The Sum-Product Conjecture over the Reals
Statement
Erdos and Szemeredi conjectured that every finite set of reals satisfies . False: there are arbitrarily large , of algebraic integers in a number field of degree , with for an absolute . Variants give counterexamples in function fields of fixed positive characteristic.
Context
the model's contribution is one simplifying lemma; the authors state the main ideas are human
The Erdos-Szemeredi sum-product conjecture is one of the organizing conjectures of additive combinatorics, with a large literature of partial exponents built directly on the expectation that it is true.
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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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