{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R699",
  "content_hash": null,
  "slug": "sac-claim-density-zero",
  "type": "claim",
  "title": "Subtract-a-cube losing positions have density zero",
  "summary": "The losing positions form a cube-difference-free set, so the polynomial Szemerédi theorem forces their natural density to be zero.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Let \\(C\\) be the set of P-positions. Two elements of \\(C\\) cannot differ by a positive cube, since the larger position could move to the smaller one. Suppose \\(C\\) had positive upper density. Bergelson and Leibman's polynomial extension of Szemerédi's theorem, applied to \\(p(t)=t^3\\), would give \\(x,y\\in C\\) and \\(t\\geq 1\\) with \\(x-y=t^3\\). This contradicts the defining property of P-positions. Therefore \\(\\limsup_{N\\to\\infty}|C\\cap[1,N]|/N=0\\), and the nonnegative ratios converge to zero.\n\nThe argument settles the database question. It applies more generally when the legal moves are the positive values of an integer-valued intersective polynomial with zero constant term.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the full set of losing positions in the normal-play subtract-a-cube game"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1090/S0894-0347-96-00194-4",
      "locator": "Bergelson and Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, JAMS 9 (1996), section 0.2, page 726"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1090/S0894-0347-96-00194-4",
    "locator": "Bergelson and Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, JAMS 9 (1996), section 0.2, page 726"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R697",
      "title": "Exact subtract-a-cube sweep through 200,000",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "R698",
      "title": "Determine the growth order of the greedy cube-difference-free set",
      "object_type": "attempt",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "subtract-a-cube-density",
      "title": "subtract a cube density",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}