{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0230",
  "title": "Lacunary Counterexamples to a Distinct-Summand Freiman Container Problem",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Fix K > 1. We construct finite nonempty sets A,B ⊆ ℤ with |A| > |B| and |A+B| < K|A| for which B is not contained in any generalized arithmetic progression of bounded rank and size O_K(|A|). More strongly, for every prescribed rank bound d and size constant C, one counterexample defeats all progressions of rank at most d and cardinality at most C|A|. The construction is B = {0,1,N,…,N^{r−1}} and A = kB. Uniqueness of base-N digits gives exact polynomial growth |sB| = (s+r choose r) for s < N, whereas every rank-d generalized arithmetic progression has h-fold growth at most h^d. This answers Problem 2.5 from the 2004 AIM list on recent trends in additive combinatorics in the negative for every K > 1.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "additive combinatorics",
    "Freiman theory",
    "generalized arithmetic progressions",
    "sumset growth",
    "lacunary sets",
    "sumset",
    "generalized arithmetic progression",
    "Freiman theorem",
    "lacunary set",
    "AIM-COMBINATORICS-0230",
    "open mathematics",
    "mathematical proof",
    "math.CO"
  ],
  "manuscript_version_date": "2026-08-30",
  "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-0230/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0230/paper.pdf?v=6881b8037de2",
  "doi": "10.5281/zenodo.22245483",
  "zenodo_record_url": "https://zenodo.org/records/22245483",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
    "paper.pdf": {
      "sha256": "6881b8037de26bcb54ff0bbdb22b06bfe46ccc851ab9fc486a5610bac1a55ff1"
    },
    "source.zip": {
      "sha256": "ac8ac7fd82e55e4d51693d1612054d796aae112c34ca019a8ce6f2769c499332"
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    "verification_report.md": {
      "sha256": "3183d8e18d312504fcf58ca3993318ac89ce277b8da4fc3a7769a75bad409b71"
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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."
}