{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0233",
  "title": "Power-Saving Lower Bounds for Additive Bases of Polynomial Sequences",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let h ≥ 2 be fixed, let f be a natural-valued polynomial of degree d ≥ 2, and suppose that A ⊆ ℕ satisfies f(ℕ) ⊆ hA. We prove |A ∩ [0,X]| ≥ X^{4/(3hd)−o(1)} when h is even, and |A ∩ [0,X]| ≥ X^{4/((3h+1)d)−o(1)} when h is odd. Both exponents are strictly larger than the elementary exponent 1/(hd). This gives an affirmative qualitative answer, for every fixed h ≥ 2, to Problem 2.8 from the 2004 AIM problem list on recent trends in additive combinatorics. The argument combines a self-contained graph-path form of an Erdős–Newman two-basis estimate with a divisor bound for repeated differences of polynomial values. For odd h, one summand is first frozen on a large fiber and the remaining summands are then split into two equal blocks. No optimality of the displayed exponents is asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "additive bases",
    "polynomial sequences",
    "sumsets",
    "repeated differences",
    "Erdős–Newman estimate",
    "additive basis",
    "polynomial sequence",
    "sumset",
    "additive",
    "combinatorics",
    "AIM-COMBINATORICS-0233",
    "open mathematics",
    "mathematical proof",
    "math.CO"
  ],
  "manuscript_version_date": "2026-08-29",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0233/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0233/paper.pdf?v=aa349ce9ac56",
  "doi": "10.5281/zenodo.22245345",
  "zenodo_record_url": "https://zenodo.org/records/22245345",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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    "source.zip": {
      "sha256": "4232b7764cc4365402e36326b26895dbfe703b4b54c62d2be6979433a03f44dc"
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    "verification_report.md": {
      "sha256": "6e96dbac0968ead4534dd07111904ff95b3fef35580828b3f97ca51fa20b0c3d"
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  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}