Nathanson's Problems on Product Intersection Sets
Statement
Nathanson asked which subsets of can occur as product intersection sets of a family of semigroup subsets, for arbitrary and for decreasing families (his Problems 10 and 11). Both are solved by complete classifications.
Context
Recently posed numbered problems of Nathanson, a central figure in additive number theory, with no accumulated literature yet.
People
Attempts
No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.
"Both classifications were autonomously discovered and formally verified in Lean by Aristotle." The appendix documents the prompts: Aristotle was asked to characterize the two cases separately, then combine them; it even adopted a stronger definition than the source paper's and the Lean code verifies the equivalence explicitly.
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1 machine checkNo person has reviewed this attempt. 1 machine check below — a machine check is not human verification.
Machine check · not human verification
machine: partially checkedscope Lean formalization of the core argument; statement correspondence not independently audited
Discovered and kernel-checked in Lean by the same system; the human authors audited the informal-to-formal correspondence. Tier: Aristotle wrote both proof and formal statements (and at one point adopted a stronger definition than the source paper's, caught by the authors) - exactly the failure mode an independent statement audit exists for, and none has happened.
No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.
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