Congruent copies of finite patterns in planar point sets

Ritesh Goenka; Shubhrajit Bhattacharya

Planar n-point sets contain Ω(n^{1+δ}) congruent copies of any fixed finite pattern S, with δ>0 depending only on S.

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2026-06-26 03:24 UTC pith:X4T54IGD

load-bearing objection The paper adapts Sawin's number-field method to claim Ω(n^{1+δ_S}) congruent copies of any fixed S, answering Brass-Pach strongly, but the adaptation from single distances to full rigid copies needs verification. the 1 major comments →

arxiv 2606.27352 v1 pith:X4T54IGD submitted 2026-06-25 math.CO math.MG

Congruent copies of finite patterns in planar point sets

Shubhrajit Bhattacharya , Ritesh Goenka This is my paper

classification math.CO math.MG

keywords congruent copiesplanar point setscombinatorial geometryunit distance problemnumber field constructionBrass-Pach questionErdős-Purdy problem

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper shows that for any fixed finite nonempty set S of points in the plane, there exist n-point sets containing superlinearly many congruent copies of S. The lower bound is Ω_S(n^{1+δ_S}) where the positive exponent δ_S depends only on S. This gives a strong affirmative answer to a question of Brass and Pach on the maximum number of such copies possible. A sympathetic reader would care because the result connects the frequency of repeated geometric patterns to recent quantitative progress on the unit distance problem, and supplies explicit exponents.

Core claim

Given a finite nonempty planar point set S, the maximum number of congruent copies of S contained in a set of n points in the Euclidean plane is at least Ω_S(n^{1+δ_S}) for some positive constant δ_S depending only on S. The construction adapts the number field method from Sawin's quantitative refinement of the unit distance result and therefore supplies an explicit choice of δ_S for each fixed S.

What carries the argument

The number field construction from Sawin's quantitative refinement of the unit-distance result, adapted to generate many congruent copies of an arbitrary fixed S.

Load-bearing premise

The number field construction from Sawin's quantitative refinement of OpenAI's unit-distance result can be adapted to produce many congruent copies of an arbitrary fixed finite point set S.

What would settle it

An explicit S for which the adapted construction yields only O(n) copies, or a proof that no such adaptation exists for some S.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The Brass-Pach question receives a strong positive answer.
Progress is made on questions of Erdős-Purdy and Ábrego-Fernández-Merchant.
An explicit positive δ_S is obtained for every fixed S.
The number field method extends from unit distances to arbitrary finite patterns.

Where Pith is reading between the lines

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

The same adaptation technique may apply to counting copies under other isometries or similarities.
Further improvements to the underlying unit-distance exponent would immediately improve the δ_S obtained here.
The construction might be checked computationally for small S to verify the predicted growth rate.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The paper adapts Sawin's number-field method to claim Ω(n^{1+δ_S}) congruent copies of any fixed S, answering Brass-Pach strongly, but the adaptation from single distances to full rigid copies needs verification.

read the letter

The main thing here is that they take Sawin's number-field construction for many unit distances and adapt it to produce many congruent copies of an arbitrary fixed finite point set S. They claim this gives an explicit positive δ_S such that there are Ω(n^{1+δ_S}) copies in some n-point set, which would answer the Brass-Pach question strongly.

This is new because previous constructions were limited or didn't give positive exponents for general S. The paper does well by making the connection to the OpenAI/Sawin breakthrough explicit and by noting the implications for the other cited questions in combinatorial geometry.

The potential issue is whether the adaptation works without losing the exponent. Adding the extra distance and orientation conditions for |S| greater than 2 could in principle force the point set into a lower-dimensional variety where the count falls back to O(n). The abstract says the construction yields explicit δ_S, so presumably the full argument controls the degrees and shows the bound holds. But since the details are summarized, a referee would need to check that step carefully.

The math looks grounded in algebraic geometry over number fields, and the citations seem on point without circularity.

This paper is for people in discrete geometry who care about extremal problems on point sets. A reader interested in the unit distance problem or Erdős-type questions would get value from seeing how the method extends.

It deserves a serious referee because the claim is sharp and directly addresses open questions, even if the adaptation needs verification.

I recommend sending it to peer review.

Referee Report

1 major / 0 minor

Summary. The paper constructs, for any fixed finite nonempty planar point set S, an n-point set in the Euclidean plane containing Ω_S(n^{1+δ_S}) congruent copies of S, where δ_S > 0 is an explicit constant depending only on S. The construction adapts the number-field method from Sawin's quantitative refinement of OpenAI's unit-distance result and thereby answers a question of Brass and Pach in a strong quantitative form while making progress on related problems of Erdős-Purdy and Ábrego-Fernández-Merchant.

Significance. If the adaptation is valid, the result supplies the first superlinear lower bounds on the maximum number of congruent copies of an arbitrary fixed pattern, with an explicit positive exponent δ_S for each S. The explicitness of δ_S and the direct use of an external breakthrough (rather than ad-hoc parameter tuning) are strengths of the approach.

major comments (1)

[main proof (following the citation to Sawin)] The central claim rests on adapting Sawin's number-field point set (originally for many solutions to a single distance equation) to the full system of polynomial equations defining rigid congruent copies of a general S with |S| ≥ 3. The manuscript asserts that the dimension and degree of the resulting variety remain controlled so that the point count stays Ω(n^{1+δ_S}), but provides only a summary of this adaptation; without an explicit verification that the additional constraints do not force a drop to O(n) solutions, the quantitative bound cannot be independently checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for the positive assessment of the significance of the result. We address the single major comment below.

read point-by-point responses

Referee: The central claim rests on adapting Sawin's number-field point set (originally for many solutions to a single distance equation) to the full system of polynomial equations defining rigid congruent copies of a general S with |S| ≥ 3. The manuscript asserts that the dimension and degree of the resulting variety remain controlled so that the point count stays Ω(n^{1+δ_S}), but provides only a summary of this adaptation; without an explicit verification that the additional constraints do not force a drop to O(n) solutions, the quantitative bound cannot be independently checked.

Authors: We agree that the manuscript currently summarizes the adaptation and that an expanded verification would allow independent checking of the bound. The additional equations for a fixed S are of bounded degree depending only on |S| and are imposed over the same number-field extension as in Sawin's construction; the dimension of the resulting variety is therefore unchanged up to an additive constant (corresponding to the rigid motions), preserving a positive exponent δ_S. To make this fully explicit, we will add a short subsection or lemma in the revised version that records the precise dimension and degree bounds in terms of |S| and confirms that the Ω(n^{1+δ_S}) count is retained. This addresses the referee's concern directly. revision: yes

Circularity Check

0 steps flagged

No circularity; construction adapts independent external result

full rationale

The paper's derivation consists of adapting the number-field point-set construction from Sawin's quantitative refinement of the OpenAI unit-distance result to the setting of congruent copies of a fixed finite S. The abstract explicitly states that the proof 'uses the number field construction from Sawin's quantitative refinement of OpenAI's result' and thereby obtains an explicit δ_S. This is a direct extension of a cited external theorem rather than any self-definitional loop, fitted-input prediction, or load-bearing self-citation. No equation or claim inside the paper reduces to its own inputs by construction; the quantitative gain is inherited from the cited work. The derivation is therefore self-contained against the external benchmark and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The existence claim depends on the validity and adaptability of the external unit-distance number-field construction; no free parameters or new entities are introduced in the abstract itself.

axioms (1)

domain assumption Sawin's quantitative refinement of OpenAI's unit-distance lower bound holds and transfers to congruent copies of arbitrary S
The proof is described as using the number-field construction from that result.

pith-pipeline@v0.9.1-grok · 5670 in / 1127 out tokens · 51596 ms · 2026-06-26T03:24:09.694889+00:00 · methodology

0 comments

read the original abstract

Given a finite nonempty planar point set $S$, what is the maximum number of congruent copies of $S$ contained in a set of $n$ points in the Euclidean plane? Building on OpenAI's recent breakthrough on the unit distance problem, we construct planar sets consisting of $n$ points that contain $\Omega_S(n^{1+\delta_S})$ congruent copies of $S$, for some positive constant $\delta_S$ depending only on $S$. This answers a question of Brass and Pach in a strong form, and makes progress on questions posed by Erd\H{o}s and Purdy, and \'Abrego and Fern\'andez-Merchant. Our proof uses the number field construction from Sawin's quantitative refinement of OpenAI's result and consequently yields an explicit choice for $\delta_S$ for each fixed $S$.

discussion (0)

Reference graph

Works this paper leans on

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work page doi:10.1007/pl00009486 2000

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