Erdős–Szemerédi sum-product problem

Open

erdos-sum-product

Statement

For finite

A \subseteq \mathbb{R}

, prove

\max(|A+A|,\, |AA|) \gg |A|^{2 - \varepsilon}

for every

\varepsilon > 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