Let $A$ be a finite set of integers. Is it true that for every $ϵ > 0$ $max (∣ A + A ∣, ∣ AA ∣) ≫_{ϵ} ∣ A ∣^{2 - ϵ} ?$

Worked, still open.

number theory · open · prize $250 · formalized (Lean) · 0 attempts

machinery: additive-combinatorics,sum-product,multiplicative-energy,Szemeredi-Trotter-incidences,geometric

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

This is **open** in full generality. In fact, your inequality is essentially the **Erdős–Szemerédi sum–product conjecture** (for (h=2)), which asserts that for every (\epsilon>0), [ \max\bigl(|A+A|,\ |AA|\bigr)\ \ge c(\epsilon),|A|^{2-\epsilon} \qquad (A\subset\mathbb Z\ \text{finite}), ] equivalently (\max(|A+A|,|AA|)…

candidate solution ↗

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

3 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_52 : answer(sorry) ↔ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∃ (C : ℝ), 0 < C ∧ ∀ (A : Finset ℤ),
    (max (A + A).card (A * A).card : ℝ) ≥ C * (A.card : ℝ) ^ (2 - ε)

formal-conjectures/52.lean ↗

oeis

A263996 — Smallest possible cardinality of the union of the set of pairwise sums and the set of pairwise products from a set of n positive integers.1,4,7,11,15,20,26,30,36,44,49,57,64,71,80,86,96,104,112,121,131,141,150,160,169,179,190,200,212,222,235,248,260,272,283,

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