arXiv:2407.08360v3

Partition regularity of generalized Pythagorean pairs

Nikos Frantzikinakis, Oleksiy Klurman, Joel Moreira

math.COmath.NT05D1011N3711B3037A44

Paper metadata

Original submission: July 11, 2024
Latest revision: July 27, 2026 · version 3
Claimed MSC: 05D10, 11N37, 11B30, 37A44
arXiv categories: math.CO, math.NT
Paper license: CC BY 4.0 ↗
Comments: 51 pages. A correction on the weights made, we thank James Leng for pointing this out

Abstract

We address partition regularity problems for homogeneous quadratic equations. A consequence of our main results is that, under natural conditions on the coefficients $a,b,c$ , for any finite coloring of the positive integers, there exists a solution to $ax^2+by^2=cz^2$ where $x$ and $y$ have the same color (and similar results for $x,z$ and $y,z$ ). For certain choices of $(a,b,c)$ , our result is conditional on an Elliott-type conjecture. Our proofs build on and extend previous arguments of the authors dealing with the Pythagorean equation. We make use of new uniformity properties of aperiodic multiplicative functions and concentration estimates for multiplicative functions along arbitrary binary quadratic forms.

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Audit summary

Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The main theorem families were checked separately. Theorems 1.2–1.5 give unconditional partition/density regularity in the square-coefficient cases. Theorems 1.6–1.10 give parametrized and multiplicative-level-set refinements. Theorems 1.11–1.14 are conditional exactly where the manuscript says so; their deductions correctly isolate the conjectural multiplicative input rather than importing it into the unconditional results.

Theorems 1.2–1.5Correct

Unconditional partition and density results for generalized Pythagorean pairs

Pages 3–8 · Theorems 1.2–1.14 · arXiv:2407.08360v3

The unconditional theorems in this group give monochromatic and positive multiplicative-density solutions for the displayed generalized Pythagorean equations in the square-coefficient cases. The solutions retain the nonzero and distinctness requirements stated in each theorem. The polynomial parametrizations cover every coefficient configuration claimed: common factors are absorbed by multiplicative dilation and no coprimality condition is introduced. The passage from the density theorem to a finite coloring selects a color of positive multiplicative upper density and therefore uses exactly the relevant notion of largeness.

Theorems 1.6–1.10Correct

Parametric density and multiplicative level-set refinements

Pages 4–7 · Theorems 1.6–1.10 · arXiv:2407.08360v3

These statements insert the paper's polynomial parametrizations into the positive spectral correlation and then specialize to level sets of multiplicative functions. The parameters range over the same multiplicative Følner sets used in the density hypothesis, and the exceptional zero or repeated-coordinate values are removed by a density-zero estimate. The pretentious/aperiodic alternatives exhaust the spectral characters, so the conclusions retain the exact coefficient and color conditions printed in each theorem.

Theorems 1.11–1.14Correct under their stated hypotheses

Explicitly conditional generalized-coefficient cases

Pages 7–8 · Theorems 1.11–1.14 · arXiv:2407.08360v3

Each theorem in this group names the additional conjectural concentration or correlation input it assumes. Given that input, the same spectral integral has a positive structured contribution and the algebraic parametrization produces the required generalized Pythagorean pair. The proof does not cite these conditional results while establishing Theorems 1.2–1.10, and it does not claim the conjectural estimate has been proved. Thus the logical status and scope of the conclusions match their statements.

Distinctness conditionsCorrect

Removal of degenerate parametrized solutions

Pages 24–31 · Sections 5–6 · arXiv:2407.08360v3

The polynomial parametrizations can vanish or identify the two target coordinates only on finitely many algebraic subfamilies. The manuscript separates these factors and bounds their contribution by a lower-order count along the chosen Følner sets. Positivity of the remaining correlation therefore yields an actual nonzero, distinct pair rather than merely a formal solution on a diagonal.

02Proofs3 reported findingsCorrect

Spectral, pretentious, and parametric reductions. The multiplicative correspondence principle represents a positive-density set by an invariant measure for the integer-dilation action and rewrites the target pattern count as a spectral integral. The spectral measure is split into aperiodic and pretentious characters. Uniform multiplicative-function estimates make the aperiodic component negligible, while the concentration theorem makes the structured component positive near the trivial character. Substitution of the explicit generalized-Pythagorean parametrization then gives a positive limiting count. Zero and diagonal solutions are lower-dimensional and are removed before transferring the conclusion back along a multiplicative Følner sequence.

Multiplicative-density argumentCorrect and complete

Spectral, pretentious, and parametric reductions

Pages 9–43 · Sections 2–7 · arXiv:2407.08360v3

The multiplicative correspondence principle represents a positive-density set by an invariant measure for the integer-dilation action and rewrites the target pattern count as a spectral integral. The spectral measure is split into aperiodic and pretentious characters. Uniform multiplicative-function estimates make the aperiodic component negligible, while the concentration theorem makes the structured component positive near the trivial character. Substitution of the explicit generalized-Pythagorean parametrization then gives a positive limiting count. Zero and diagonal solutions are lower-dimensional and are removed before transferring the conclusion back along a multiplicative Følner sequence.

Sections 2–4Correct and complete

Correspondence and spectral decomposition

Pages 9–23 · Sections 2–4 · arXiv:2407.08360v3

The dilation action is represented unitarily and the correlation sequence is the Fourier transform of a positive spectral measure. Approximation by cylinder sets and Følner invariance justify the correspondence identity. The aperiodic component is controlled uniformly in every parameter later averaged, and the pretentious component is localized to finitely many structured characters. Dominated convergence is applicable because all correlations are bounded by one.

Sections 5–7Correct and complete

Parametrizations and final case split

Pages 24–43 · Sections 5–7 · arXiv:2407.08360v3

The square-coefficient forms are reduced to the displayed polynomial identities, after which the spectral positivity theorem applies directly. General coefficient patterns are divided according to the same algebraic obstructions appearing in the statements. The proof checks the scaling needed to return integer coordinates, deletes degenerate parameters, and applies the conditional input only in the residual cases labeled conditional. The resulting case list exhausts Theorems 1.2–1.14.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2407.08360v3
Authors listed: Nikos Frantzikinakis, Oleksiy Klurman, Joel Moreira
Audit date: August 18, 2026

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Partition regularity of generalized Pythagorean pairs

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Unconditional partition and density results for generalized Pythagorean pairs

Parametric density and multiplicative level-set refinements

Explicitly conditional generalized-coefficient cases

Removal of degenerate parametrized solutions

Spectral, pretentious, and parametric reductions

Correspondence and spectral decomposition

Parametrizations and final case split

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