{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1247",
  "content_hash": null,
  "slug": "erdos-problem-52-attempt-resolution-route",
  "type": "attempt",
  "title": "Resolve the stated acceptance condition",
  "summary": "Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 52: Let $A$ be a finite set of integers. For every real number $\\varepsilon$ with $0 < \\varepsilon < 1$, does there exist a real constant $C > 0$ such that\n\\[\n\\max(|A+A|, |A \\cdot A|) \\geq C \\cdot |A|^{2-\\varepsilon}\n\\]\nholds for all such sets $A$? Here $A+A$ denotes the sumset $\\{a+b : a,b \\in A\\}$ and $A \\cdot A$ denotes the product set $\\{a \\cdot b : a,b \\in A\\}$.",
  "relevance": "For Erdős's sum-product growth problem for finite sets of integers, record erdos-problem-52-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 52: Let $A$ be a finite set of integers.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 52: Let $A$ be a finite set of integers. For every real number $\\varepsilon$ with $0 < \\varepsilon < 1$, does there exist a real constant $C > 0$ such that\n\\[\n\\max(|A+A|, |A \\cdot A|) \\geq C \\cdot |A|^{2-\\varepsilon}\n\\]\nholds for all such sets $A$? Here $A+A$ denotes the sumset $\\{a+b : a,b \\in A\\}$ and $A \\cdot A$ denotes the product set $\\{a \\cdot b : a,b \\in A\\}$. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.",
  "status": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://www.erdosproblems.com/52",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/52",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1248",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-52",
      "title": "erdos problem 52",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}