arXiv:2603.27258v2

A short proof of Erdős's $B+C$ conjecture

Bryna Kra, Joel Moreira, Florian K. Richter, Donald Robertson

math.DSmath.COmath.NT05D1011B1337A4411B30

Paper metadata

Original submission: March 28, 2026
Latest revision: August 4, 2026 · version 2
Claimed MSC: 05D10, 11B13, 37A44, 11B30
arXiv categories: math.DS, math.CO, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 7 pages; minor revisions following the referee's report

Abstract

We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.

AI-generated audit

Audit summary

Audited against arXiv v2

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.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements2 reported findingsCorrect

The main dynamical theorem is correct and yields the stated $B+C$ theorem for every set of positive upper Banach density.

Theorem 3.1Correct

Two iterated orbit limits land in the positive-measure clopen set

Pages 4–6 · Theorem 3.1 and Section 4 · arXiv:2603.27258v2

The self-joining generated by a generic pair $(x,y)$ has both marginals $\mu$ . Splitting $1_E$ into its Kronecker-factor projection and weak-mixing component yields infinitely many return times $b$ that bring $T^b x$ close to $y$ while retaining a positive joining intersection. Iteration produces $T^{b_i}x\to y$ and positive intersections for every finite initial segment. Compactness then supplies subsequences whose two iterated limits both lie in the closed set $E$ , with every required quantifier preserved.

Theorem 1.1Correct

Every positive-upper-Banach-density set contains an infinite sumset

Pages 1 and 4 · Theorem 1.1 and the reduction following Theorem 3.1 · arXiv:2603.27258v2

The correspondence principle represents $A$ as the return set of a generic point to a positive-measure clopen set. The two iterated limits from Theorem 3.1 permit an alternating diagonal choice of subsequences $(b_i')$ and $(c_j')$ with $T^{b_i'+c_j'}x\in E$ for every pair $(i,j)$ . Property (3) of the correspondence principle then gives $B+C\subset A$ for the two infinite sets $B=\{b_i'\}$ and $C=\{c_j'\}$ .

02Proofs3 reported findingsCorrect

The joining argument, Kronecker/weak-mixing decomposition, iterative construction, and combinatorial extraction are correct and complete. One harmless tensor-factor typo is uniquely determined by the displayed estimate and does not lower the proof status.

Claim 1Correct and complete

Structured and weak-mixing estimates give infinitely many admissible return times

Pages 5–6 · Claim 1 and Equations (4.1)–(4.4) · arXiv:2603.27258v2

The support property of $f_c=\mathbb E(1_E\mid\mathcal K)$ makes $\eta=\langle1\otimes f_c,1_Y\rangle$ positive. Topological eigenfunctions approximate $f_c$ and are equicontinuous along the orbit, so every $b$ with $T^b x$ sufficiently close to $y$ satisfies $\|(T^bf_c\otimes1)-(1\otimes f_c)\|_2<\eta/2$ . Such $b$ have positive upper density along the chosen Følner sequence. Since $f_{wm}\otimes1$ is weak mixing for the joining—its self-correlation is the self-correlation of $f_{wm}$ under the first marginal—infinitely many of those $b$ also make the weak-mixing contribution smaller than $\eta/2$ . The resulting joining intersection is strictly positive.

Claim 1 lower boundTypo

The comparison term has its tensor factors reversed in one displayed line

Page 5 · paragraph immediately after conditions [1]–[3] · arXiv:2603.27258v2

The displayed lower bound prints $\langle f_c\otimes1,1_Y\rangle-\eta$ . It should read $\langle1\otimes f_c,1_Y\rangle-\eta$ , because condition [2] compares $T^bf_c\otimes1$ with $1\otimes f_c$ and $\eta$ was defined as $\langle1\otimes f_c,1_Y\rangle$ . With the two strict error bounds $<\eta/2$ , the corrected line is positive. The intended correction is unique and changes no subsequent argument.

Theorem 2.5Correct and complete

The enhanced correspondence principle has the required topological model

Page 3 · Theorem 2.5 · arXiv:2603.27258v2

The cited correspondence theorem supplies an ergodic system, a generic point, and a clopen return set representing $A$ . The cited topological-model lemma then realizes every measurable eigenfunction by a topological eigenfunction, which is precisely the additional hypothesis used in the equicontinuity estimate of Claim 1.

Kra–Moreira–Richter–Robertson, Infinite sumsets in sets with positive density ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2603.27258v2
Authors listed: Bryna Kra, Joel Moreira, Florian K. Richter, Donald Robertson
Audit date: August 18, 2026

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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. 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P(delta)

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A short proof of Erdős's B+CB+CB+C conjecture

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Two iterated orbit limits land in the positive-measure clopen set

Every positive-upper-Banach-density set contains an infinite sumset

Structured and weak-mixing estimates give infinitely many admissible return times

The comparison term has its tensor factors reversed in one displayed line

The enhanced correspondence principle has the required topological model

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A short proof of Erdős's $B+C$ conjecture