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Ben Green's Open Problem 90

Combinatorics · posed by Ben Green · solved

1 attempt

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).

Context

A numbered problem from Ben Green's published list of open problems; the catalog already tracks Problem 57 from the same list.

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Attempts

1 attempt

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  • #1

    Attempt 1

    proof attemptChatGPT 5.4 with Jie Ma, Quanyu Tang, Max Wenqiang Xu ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    ChatGPT 5.4
    people
    Jie Ma, Quanyu Tang, Max Wenqiang Xu

    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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