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The Sum-Product Conjecture over the Reals

Combinatorics · posed by Paul Erdos, Endre Szemeredi, 1983 · disproved

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Statement

Erdos and Szemeredi conjectured that every finite set of reals satisfies max⁡(∣A+A∣,∣AA∣)≥∣A∣2−o(1)\max(|A+A|,|AA|) \ge |A|^{2-o(1)}. False: there are arbitrarily large A⊂RA \subset \mathbb{R}, of algebraic integers in a number field of degree ≍log⁡∣A∣\asymp \log|A|, with max⁡(∣A+A∣,∣AA∣)≤∣A∣2−c\max(|A+A|,|AA|) \le |A|^{2-c} for an absolute c>0c > 0. Variants give counterexamples in function fields of fixed positive characteristic.

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

  1. construction · #1

    Thomas F. Bloom, Will Sawin, Carl Schildkraut and Dmitrii Zakharov, using GPT-5.5 Pro

    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 assisted
    — a person led the work and used a model along the way.

    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.

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