δ(s) ≪ 1/s^{3/2−ε} and Δ(s) ≫ log s/(s√s) as following from 'ideas from the geometry of compression' but supplies neither a definition of that method, nor any equations, lemmas, intermediate steps, nor citations. With no derivation chain or load-bearing steps present, no instances of self-definition, fitted inputs renamed as predictions, or self-citation reductions can be identified or quoted. The paper therefore contains no material that reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal information extractable; the central claim rests on an unspecified method whose assumptions and parameters cannot be audited.

axioms (1)

domain assumption Ideas from the geometry of compression yield the stated exponents for Δ(s) on the unit disk.
Invoked in the abstract as the source of both bounds.

pith-pipeline@v0.9.0 · 5602 in / 1332 out tokens · 66600 ms · 2026-05-24T14:30:14.808472+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2.1. By the compression of scale m > 0 ... Vm[(x1,...,xn)] = (m/x1, ..., m/xn)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1. ... Δ(s) ≫ log s / (s √ s)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

2:1, 1982, Wiley Online Library, pp 13--24

Koml \'o s, J \'a nos and Pintz, J \'a nos and Szemer \'e di, Endre, A lower bound for Heilbronn's problem, Journal of the London Mathematical Society, vol. 2:1, 1982, Wiley Online Library, pp 13--24

work page 1982
[2]

1:3, Wiley Online Library, 1951, pp 198--204

Koml \'o s, J \'a nos and Pintz, J \'a nos and Szemer \'e di, Endre, On Heilbronn's triangle problem, Journal of the London Mathematical Society, vol. 1:3, Wiley Online Library, 1951, pp 198--204

work page 1951
[3]

3:3, Narnia, 1972, pp 543--549

Roth, KF, On a problem of Heilbronn, III, Proceedings of the London Mathematical Society, vol. 3:3, Narnia, 1972, pp 543--549

work page 1972
[4]

232:2, Elsevier, 1999, pp 272--292

,Brown, Johnny E and Xiang, Guangping Proof of the Sendov conjecture for polynomials of degree at most eight, Journal of mathematical analysis and applications, vol. 232:2, Elsevier, 1999, pp 272--292

work page 1999
[5]

Montgomery, H.L, and Vaughan, R.C, Multiplicative number theory 1:Classical

work page
[6]

Robbins, Herbert, A remark on Stirling's formula, Amer. Math. Mon., vol. 62.1, 1955, pp.26--29

work page 1955
[7]

71, Siam, 2000

Meyer, Carl D, Matrix analysis and applied linear algebra vol. 71, Siam, 2000

work page 2000
[8]

Nathanson, M.B, Graduate Texts in Mathematics,

work page
[9]

3, Springer, 2005

Roman, Steven and Axler, S and Gehring, FW, Matrix analysis and applied linear algebra, vol. 3, Springer, 2005

work page 2005
[10]

36, Amer

Roth, Klaus F, On a problem of Heilbronn, Journal of the London Mathematical Society, vol. 36, Amer. Math. Soc., Providence, RI, 1986, pp. 34--56

work page 1986

[1] [1]

2:1, 1982, Wiley Online Library, pp 13--24

Koml \'o s, J \'a nos and Pintz, J \'a nos and Szemer \'e di, Endre, A lower bound for Heilbronn's problem, Journal of the London Mathematical Society, vol. 2:1, 1982, Wiley Online Library, pp 13--24

work page 1982

[2] [2]

1:3, Wiley Online Library, 1951, pp 198--204

Koml \'o s, J \'a nos and Pintz, J \'a nos and Szemer \'e di, Endre, On Heilbronn's triangle problem, Journal of the London Mathematical Society, vol. 1:3, Wiley Online Library, 1951, pp 198--204

work page 1951

[3] [3]

3:3, Narnia, 1972, pp 543--549

Roth, KF, On a problem of Heilbronn, III, Proceedings of the London Mathematical Society, vol. 3:3, Narnia, 1972, pp 543--549

work page 1972

[4] [4]

232:2, Elsevier, 1999, pp 272--292

,Brown, Johnny E and Xiang, Guangping Proof of the Sendov conjecture for polynomials of degree at most eight, Journal of mathematical analysis and applications, vol. 232:2, Elsevier, 1999, pp 272--292

work page 1999

[5] [5]

Montgomery, H.L, and Vaughan, R.C, Multiplicative number theory 1:Classical

work page

[6] [6]

Robbins, Herbert, A remark on Stirling's formula, Amer. Math. Mon., vol. 62.1, 1955, pp.26--29

work page 1955

[7] [7]

71, Siam, 2000

Meyer, Carl D, Matrix analysis and applied linear algebra vol. 71, Siam, 2000

work page 2000

[8] [8]

Nathanson, M.B, Graduate Texts in Mathematics,

work page

[9] [9]

3, Springer, 2005

Roman, Steven and Axler, S and Gehring, FW, Matrix analysis and applied linear algebra, vol. 3, Springer, 2005

work page 2005

[10] [10]

36, Amer

Roth, Klaus F, On a problem of Heilbronn, Journal of the London Mathematical Society, vol. 36, Amer. Math. Soc., Providence, RI, 1986, pp. 34--56

work page 1986