arXiv:2607.14071v3

Strongly complete sets and a conjecture of Erdős

Steve Fan

math.NTmath.CO11B1311B7511J71

Paper metadata

Original submission: July 15, 2026
Latest revision: July 25, 2026 · version 3
Claimed MSC: 11B13, 11B75, 11J71
arXiv categories: math.NT, math.CO
Paper license: CC BY-NC-ND 4.0 ↗
Comments: 16 pages

Abstract

A set $A\subseteq\mathbb{N}$ is called $\textit{complete}$ if every sufficiently large integer can be written as a sum of distinct elements of $A$ . It is $\textit{strongly complete}$ if it remains complete after one deletes finitely many elements from it. We show that $A\subseteq\mathbb{N}$ is strongly complete whenever $\big|A\cap(2^k,2^{k+1}]\big|\ge6$ for every sufficiently large $k\in\mathbb{N}$ , and $\sum_{a\in A}\|aθ\|=\infty, \quad\forallθ\in\mathbb{R}\setminus\mathbb{Z}.$ In particular, this resolves a 1961 conjecture of Erdős. The proof builds on previous work of Bergelson and Simmons. In fact, our approach allows us to establish a more general strong-completeness criterion with suitable ordered blocks in place of dyadic intervals. We also discuss some applications of our results as well as their connections to a few other interesting problems, including two completeness problems of Erdős and Graham.

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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.

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Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The strong-completeness criterion, its dyadic specialization resolving the stated Erdős conjecture, the translated-block application, and the ordered-block extension are correct under their printed hypotheses.

Theorem 1.1 and Corollary 1.2Correct

Strong completeness from block density and norm-series divergence

Pages 2 and 10–12 · Theorem 1.1, Theorem 4.1, and Corollary 1.2 · arXiv:2607.14071v3

The constants $u_\rho=\lceil\rho(\rho-1)\rceil$ , $v_\rho=\max\{u_\rho,2\}$ , and $M_\rho=2u_\rho+v_\rho$ leave two $u_\rho$ -element components and a residual component with at least $v_\rho$ elements in every sufficiently late block. Lemma 3.3 makes all three finite-sum sets syndetic, while Lemmas 3.1–3.2 preserve divergence of the residual norm series at the countable exceptional set. Corollary 2.5 then gives strong completeness. At $\rho=2$ the constant is $M_2=6$ , exactly the stated dyadic specialization.

Full paper, version 3 ↗

Corollary 1.3Correct

Translated finite patterns

Pages 3 and 12 · Corollary 1.3 · arXiv:2607.14071v3

If the norm series converged at a nonzero $\theta$ , then for every $s,s'\in S$ the terms $\|(b_k+s)\theta\|$ and $\|(b_k+s')\theta\|$ would tend to zero, forcing $\|(s-s')\theta\|=0$ . The hypothesis $\gcd(S-S)=1$ then forces $\theta=0$ , a contradiction. Thus Theorem 1.1 applies with the required block count.

Full paper, version 3 ↗

Theorem 5.1 and Corollary 5.2Correct

Ordered-block generalization

Pages 13–15 · Theorem 5.1 and Corollary 5.2 · arXiv:2607.14071v3

The bounded ratio of the block minima gives the countable exceptional set needed by Lemma 3.1. The quantitative block hypothesis permits deletion of $2r$ elements per block while retaining every required divergent series. The displayed bound on $\beta_k-r\sum_{\ell<k}\alpha_\ell$ directly bounds $\Delta$ for both selected components and for the residual component. These facts verify the general theorem and its consecutive-integer-block corollary.

Full paper, version 3 ↗

02Proofs3 reported findingsCorrect

The proofs are correct and complete. The finite-support refinement of the cited Bergelson–Simmons argument uses only the hypotheses identified in that source, and the new countability, deletion, and block estimates close every downstream step.

Lemma 2.2 and Corollary 2.5Correct and complete

Finite-support refinement and strong-completeness criterion

Pages 5–7 · Lemma 2.2 through Corollary 2.5 · arXiv:2607.14071v3

The cited construction first obtains a thick set inside $\operatorname{FS}(B_1\cup B_2)$ after adding finitely many subset sums from $C$ . Taking the union of the finitely many summands produces a finite set $E\subset C$ , so disjointness is preserved. Removing $E$ from $C$ retains syndeticity and both divergence and modular completeness. The sum of the resulting thick and syndetic finite-sum sets is cofinite, and the same argument survives every finite deletion.

Bergelson–Simmons, Theorem 2.1 and Claims 2.13–2.14 ↗

Lemmas 3.1–3.3Correct and complete

Exceptional-set, deletion, and block estimates

Pages 8–10 · Section 3 · arXiv:2607.14071v3

For a bounded-ratio unbounded integer sequence, two points whose multiples are eventually small have a uniform separation, making each eventual-smallness set finite and their union countable. The deletion lemma retains the largest function values on successive finite blocks assigned to a repeating enumeration of the countable constraints, forcing every retained series to diverge. Finally, $n\geq\rho(\rho-1)$ makes the contribution from earlier $\rho$ -adic blocks dominate the current element up to a fixed constant, which proves the required finite $\Delta$ bounds.

Full paper, version 3 ↗

Theorems 4.1 and 5.1Correct and complete

Assembly of the three-component partitions

Pages 10–14 · Proofs of Theorems 4.1 and 5.1 · arXiv:2607.14071v3

In each construction the chosen block minima form an unbounded sequence with bounded successive ratios. The deleted elements split into two infinite components of the prescribed size, the residual component contains the chosen minima, and divergence is checked separately inside and outside the countable exceptional set. The finite- $\Delta$ estimates then match all hypotheses of Corollary 2.5, with finite initial blocks harmless throughout.

Full paper, version 3 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.14071v3
Authors listed: Steve Fan
Audit date: August 15, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

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P(delta)

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Strongly complete sets and a conjecture of Erdős

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Strong completeness from block density and norm-series divergence

Translated finite patterns

Ordered-block generalization

Finite-support refinement and strong-completeness criterion

Exceptional-set, deletion, and block estimates

Assembly of the three-component partitions

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