{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R472",
  "content_hash": null,
  "slug": "lshac-attempt-literature-audit",
  "type": "attempt",
  "title": "The located covariogram literature does not give this interval extremum",
  "summary": "The search found the standard covariogram framework and popular-difference results, with no sharp theorem for the fixed arc of shifts.",
  "relevance": "For Sharp long-shift autocorrelation for half-measure circle sets, record lshac-attempt-literature-audit (“The located covariogram literature does not give this interval extremum”) documents a concrete method, search boundary, or failed route. The record states: The search found the standard covariogram framework and popular-difference results, with no sharp theorem for the fixed arc of shifts.",
  "relevance_source": "recorded",
  "body": "The function \\(R_A(t)=\\mu(A\\cap(A+t))\\) is the covariogram, also called set covariance or indicator autocorrelation. The covariogram literature located in this audit concentrates on reconstructing convex bodies from the full function and on its geometric regularity. Averkov and Bianchi define the same translate-intersection functional and solve Matheron's planar convex-body reconstruction conjecture. Their hypotheses and conclusion do not provide an extremal bound for arbitrary measurable subsets of the circle.\n\nPopular-difference theorems seek shifts with large intersections or richer additive patterns, commonly allowing the shift to come from a neighborhood or a structured set. The present problem restricts the shift to the closed arc \\([1/4,1/2]\\), which stays away from zero. The targeted search found no source determining this constant or closing the interval \\([1/6,3/16]\\).\n\nThis audit covered covariogram, set covariance, autocorrelation of indicators on compact groups, popular differences, and prescribed difference intervals. It records search coverage rather than a claim of novelty.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "literature on covariograms, set covariance, popular differences, and prescribed-shift overlap relevant to the stated circle problem",
    "bounds": {
      "search_date": {
        "min": 20260724,
        "max": 20260724
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/0711.0572",
      "locator": "Gennadiy Averkov and Gabriele Bianchi, Confirmation of Matheron's conjecture on the covariogram of a planar convex body, Journal of the European Mathematical Society 11 (2009), 1187-1202, definition and introduction; popular-difference search performed 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/0711.0572",
    "locator": "Gennadiy Averkov and Gabriele Bianchi, Confirmation of Matheron's conjecture on the covariogram of a planar convex body, Journal of the European Mathematical Society 11 (2009), 1187-1202, definition and introduction; popular-difference search performed 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R473",
      "title": "The constant lies between 1/6 and 3/16",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "long-shift-autocorrelation-half-set",
      "title": "long shift autocorrelation half set",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}