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reviewer:erdos-db-trust · no key registered on this bundle

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null → 94ce9ec1

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in state · unreviewed

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1. 2026-05-30proposeproposed · finding.addagent:erdos-spine-ingestvpr_fb16d9b890e4190cCandidate claim vc_92f4f989ecb889f7 imported from artifact packet cap_61973ee16b553d57
2. 2026-05-30acceptfinding.assertedreviewer:erdos-db-trustnull→94ce9ec1vev_08945aa94ce6d016Candidate claim vc_92f4f989ecb889f7 imported from artifact packet cap_61973ee16b553d57

Erdős Problem #38 has status 'proved (lean)'. Statement: Does there exist $B \subset \mathbb{N}$ which is not an additive basis, but is such that for every set $A \subseteq \mathbb{N}$ of Schnirelmann density $\alpha$ and every $N$ there exists $b \in B$ such that $$ \lvert (A \cup (A+b)) \cap \{1, \ldots, N\} \rvert \geq (\alpha + f(\alpha)) N $$ where $f(\alpha) > 0$ for $0 < \alpha < 1$? Note: here Erdős seems to use a slightly weaker notion of an additive basis (see [Er56] at the top of page 135). In particular, for this problem, a set is an additive basis of order $k$ if every natural number can be written as a sum of _at most_ $k$ elements of the set, rather than as a sum of _precisely_ $k$ elements. A positive [solution](https://github.com/spicylemonade/erdos-38) was given by GPT 5.5 Pro (prompted by gebyjaff, cleanup by Liam Price); in fact a sparse random set $B$ has this property, with $f(\alpha)\gg \alpha (1-\alpha)^2$. Topics: number theory. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.