{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1237",
  "content_hash": null,
  "slug": "erdos-problem-40-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 40: For a set $A \\subseteq \\mathbb{N}$ and a positive integer $N$, let $|A \\cap \\{1, \\ldots, N\\}|$ denote the cardinality of the intersection. For $n \\in \\mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \\in A \\times A$ such that $a + b = n$. We say that a property holds for sets $A$ satisfying $|A \\cap \\{1, \\ldots, N\\}| \\gg \\frac{\\sqrt{N}}{g(N)}$ if there exists a positive constant $C$ such that $|A \\cap \\{1, \\ldots, N\\}| \\geq C \\cdot \\frac{\\sqrt{N}}{g(N)}$ for all sufficiently large $N$. A function $g: \\mathbb{N} \\to \\mathbb{R}$ tends to infinity if for every $M > 0$, there exists $N_0$ such that $g(N) > M$ for all $N \\geq N_0$. Determine the set of all functions $g: \\mathbb{N} \\to \\mathbb{R}$ tending to infinity such that for every set $A \\subseteq \\mathbb{N}$ satisfying $|A \\cap \\{1, \\ldots, N\\}| \\gg \\frac{\\sqrt{N}}{g(N)}$, we have $\\limsup_{n \\to \\infty} (1_A * 1_A)(n) = \\infty$.",
  "relevance": "For Erdős Problem 40 on additive bases with slow growth, record erdos-problem-40-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 40: For a set $A \\subseteq \\mathbb{N}$ and a positive integer $N$, let $|A \\cap \\{1, \\ldots, N\\}|$ denote the cardinality of the intersection.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 40: For a set $A \\subseteq \\mathbb{N}$ and a positive integer $N$, let $|A \\cap \\{1, \\ldots, N\\}|$ denote the cardinality of the intersection. For $n \\in \\mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \\in A \\times A$ such that $a + b = n$. We say that a property holds for sets $A$ satisfying $|A \\cap \\{1, \\ldots, N\\}| \\gg \\frac{\\sqrt{N}}{g(N)}$ if there exists a positive constant $C$ such that $|A \\cap \\{1, \\ldots, N\\}| \\geq C \\cdot \\frac{\\sqrt{N}}{g(N)}$ for all sufficiently large $N$. A function $g: \\mathbb{N} \\to \\mathbb{R}$ tends to infinity if for every $M > 0$, there exists $N_0$ such that $g(N) > M$ for all $N \\geq N_0$. Determine the set of all functions $g: \\mathbb{N} \\to \\mathbb{R}$ tending to infinity such that for every set $A \\subseteq \\mathbb{N}$ satisfying $|A \\cap \\{1, \\ldots, N\\}| \\gg \\frac{\\sqrt{N}}{g(N)}$, we have $\\limsup_{n \\to \\infty} (1_A * 1_A)(n) = \\infty$. 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/40",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/40",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1238",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-40",
      "title": "erdos problem 40",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}