{
  "schema_version": 1,
  "problem_number": "OWR-14299577-018",
  "title": "Differences of Signed Representatives: Sharp Answers to Two Questions of Bernert and Arala Santos",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let A ⊂ {−n,…,n}∖{0} contain exactly one of k and −k for every 1 ≤ k ≤ n. At the 2025 Oberwolfach workshop on analytic number theory, C. Bernert and N. Arala Santos asked two questions about such sets. First, must A−A contain (1−o(1))n elements of {1,…,n}? Second, for every B ⊆ {1,…,n} with |B| ≥ εn, must the number of pairs (a,b) ∈ A² with a−b ∈ B be ≫_ε n²? We answer both questions affirmatively, with sharp bounds. At most two elements of {1,…,n} are missing from A−A. For every B ⊆ {1,…,n}, the number of pairs (a,b) ∈ A² with a−b ∈ B is at least ⌊(|B|−1)²/4⌋. Both bounds are attained by sets of the form {1,…,a} ∪ {−(a+1),…,−n}, the second for all |B| ≤ ⌊2n/3⌋+1. The proof is a short double-counting argument. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "keywords": [
    "additive combinatorics",
    "difference sets",
    "signed sets",
    "representation function",
    "double counting",
    "Oberwolfach Reports",
    "OWR-14299577-018",
    "open mathematics",
    "mathematical proof",
    "math.CO",
    "math.NT"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299577-018/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299577-018/paper.pdf?v=25a5474f2651",
  "doi": "10.5281/zenodo.23006720",
  "zenodo_record_url": "https://zenodo.org/records/23006720",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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    "source.zip": {
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}