Erdős problems frontier

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  "text": "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.",
  "type": "open_question"
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  "text": "Agent-imported candidate claim; scope requires review."
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  "extraction_confidence": 0.7,
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 "created": "2026-05-30T00:42:06.826507+00:00",
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    "artifact_packet_id": "cap_61973ee16b553d57",
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  "authors": [
   {
    "name": "Erdős Open-Problem spine ingest",
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   }
  ],
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  "extraction": {
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  "source_type": "model_output",
  "title": "cap_61973ee16b553d57 · vc_92f4f989ecb889f7",
  "url": "https://www.erdosproblems.com/38",
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