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Precise statement
Erdos and Szemeredi conjectured that every finite set of reals satisfies $\max(|A+A|,|AA|) \ge |A|^{2-o(1)}$. False: there are arbitrarily large $A \subset \mathbb{R}$, of algebraic integers in a number field of degree $\asymp \log|A|$, with $\max(|A+A|,|AA|) \le |A|^{2-c}$ for an absolute $c > 0$. Variants give counterexamples in function fields of fixed positive characteristic.
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fidelity, correctness, priority, or novelty.
What AI did
GPT-5.5 Pro
The limits here matter more than the headline, and the authors state them plainly: GPT-5.5 Pro was a sounding board in the early stages, but the final proof including all the main ideas was almost entirely human-generated, and everything in the paper was written by the authors. The single exception they name is Lemma 3.4, suggested by the model, which replaced a more complicated result of Schinzel with a short elementary argument. There is a second, indirect AI thread: the authors say they were inspired to revisit number fields of large degree by OpenAI's counterexample to the unit distance conjecture, and note their construction needed far less number-theoretic input than that one did.
arXiv preprint by four established additive combinatorialists; not yet peer-reviewed. Given the size of the claim this one deserves refereeing before it is treated as settled.