{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R474",
  "content_hash": null,
  "slug": "lshac-claim-one-sixth-lower-bound",
  "type": "claim",
  "title": "The shift 1/3 always has overlap at least 1/6",
  "summary": "Counting pairs among three one-third translates proves R_A(1/3) >= 1/6.",
  "relevance": "For Sharp long-shift autocorrelation for half-measure circle sets, record lshac-claim-one-sixth-lower-bound (“The shift 1/3 always has overlap at least 1/6”) records a bound, answer, status fact, or structural consequence. The record states: Counting pairs among three one-third translates proves R_A(1/3) >= 1/6.",
  "relevance_source": "recorded",
  "body": "Let\n\\[\nA_0=A,\\qquad A_1=A+\\frac13,\\qquad A_2=A+\\frac23\n\\]\nand define the integer-valued multiplicity\n\\[\nN(x)=\\mathbf1_{A_0}(x)+\\mathbf1_{A_1}(x)+\\mathbf1_{A_2}(x).\n\\]\nFor \\(N\\in\\{0,1,2,3\\}\\), the pointwise inequality\n\\[\n\\binom N2\\geq N-1\n\\]\nholds. Integration gives\n\\[\n\\sum_{0\\leq i<j\\leq2}\\mu(A_i\\cap A_j)\n=\\int_{\\mathbb T}\\binom{N(x)}2\\,dx\n\\geq\\int_{\\mathbb T}(N(x)-1)\\,dx\n=\\frac12.\n\\]\nTranslation invariance and the evenness \\(R_A(t)=R_A(-t)\\) make all three intersections on the left equal to \\(R_A(1/3)\\). Therefore\n\\[\n3R_A(1/3)\\geq\\frac12,\n\\qquad R_A(1/3)\\geq\\frac16.\n\\]\nSince \\(1/3\\in[1/4,1/2]\\), every admissible set satisfies \\(M(A)\\geq1/6\\).",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "universal",
    "statement": "every measurable A in the circle with measure 1/2"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/0711.0572",
      "locator": "Independent three-translate counting argument, 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/0711.0572",
    "locator": "Independent three-translate counting argument, 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R473",
      "title": "The constant lies between 1/6 and 3/16",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "long-shift-autocorrelation-half-set",
      "title": "long shift autocorrelation half set",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}