What was already known
Positive-density recurrence was known for intersective polynomials, and later quantitative bounds saved logarithmic or stretched-exponential factors. The immediately preceding fixed-power result concerned only square differences.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Extends a fixed power saving from squares to every intersective polynomial, with an exponent depending only on degree.
Limitations
The polynomial h is fixed as N grows; constants other than c_k may depend on h. The linked Lean scope checks the special polynomial h(m)=m², not all intersective polynomials or degree-uniform exponents.
Technical verification
OpenAI reports a Lean check of a narrower theorem, case, or component; the selected claim extends beyond that scope.
Formal checking
The pinned OpenAI Lean scope note verifies only this narrower statement, not the full selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For h(m)=m², there are absolute c,C>0 such that every A⊆{1,…,N} with no nonzero square difference has |A|≤C N^{1−c}.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and precise Lean-scope or conditional-input review for non-notifying local inclusion.
Read this published revision