arXiv:2609.30410v2

Density regularity of $\{x,x+y,xy\}$ in the integers

math.CO05D1037A4428D05

Paper metadata

Original submission: September 24, 2026
Latest revision: October 7, 2026 · version 2
Claimed MSC: 05D10, 37A44, 28D05
arXiv categories: math.CO
Paper license: CC BY 4.0 ↗
Comments: 24 pages, This version corrects a mistake appearing in Proposition 2.3 of the previous version

Abstract

Fix $k,s\in\mathbb{N}$ . We prove that there exists a subadditive density on $\mathbb{N}$ such that, for every polynomial $P\in\mathbb{Z}[y]$ of degree $k$ and with $P(0)=0$ , every set of positive density contains configurations $\{x,x+P(y),xy^s\}$ for arbitrarily large $x>y\geq 2$ . This provides a density strengthening of Moreira's result on partition regularity for $\{x,x+y,xy\}$ .

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Audited against arXiv v2

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.

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Generated October 24, 2026

01Statements2 reported findingsNo statement issues found

The constructed subadditive density supports the polynomial sum-product configurations claimed in the current version. The familiar partition-regularity consequences are explicitly identified as consequences of known results, not separate new claims.

Theorem 1.1; Definitions 1.5–1.7No issue found

The localized invariant-mean density has the required positivity and subadditivity

PDF pp. 2–5, 14–17 · arXiv:2609.30410v2

Polynomial Bohr neighborhoods have a positive uniform interval density, so conditioning a shift-invariant mean on them is defined. Directed limsup preserves monotonicity, normalization and the subadditive bound. Ergodicity of the inner mean passes to the separable affine algebra; its Gelfand system and natural extension retain all finite correlation identities. Positive recurrence therefore gives $x=a^sn$ , $y=m/a$ , and $xy^s=m^sn$ in the same set.

Proposition 2.3; Theorem 3.2No issue found

The corrected polynomial characteristic-factor reduction handles both coefficient regimes

PDF pp. 8–10, 15, 20–22 · arXiv:2609.30410v2

For linearly independent polynomial iterates, adjoining rotations absorbs the polynomial phase and weak mixing gives orthogonality to every rational eigenfactor. For dependent iterates, the family $r_1P,r_2P,m,\ldots,m^{k-1}$ has algebraic complexity at most one: the coefficient of $x$ removes the lower-degree quadratic coefficients, and the coefficient of degree $2k$ then forces the two remaining ones to vanish because $r_1r_2(r_1-r_2)\ne0$ . The corresponding characteristic-factor theorem applies to the ergodic extension.

02Proofs2 reported findingsNo proof issues found

The current proof closes the compact/uniform decomposition, twisted polynomial averages, and the final fourth-power lower bound. The order of limits is consistent with the divisibility restrictions.

Lemma 2.7; Proposition 3.1No issue found in the reviewed proof

Ergodicity survives the algebra restriction and correspondence construction

PDF pp. 12–16 · arXiv:2609.30410v2

A proposed nontrivial decomposition of the restricted mean can be extended by a positive functional dominated by the original mean. Averaging its shifts preserves domination and the restriction, contradicting extremality after normalization. The Gelfand representation then identifies extremal invariant means with ergodic invariant probability measures; the natural extension supplies the negative polynomial iterates without changing the correlations.

Proposition 4.1No issue found in the reviewed proof

The weak-mixing contribution vanishes and the structured part gives a positive lower bound

PDF pp. 18–22 · arXiv:2609.30410v2

Bohr restrictions simultaneously enforce divisibility and near-invariance of finitely many compact components. Cauchy–Schwarz eliminates the moving bounded function; after the negligible diagonal is removed, distinct indices give polynomial iterates with distinct leading coefficients, permitting Proposition 2.3. Approximation of the Bohr indicator by polynomial phases is legitimate because its boundary has zero limiting measure. Ramsey stabilization and a second Cauchy–Schwarz step compare the pair average with the square of the single average. Orthogonality and conditional expectation then give the lower bound $\delta^4$ with the stated adjustable error.

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Paper: arXiv:2609.30410v2
Authors listed: Florian K. Richter, Konstantinos Tsinas
Audit date: October 24, 2026

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Density regularity of {x,x+y,xy}\{x,x+y,xy\}{x,x+y,xy} in the integers

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The localized invariant-mean density has the required positivity and subadditivity

The corrected polynomial characteristic-factor reduction handles both coefficient regimes

Ergodicity survives the algebra restriction and correspondence construction

The weak-mixing contribution vanishes and the structured part gives a positive lower bound

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Density regularity of $\{x,x+y,xy\}$ in the integers