Erdős–Szemerédi sum-product problem

Open

erdos-sum-product

Statement

For finite $A \subseteq R$ , prove $max (∣ A + A ∣, ∣ AA ∣) ≫ ∣ A ∣^{2 - ε}$ for every $ε > 0$ : a set cannot be both additively and multiplicatively structured.

Current frontier

erdosproblems.com/52, OPEN. The exponent reached $4/3$ (Solymosi 2009) and now $1962/1469 - o (1) \approx 1.3356$ for the reals and integers (Cushman 2025). The conjectured exponent $2 - o (1)$ is far off.

When this counts as solved

QUANTITATIVE

PROOF_COMPLETE for the $2 - o (1)$ exponent, or a barrier ruling it out. BREAKTHROUGH for any rigorous improvement of the best integer/real exponent above the current best $1962/1469 \approx 1.3356$ .

Classification

Quantitative

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