arXiv:2508.09778v3

Recurrence for pretentious systems along generalized Pythagorean triples

math.DSmath.COmath.NT37A4405D1011N37

Paper metadata

Original submission: August 13, 2025
Latest revision: October 11, 2026 · version 3
Claimed MSC: 37A44, 05D10, 11N37
arXiv categories: math.DS, math.CO, math.NT
Paper license: CC BY 4.0 ↗
Comments: 44 pages. Revised in response to the referee's comments, with additional changes necessitated by a modification of the weight sequences

Abstract

We establish multiple recurrence results for pretentious measure-preserving multiplicative actions along generalized Pythagorean triples, that is, solutions to the equation $ax^2 + b y^2 = c z^2$ . This confirms the ergodic-theoretic form of the generalized Pythagorean partition regularity conjecture in this critical case of structured measure-preserving actions. As a consequence of our main theorem, any finite coloring of $\mathbb{N}$ generated by the level sets of finitely many pretentious completely multiplicative functions, must contain a monochromatic generalized Pythagorean triple.

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Audited against arXiv v3

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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 October 24, 2026

01Statements3 reported findingsNo statement issues found

The recurrence theorem for every Rado triple, its quantitative lower bounds in the separate parametrization cases, and the multiplicative-function corollary are supported with the stated hypotheses. The result concerns pretentious multiplicative actions and does not claim the full partition-regularity conjecture.

Theorem 1.1No issue found

All three Rado cases are covered by positive-density parameter families

PDF pp. 3, 17–42 · Theorems 4.1 and 5.1 · arXiv:2508.09778v3

When $a=c$ (and symmetrically $b=c$ ), the parametrization in (4.1) reduces the claim to Theorem 4.1; the reducible and irreducible cases for the second quadratic form are both treated. When $a+b=c$ , (5.1) gives the corresponding three forms and Sections 5.1–5.3 cover all reducibility combinations. Each averaged lower bound is transferred from a fixed affine grid to a positive-upper-density set of original parameters. Polynomial equalities that could make two of $x,y,z$ coincide occupy only zero-density subsets, so a pairwise-distinct triple can be selected.

Remark 1; Theorems 4.1 and 5.1No issue found

The quantitative exponents and their conditional improvement match the proofs

PDF pp. 3, 17–18, and 30–31 · arXiv:2508.09778v3

The $a=c$ or $b=c$ argument ends with three conditional-expectation factors and gives the cubic lower bound. In the general $a+b=c$ irreducible case, three separate characteristic-factor projections plus the original bounded function give the fourth-power bound after normalization. Reducibility, or equality of two root-count functions at all but finitely many primes, permits a common factor reduction and restores the cubic bound exactly under the alternatives stated in Theorem 5.1.

Corollary 1.2No issue found

The rotation argument produces simultaneous values in the requested arc

PDF pp. 3–4 · arXiv:2508.09778v3

The product rotation has spectral support on finite products of the given pretentious functions and hence is a pretentious action. Recurrence supplies three values lying in common small arc-translates. A power of the first point is placed near the identity by a finite torus pigeonhole argument; homogeneous rescaling preserves the equation and distinctness, while the three-arc containment places every function value in the prescribed arc.

02Proofs4 reported findingsNo proof issues found

The exact current-version proofs coherently combine congruence-selected grids, concentration estimates, spectral factor reductions, and conditional-expectation inequalities. The auxiliary averaging introduced in the most difficult irreducible case controls the previously uncancelled oscillatory factors.

Proposition 2.4; Lemmas 4.2 and 4.5No issue found in the reviewed proof

The characteristic factors and congruence grids are valid

PDF pp. 10–13, 18–20, and 27 · arXiv:2508.09778v3

The factor descriptions follow in both directions from multiplicative Følner invariance and the relevant prime-supported averages. In the grid constructions, Hensel lifting fixes exact prime-power valuations and the Chinese remainder conditions lie on disjoint prime ranges. The exclusions on the primes prevent simultaneous roots of independent forms, so the asserted gcd identities and residue conditions all follow.

Propositions 4.3 and 4.6No issue found in the reviewed proof

Spectral decompositions close both cases for equal coefficients

PDF pp. 20–30 · arXiv:2508.09778v3

The independence lemma first separates the three multiplicative averaging parameters. Orthogonal components vanish because a prime dilation on the appropriate Følner set has a nontrivial eigenvalue, while the retained components concentrate to $T_{1/q}F_1$ and $F_2$ . The ordered limits remove the Archimedean error, the mean ergodic theorem identifies the invariant projection, and the conditional-expectation inequality yields the cubic lower bound. The irreducible variant uses the intersection of precisely the two needed prime-support factors.

Proposition 5.3; equations (5.24)–(5.32)No issue found in the reviewed proof

The auxiliary linear average correctly handles three irreducible forms

PDF pp. 33–38 · arXiv:2508.09778v3

After projecting separately to the three form-dependent factors, each polynomial iterate is compared in averaged $L^2$ norm with the same transformation involving an auxiliary linear variable. The large-prime divisor error for $m^2$ has mean $O_\delta(L^{-1})$ , the pretentious tail tends to zero, and the cutoff set makes the logarithmic ratios of $m^2$ and all three forms $O(\delta)$ . Invariance then removes the common transformation and Proposition 2.8 supplies the fourth-power bound.

Equations (5.33)–(5.38); Propositions 5.6 and 5.8No issue found in the reviewed proof

The special cubic cases use exactly the additional form relation available

PDF pp. 38–42 · arXiv:2508.09778v3

Equality of two root-count functions makes a single finite-support condition control both corresponding prime averages, allowing their quotient to concentrate at the identity. If a form is reducible, its two linear factors give the same reduction through the linear concentration estimate. The remaining cases differ only by these explicitly listed factor substitutions, and the preceding full arguments apply without changing the ordered-limit or positivity steps.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2508.09778v3
Authors listed: Nikos Frantzikinakis, Andreas Mountakis
Audit date: October 24, 2026

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Recurrence for pretentious systems along generalized Pythagorean triples

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

All three Rado cases are covered by positive-density parameter families

The quantitative exponents and their conditional improvement match the proofs

The rotation argument produces simultaneous values in the requested arc

The characteristic factors and congruence grids are valid

Spectral decompositions close both cases for equal coefficients

The auxiliary linear average correctly handles three irreducible forms

The special cubic cases use exactly the additional form relation available

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