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Ben Green's Open Problem 90

Combinatorics · posed by Ben Green · solved

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Statement

For A⊂FpA \subset \mathbb{F}_p of density 1/21/2, call AA almost affine invariant under φ(x)=ax+b\varphi(x) = ax+b if ∣A△φ(A)∣=o(p)|A \triangle \varphi(A)| = o(p). Problem 90 asks for the threshold KK below which AA can be almost affine invariant simultaneously under all such φ\varphi with ∣a∣,∣b∣≤K|a|, |b| \le K and a≠0a \ne 0. The threshold is K=o(log⁡p)K = o(\log p).

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Jie Ma, Quanyu Tang and Max Wenqiang Xu, using ChatGPT 5.4

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    One of the more honest disclosures in the catalog, because it itemises what the model got right and says plainly what it got wrong. The authors credit two specific ideas as mostly due to AI: considering the qq-adic valuation formulation, which is what yields the sharp o(log⁡p)o(\log p) upper bound in the final step, and using the amenability of the affine group in an earlier version of one lemma. They also record that the original AI-produced arguments contained many logical mistakes and gaps across the iterative process. Both halves belong in the record: real mathematical ideas, arriving inside output that needed human repair.

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open in Claude →open in ChatGPT →

This opens a third-party site. Nothing is posted back to ProbXiv and nothing you write there is recorded here — what a model gives you is an attempt, which a person still has to check.