Problem 37

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claims/: The 1 claim page of Problem 37, one per claimant's result; the problem's standing derives from them.

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Statement. We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0<d_s(B)<1$ where $d_s$ is the Schnirelmann density.

Can a lacunary set $A\subset\mathbb{N}$ be an essential component?

Status. DISPROVED (LEAN), the site's label; its suffix is a catalog label explained under Formalization. The status-defining source is Ruzsa's theorem (Proc. London Math. Soc. (3) 54 (1987), 38--56, refereed): an essential component $A$ satisfies $\lvert A\cap\{1,\ldots,N\}\rvert\ge(\log N)^{1+c}$ for some $c>0$ and all large $N$, while a lacunary set has only $O(\log N)$ elements up to $N$, so the answer is no. The claim page is Ruzsa, accepted on the site curator's credit and the refereed publication; the 2026 Lean development that declares itself a formalization of his theorem is linked there and gives no formalized evidence, since this corpus has not built or audited it.

Source. erdosproblems.com/37, accessed 2026-09-04 and 2026-10-07 (page last edited 23 January 2026; empty proof-claim tab). Cite as: T. F. Bloom, Erdős Problem #37, https://www.erdosproblems.com/37.

References.

• [Ru87] Ruzsa, I., Essential Components. Proc. London Math. Soc. (3) 54 (1987), no. 1, 38-56, doi:10.1112/plms/s3-54.1.38; not held.

Formalization. The Lean qualification in the site's label is a catalog label. formal-conjectures has no file for Problem 37, and the site's indicator reads "Formalised statement? No" (2026-10-07). The community database (teorth/erdosproblems, file commit of 2026-09-28) lists status "disproved (Lean)", formal_status Lean and formalized "no" as of its last update on 2026-08-24, without recording when that state was set, and names no artifact. The locatable artifact is the Lean development src/latest/ErdosProblems/Erdos37.lean of Boris Alexeev's lean-proofs repository (added 2026-08-17; pinned on the Ruzsa claim page as a formalization of his theorem), which states the question under its own definitions. This corpus has not built or checked it, and no local kernel credit is claimed.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• jin_2014_density_versions_plunnecke_inequality
• jin_2014_density_versions_plunnecke_inequality / theorem_2
• nathanson_2014_paul_erdos_additive_bases
• nathanson_2014_paul_erdos_additive_bases / theorem_p2_essential_component
• erdos_1956_problems_results_additive_number_theory
• erdos_1956_problems_results_additive_number_theory / conjecture_p136