{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1213",
  "content_hash": null,
  "slug": "erdos-problem-153-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 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \\leq b$ in $A$ are distinct. For a finite set $S = \\{s_1 < s_2 < \\cdots < s_t\\}$ of natural numbers, define the mean square gap to be \\[\\frac{1}{t}\\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \\{a+b : a,b \\in A\\} = \\{s_1 < s_2 < \\cdots < s_t\\}$ is the sumset. Does $f(n) \\to \\infty$ as $n \\to \\infty$?",
  "relevance": "For Asymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets, record erdos-problem-153-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 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \\leq b$ in $A$ are distinct.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \\leq b$ in $A$ are distinct. For a finite set $S = \\{s_1 < s_2 < \\cdots < s_t\\}$ of natural numbers, define the mean square gap to be \\[\\frac{1}{t}\\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \\{a+b : a,b \\in A\\} = \\{s_1 < s_2 < \\cdots < s_t\\}$ is the sumset. Does $f(n) \\to \\infty$ 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/153",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/153",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1214",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-153",
      "title": "erdos problem 153",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}