{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1201",
  "content_hash": null,
  "slug": "erdos-problem-14-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 14: For a set $A \\subseteq \\mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\\{1, 2, \\ldots, N\\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \\subseteq \\mathbb{N}$ such that $f_A(N) = o(\\sqrt{N})$ as $N \\to \\infty$? Equivalently, is there a set $A \\subseteq \\mathbb{N}$ for which $f_A(N)/\\sqrt{N} \\to 0$ as $N \\to \\infty$?",
  "relevance": "For Erdős's Problem on Non-Unique Representability as Sums, record erdos-problem-14-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 14: For a set $A \\subseteq \\mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 14: For a set $A \\subseteq \\mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\\{1, 2, \\ldots, N\\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \\subseteq \\mathbb{N}$ such that $f_A(N) = o(\\sqrt{N})$ as $N \\to \\infty$? Equivalently, is there a set $A \\subseteq \\mathbb{N}$ for which $f_A(N)/\\sqrt{N} \\to 0$ as $N \\to \\infty$? 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/14",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/14",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1202",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-14",
      "title": "erdos problem 14",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}