Ben Green's Open Problem 90

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

Result: Proved
Status: Resolved
AI contribution: AI co-developed
Method: Argument
Field: Additive combinatorics
Posed by: Ben Green
Year posed: —
Years open: —
Solved: 2026-05-13
Model: ChatGPT 5.4
Vendor: OpenAI
Collaborators: Jie Ma, Quanyu Tang, Max Wenqiang Xu
Verification: Unreviewed
Publication: Preprint
Significance: 30 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

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 $q$ -adic valuation formulation, which is what yields the sharp $o(\log p)$ 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.

Verification

arXiv preprint, not peer-reviewed. The authors state the model's original arguments contained many logical mistakes and gaps, so the published proof is their reconstruction rather than model output.

Source

arXiv:2605.13454 - Almost Affine Invariance Over Prime Fields: Green Problem 90

Ben Green's Open Problem 90

What the AI did

Verification

Source

Discussion