arXiv:2609.20522v1

Infinite prime sumsets in structured and $U^k(Φ)$ -uniform sets

math.DSmath.COmath.NT37A0537A4411B3011B1311P32

Paper metadata

Original submission: September 17, 2026
Claimed MSC: 37A05, 37A44, 11B30, 11B13, 11P32
arXiv categories: math.DS, math.CO, math.NT
Comments: Comments are welcome!

Abstract

By introducing new ergodic-theoretic techniques in nilsystems, we determine which infinite sumset configurations occur in $U^k(Φ)$ -uniform and Nil-Bohr sets. To be more precise, our first result associates the degree $k$ of a $U^k(Φ)$ -uniform set with the variety of sumsets it contains, solving a conjecture of Kra, Moreira, Richter and Robertson. Restricting to Nil-Bohr sets we show the existence of infinite sumsets with summands in the shifted primes $\mathbb{P}-1$ . As a consequence, we show that for any real polynomial $Q(n)$ with leading irrational coefficient of degree $k$ , and any natural numbers $\ell_1, \cdots, \ell_k$ there is an infinite set $P\subset \mathbb{P}$ such that $Q\Big(\sum_{p \in I} p\Big) \in U \pmod 1 \quad \text{ for all } I \subset P , |I| = \ell_1, \ldots, \ell_k.$

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Audited against arXiv v1

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 September 19, 2026

01Statements6 reported findingsSuspected unsupported statements

The arbitrary-level sumset theorem for $U^k(\Phi)$ -uniform sets, the shifted-prime Nil-Bohr theorem, the polynomial prime-sumset theorem, and the principal nilsystem results are supported. Theorem 7.5, however, is not verified in its stated generality: its proof does not construct a single nilsystem model that simultaneously preserves the required uniformity, and it applies a positivity identity for the full Hall-Petresco fiber measure to one selected connected component without justification.

Theorem 7.5Not independently verified

Prime sumsets in sets that are simultaneously basic Nil-Bohr and $U^k(\Phi)$ -uniform

PDF pages 35-36 · Theorem 7.5, Equation (35), and its proof · arXiv:2609.20522v1

The theorem claims that for $k\geq2$ , sets $A_1,\ldots,A_k$ that are each both basic Nil-Bohr and $U^k(\Phi)$ -uniform, and distinct natural numbers $\ell_1,\ldots,\ell_k$ , admit one infinite set of primes whose $\ell_i$ -term sums all lie in $A_i$ . The proof's common-nilsystem claim and its connected-component positivity claim are not established by the cited results; both defects are detailed in the Proofs panel. No complete repair or counterexample was found, so the theorem is treated as unsupported rather than false.

Exact manuscript, arXiv:2609.20522v1 ↗

Theorem 1.1No issue found

Polynomial prime sumsets

PDF page 2 and page 36 · Theorem 1.1 and its proof · arXiv:2609.20522v1

The printed proof invokes the unsupported Theorem 7.5, but the statement has a specialized independent repair. Use the standard $k$ -step affine skew-shift model for $Q$ and an interval cylinder $E$ whose conditional Haar measure over $Z_{k-1}$ is constant. For each prescribed $\ell_i$ , choose $z_i\in T^{-\ell_i}E$ in the $Z_{k-1}$ -fiber of the base point and apply Proposition 6.5; its shifted-prime set $B$ gives the required prime set $P=B+1$ . For degree one, choose a sufficiently small interval $V$ with $\ell_1V+\beta\subset U$ and apply Vinogradov equidistribution along the primes. This proves the exact stated conclusion without Theorem 7.5.

Exact manuscript, Proposition 6.5 ↗

Theorem 1.2No issue found

Arbitrary prescribed sumset levels in $U^k(\Phi)$ -uniform sets

PDF page 2 and page 34 · Theorem 1.2, Equation (2), and its proof · arXiv:2609.20522v1

Proposition 2.4 gives a single ergodic Furstenberg model with clopen representatives preserving all $U^k$ -uniformity hypotheses. The progressive Hall-Petresco measure and its good-projection identity place the sets in the specified coordinates $\ell_i$ and reduce the conditional expectations to their positive means. Lemma 7.1 then supplies one infinite set $B$ with every required sumset level and no shift.

Theorem 1.3No issue found

Shifted-prime finite-sum levels in Nil-Bohr sets

PDF page 3 and page 35 · Theorem 1.3, Equation (3), and its proof · arXiv:2609.20522v1

The basic Nil-Bohr subset is represented in an ergodic nilsystem. Multiple recurrence supplies a translate with positive mass in the connected Hall-Petresco component, and Theorem 5.13 plus the shifted-prime progressive lemma produces an infinite set in $\mathbb P-1$ . Restoring the translate gives all sum sizes from one through $k$ with the stated shift.

Theorems 1.4 and 6.6No issue found

Dynamical characterizations by finite sums and shifted-prime sums

PDF page 4 and pages 32-33 · Theorems 1.4 and 6.6 · arXiv:2609.20522v1

Proposition 6.5 proves the strongest shifted-prime implication for nilsystems. The finite and unrestricted implications follow immediately, while the established Host-Kra-Maass cube-completion criterion gives the converse. Passing through a nilsystem stage of a pronilsystem inverse limit preserves the return-time inclusion and extends the equivalence with the same neighborhood quantifiers.

Theorems 3.10, 4.15, and 5.13No issue found

Fiberwise decomposition and progressive Hall-Petresco measures

PDF pages 14-15, 20-21, and 29-30 · Theorems 3.10, 4.15, and 5.13 · arXiv:2609.20522v1

The fiberwise continuous ergodic decomposition follows from the orbit-closure description and disintegration on nilmanifolds. Its support and genericity consequences prove progressivity for every Hall-Petresco fiber. The shifted-prime upgrade combines the connected-component decomposition with the von Mangoldt Gowers-uniformity estimate and the $W$ -trick; the limiting and uniformity steps preserve the component and all open-set quantifiers.

Green-Tao-Ziegler input, arXiv:2107.02158 ↗

02Proofs6 reported findingsSuspected incomplete or incorrect proofs

Theorem 7.5 has two substantive gaps: neither the simultaneous nilsystem and uniform-set representation nor positivity in the distinguished connected Hall-Petresco component follows from the cited results. The printed proof of Theorem 1.1 inherits that gap, although a specialized affine-nilsystem argument repairs Theorem 1.1 itself. The remaining central proof chain closes; one harmless inequality-direction typo is recorded.

Proof of Theorem 7.5, model constructionSuspected incomplete as written · no repair proposed

The common nilsystem representation is not established

PDF page 35 · sentence beginning ‘Following steps similar to the proof of Theorem 1.3 and using Remark 7.4’ · arXiv:2609.20522v1

Remark 7.4 realizes finitely many basic Nil-Bohr return sets in one minimal nilsystem. Proposition 2.4 instead gives a generally non-nil Furstenberg system with $U^k$ -uniform clopen representatives. No result identifies these two models or proves that the open return sets in the nilsystem model remain $U^k(X,\mu,T)$ -uniform. Sequence uniformity does not automatically transfer through an arbitrary open-set coding, especially when its boundary may have positive measure. This missing construction is essential to the next conditional-expectation step.

Exact manuscript, arXiv:2609.20522v1 ↗

Proof of Theorem 7.5, component positivitySuspected incomplete as written · no repair proposed

The $\rho_a^\circ$ positivity identity does not follow from Proposition 4.10

PDF pages 35-36 · displayed positivity assertion before the construction of $B$ · arXiv:2609.20522v1

Proposition 4.10 proves the good-projection identity for $\rho_a$ , not for the selected connected-component measure $\rho_a^\circ$ . The decomposition of $\rho_a$ as the average of translates of $\rho_a^\circ$ yields only average positivity and does not place positive mass in the component through $a$ . The relevant conditional-expectation identity is only almost everywhere and therefore does not settle that distinguished fiber. In addition, for arbitrary distinct $\ell_i$ , the event must put $E_i$ in coordinates $\ell_i$ of $HP_\ell(X)$ , where $\ell=\max_i\ell_i$ , with $X$ in unused coordinates; the printed consecutive-coordinate product handles only the special consecutive-level case.

Proof of Theorem 1.1Suspected incomplete as written · an apparently sound repair

The printed reduction inherits Theorem 7.5's gap

PDF page 36 · first paragraph of the proof of Theorem 1.1 · arXiv:2609.20522v1

The manuscript invokes Theorem 7.5 directly, so its printed proof does not close. A complete specialized repair uses the affine skew-shift model of the polynomial and Proposition 6.5, as detailed in the Statements panel: constant conditional fiber measure provides the required points over the base factor, shifted-prime progressivity gives $B\subset\mathbb P-1$ , and $P=B+1$ produces the prescribed prime sumsets. Thus only the printed proof, not Theorem 1.1, is adverse.

Theorems 1.2 and 1.3No issue found in the reviewed proof

The Furstenberg-model and Nil-Bohr arguments close

PDF pages 34-35 · proofs of Theorems 1.2 and 1.3 · arXiv:2609.20522v1

For Theorem 1.2, the clopen Furstenberg representatives, good-projection identity, nested conditional expectations, and progressive-measure lemma match all selected coordinates. For Theorem 1.3, the connected-component translate has positive multiple-recurrence mass and shifted-prime progressivity supplies the required infinite set. Neither proof uses the unsupported inference made later in Theorem 7.5.

Theorems 3.10, 4.15, 5.13, and 6.6No issue found in the reviewed proof

The central dynamical proof chain is complete

PDF pages 14-15, 20-21, 29-30, and 32-33 · central dynamical theorems · arXiv:2609.20522v1

The orbit-closure and disintegration arguments give the fiberwise decomposition, the continuous positivity argument proves progressivity for every fiber, the Gowers-uniform von Mangoldt comparison gives shifted-prime progressivity on connected components, and Proposition 6.5 plus inverse-limit reduction proves the dynamical equivalences. All support, component, and neighborhood transitions needed by these theorems are explicit.

Proof of Theorem 4.15Suspected typo

The final continuity sentence has the inequality direction reversed

PDF page 21 · final paragraph after Equation (22) in the proof of Theorem 4.15 · arXiv:2609.20522v1

The phrase ‘bounded by $\delta$ ’ should read ‘bounded below by $\delta$ ’ or ‘at least $\delta$ ’. Equation (21) explicitly gives the lower bound and the density and continuity argument uses that same direction, so the correction is uniquely determined and does not affect the proof.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2609.20522v1
Authors listed: Felipe Hernández, Tristán Radić
Audit date: September 19, 2026

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Infinite prime sumsets in structured and Uk(Φ)U^k(Φ)Uk(Φ)-uniform sets

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Prime sumsets in sets that are simultaneously basic Nil-Bohr and Uk(Φ)U^k(\Phi)Uk(Φ)-uniform

Polynomial prime sumsets

Arbitrary prescribed sumset levels in Uk(Φ)U^k(\Phi)Uk(Φ)-uniform sets

Shifted-prime finite-sum levels in Nil-Bohr sets

Dynamical characterizations by finite sums and shifted-prime sums

Fiberwise decomposition and progressive Hall-Petresco measures

The common nilsystem representation is not established

The ρa∘\rho_a^\circρa∘​ positivity identity does not follow from Proposition 4.10

The printed reduction inherits Theorem 7.5's gap

The Furstenberg-model and Nil-Bohr arguments close

The central dynamical proof chain is complete

The final continuity sentence has the inequality direction reversed

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Authenticate with ORCID, then require an exact arXiv match

Infinite prime sumsets in structured and $U^k(Φ)$ -uniform sets

Prime sumsets in sets that are simultaneously basic Nil-Bohr and $U^k(\Phi)$ -uniform

Arbitrary prescribed sumset levels in $U^k(\Phi)$ -uniform sets

The $\rho_a^\circ$ positivity identity does not follow from Proposition 4.10