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Updated Oct 8

Problem 818

claims/: The 1 claim page of Problem 818, one per claimant's result; the problem's standing derives from them.

Statement. Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$. Is it true that

\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{(\log \lvert A\rvert)^C}

for some constant $C>0$ ?

Status. Proved: the site labels the problem PROVED (LEAN); the corpus has not built the Lean proof, so it gives no formal evidence. The standing is derived from the claim page, accepted on the refereed publication and the site's credit.

Source. erdosproblems.com/818, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #818, https://www.erdosproblems.com/818.

References.

[So09d] Solymosi, József, Bounding multiplicative energy by the sumset. Adv. Math. 222 (2009), no. 2, 402-408.

Formalization. Statement in formal-conjectures.

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Linked library material

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Linked from (5)

Solymosi's sum-product bound answers yes problems/additive_combinatorics/E0818/claims Sumsets and Arithmetic Progressions Bounding multiplicative energy by the sumset Theorem 2.1 (p. 2): |AA||A+A|^2 >= |A|^4 / (4 ceil(log |A|)) for finite sets of positive reals

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