The problem
For a finite set A ⊂ ℝ, must max(|A+A|, |A·A|) be at least |A|^{2−o(1)}? Erdős and Szemerédi (1983) conjectured yes; the best known lower bounds were |A|^{4/3+o(1)}.
The Erdős–Szemerédi Sum–Product Conjecture over ℝ
AI-resolvedReferences
History & significance
One of additive combinatorics' central drivers, feeding into incidence geometry and the polynomial method. Elekes's connection to unit distances made it a cousin of Erdős's geometry problems.
Connected problems
The resolution (AI-assisted resolution)
Disproved over ℝ — Bloom, Sawin, Schildkraut and Zhelezov, June 2026: explicit sets with both |A+A| and |A·A| bounded by |A|^{2−δ}, built with the algebraic-integer machinery the OpenAI model introduced for unit distances days earlier — a direct intellectual descendant of the AI result. Within weeks, an agent built on GPT-5.5 Pro reproduced the disproof autonomously in seven of eight independent trials, with several proofs diverging from the published one entirely (arXiv:2607.20525). The original conjecture over ℤ remains open and is believed true.