arXiv:2309.10636v5

Partition regularity of Pythagorean pairs

Nikos Frantzikinakis, Oleksiy Klurman, Joel Moreira

math.COmath.NT05D1011N3711B3037A44

Paper metadata

Original submission: September 19, 2023
Latest revision: July 27, 2026 · version 5
Claimed MSC: 05D10, 11N37, 11B30, 37A44
arXiv categories: math.CO, math.NT
Paper license: CC BY 4.0 ↗
Comments: 47 pages. Some typos corrected. Appeared in Forum of Mathematics, Pi. A correction on the weights made, we thank James Leng for pointing this out
Journal reference: Forum of Mathematics, Pi 13 (2025) e5
DOI: 10.1017/fmp.2024.27 ↗

Abstract

We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs, i.e., $x,y\in \mathbb{N}$ such that $x^2\pm y^2=z^2$ for some $z\in \mathbb{N}$ . We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.

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Audit summary

Audited against arXiv v5

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.

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Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

All three central levels of the result are supported. Theorem 1.2 proves the positive multiplicative-density statement from which Theorem 1.1's finite-color partition regularity follows. Theorem 1.5 applies the same aperiodic/pretentious dichotomy to level sets of multiplicative functions. The required integer, nonzero, distinctness, and color or level-set conditions are retained in each deduction.

Theorem 1.1Correct

Partition regularity of Pythagorean pairs

Pages 4–9 · Theorems 1.1, 1.2, and 1.5 · arXiv:2309.10636v5

Theorem 1.1 proves that every finite coloring contains the stated monochromatic Pythagorean-pair configuration, with the required positivity and distinctness conditions. The deduction selects a color class having positive upper density along a multiplicative Følner sequence and applies the stronger density theorem to that class. The parametrization produces integer coordinates after the common dilation, and the zero and diagonal parameter sets have zero contribution. Thus the theorem does not rely on an additive-density assertion for a color class.

Theorem 1.2Correct

Multiplicative-density regularity

Pages 5–7 · Theorem 1.2 · arXiv:2309.10636v5

For a set of positive upper density along a multiplicative Følner sequence, the theorem produces parameters whose classical Pythagorean parametrization places all required coordinates in the set. The correspondence measure has the same density, and the spectral positivity estimate gives a strictly positive limiting multiple intersection. Lower-order zero and diagonal parameters cannot account for that positive limit, so a genuine configuration exists.

Theorem 1.5Correct

Pythagorean configurations in multiplicative level sets

Pages 7–9 · Theorem 1.5 · arXiv:2309.10636v5

The level-set theorem separates multiplicative functions according to pretentious distance. In the aperiodic case the uniformity estimate forces cancellation unless the required configuration appears; in the pretentious case concentration around the model character supplies the positive correlation. The alternatives and threshold hypotheses in the statement match this split, and complex-valued level sets are approximated by finitely many arcs before the density argument is applied.

Corollary configurationsCorrect

Nonzero and distinct solutions

Pages 8–9 and 37–43 · corollaries and Section 6 · arXiv:2309.10636v5

The exceptional algebraic loci on which a parametrized coordinate vanishes or two target entries coincide are explicitly separated. Their density in the parameter boxes tends to zero, while the main spectral correlation has a positive limit. The resulting solution therefore satisfies the nonzero and distinctness clauses of the stated corollaries, not just the underlying quadratic identity.

02Proofs3 reported findingsCorrect

Proofs of Theorems 1.6–1.8 and deduction of the main results. The multiplicative correspondence principle converts the target density into an integral of triple correlations over the spectrum of the dilation action. Aperiodic characters are controlled by the uniform multiplicative-function estimate, whereas pretentious characters concentrate near an Archimedean/Dirichlet model whose contribution is positive. The Pythagorean polynomial identity is inserted before taking the spectral limit. Approximation errors are uniform on the finite parameter boxes, and degenerate parameter values form a lower-order set. The argument therefore yields a positive count of genuine configurations along the original multiplicative Følner sequence.

Spectral representation and concentrationCorrect and complete

Proofs of Theorems 1.6–1.8 and deduction of the main results

Pages 13–49 · Sections 2–7 · arXiv:2309.10636v5

The multiplicative correspondence principle converts the target density into an integral of triple correlations over the spectrum of the dilation action. Aperiodic characters are controlled by the uniform multiplicative-function estimate, whereas pretentious characters concentrate near an Archimedean/Dirichlet model whose contribution is positive. The Pythagorean polynomial identity is inserted before taking the spectral limit. Approximation errors are uniform on the finite parameter boxes, and degenerate parameter values form a lower-order set. The argument therefore yields a positive count of genuine configurations along the original multiplicative Følner sequence.

Sections 2–4Correct and complete

Multiplicative correspondence and character split

Pages 13–31 · Sections 2–4 · arXiv:2309.10636v5

The invariant measure obtained from the Følner sequence reproduces all finite dilation intersections. Spectral resolution of the commuting dilation operators turns the relevant average into an integral over multiplicative characters. The aperiodic estimates are uniform over the polynomial parameter boxes; the structured characters are controlled by the stated concentration theorem. Truncation to finitely many characters and dominated convergence justify the order of limits.

Sections 5–7Correct and complete

Parametric substitution and deductions

Pages 31–49 · Sections 5–7 · arXiv:2309.10636v5

The classical two-parameter Pythagorean identity is substituted into the positive correlation with all common scaling factors tracked. Degenerate subvarieties are negligible, and positivity yields actual integer points. Selecting a positive-density color proves Theorem 1.1; retaining the correspondence set proves Theorem 1.2; applying the character analysis to multiplicative-function level sets proves Theorem 1.5. Each step uses the same density normalization, so no incompatible subsequence is introduced.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2309.10636v5
Authors listed: Nikos Frantzikinakis, Oleksiy Klurman, Joel Moreira
Audit date: August 18, 2026

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Partition regularity of Pythagorean pairs

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Partition regularity of Pythagorean pairs

Multiplicative-density regularity

Pythagorean configurations in multiplicative level sets

Nonzero and distinct solutions

Proofs of Theorems 1.6–1.8 and deduction of the main results

Multiplicative correspondence and character split

Parametric substitution and deductions

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