In 1974, Paul Erdős posed the following simple conjecture about the relationship between addition and multiplication: for a finite set A (in ℤ, or ℝ, or ℂ, say), either the set of pairwise sums or the set of pairwise products of elements of A is nearly as large as possible -- at least |A|^(2-o(1)) as |A| grows. Partial progress towards this conjecture has found applications in places such as harmonic analysis, the study of expander graphs, and exponential sum estimates.
In this talk, I will present a disproof of the sum-product conjecture over ℝ. Our construction uses high-degree number fields with bounded root discriminant and is inspired by the recent striking disproof of the Erdős unit distance conjecture by OpenAI.
Joint work with Thomas F. Bloom, Will Sawin, and Dmitrii Zhelezov.