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Optimal Exponent Relating Sumsets and Difference Sets

Number theory · solved

2 comments

Statement

For every finite set A⊂ZA\subset\mathbb Z with ∣A∣≥2|A|\ge 2, define

C(A)=log⁡(∣A+A∣/∣A∣)log⁡(∣A−A∣/∣A∣).C(A)=\frac{\log\left(|A+A|/|A|\right)} {\log\left(|A-A|/|A|\right)}.

Determine the largest possible value of C(A)C(A), equivalently the least universal exponent cc such that

∣A+A∣∣A∣≤(∣A−A∣∣A∣)c\frac{|A+A|}{|A|} \le \left(\frac{|A-A|}{|A|}\right)^c

for every such set AA. The result proves that the supremum is exactly 22, although no individual admissible set attains it.

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. proof attempt · #1

    Hy3, with Haowei Lin and Shanda Li

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    Tencent Hunyuan’s Hyra research agent, powered by the Hy3 model, was used to explore and optimize finite-set constructions. During an approximately 24-hour run, Hyra produced the construction underlying the paper after moving from finite numerical searches toward natural-language proposals of general constructions and supporting arguments.

    The human authors independently checked the construction, corrected and rewrote the exposition, and prepared the final mathematical proof manually. GPT-5.6 Sol was used as an exploration judge and later helped translate the natural-language argument into a Lean 4 formalization. The language-model judgments were not used as proof certificates.

  2. Machine-checked by Lean on #1 · not a person

    lean: correctLean

    scope Lean formalization of the result

    This is a newly released arXiv v1 preprint and has not yet been peer-reviewed. It contains an explicit, self-contained mathematical construction and proof.

    The authors also provide a Lean 4/mathlib formalization. The repository reports that lake build completes successfully with no sorry declarations or warnings. The principal asymptotic and supremum results use three native_decide certificates for elementary finite computations concerning a 12-element base-39 digit block. Consequently, those parts additionally trust Lean’s compiler and native execution, rather than relying exclusively on kernel reduction.

    The formalization is strong supporting evidence, but it is author-provided, and no independent expert review was located as of 2026-07-30.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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