combinatorics / Additive combinatorics
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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the model's contribution is one simplifying lemma; the authors state the main ideas are human
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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.
the model's contribution is one simplifying lemma; the authors state the main ideas are human
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