{
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  "category": "Combinatorics",
  "claimBoundary": "This page indexes Mathlib’s qualitative asymptotic declaration for the natural-number Roth function. Here rothNumberNat N is the maximum cardinality of a ThreeAPFree subset of Finset.range N = {0,…,N−1}, and ThreeAPFree excludes nontrivial a,b,c in the set with a+c=b+b. The exact conclusion is an IsLittleO atTop relation after coercion to ℝ. It does not state an explicit decay rate, a finite threshold, a modern quantitative bound, or a theorem about longer progressions.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem rothNumberNat_isLittleO_id : IsLittleO atTop (fun N ↦ (rothNumberNat N : ℝ)) (fun N ↦ (N : ℝ))",
  "family": "Additive combinatorics",
  "id": "library.mathlib.roth-theorem.v001",
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    "completeFormalType": "theorem rothNumberNat_isLittleO_id : IsLittleO atTop (fun N ↦ (rothNumberNat N : ℝ)) (fun N ↦ (N : ℝ))",
    "concepts": [
      "three-term arithmetic progressions",
      "asymptotic density",
      "Roth numbers",
      "corners theorem",
      "triangle removal"
    ],
    "plainLanguage": "Among the first N natural numbers, a set that avoids every nontrivial three-term arithmetic progression must occupy a proportion that tends to zero as N grows.",
    "poster": {
      "alt": "An ivory additive-combinatorics poster states rothNumberNat N equals little-o of N and traces dense sets through corners and triangle removal to an equally spaced triple.",
      "byteSize": 1955124,
      "caption": "The largest three-term-progression-free subset of {0,…,N−1} has density tending to zero, with no explicit rate claimed by the selected declaration.",
      "description": "The poster defines rothNumberNat, places the qualitative little-o conclusion at the top, and follows the retained source architecture from a dense subset through a grid corner and triangle-removal/regularity motif to three exactly equally spaced points. The exact-scope footer rules out an explicit rate.",
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      "generatedAt": "2026-07-25T20:42:04Z",
      "height": 1536,
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      "publicPath": "/assets/library-theorems/roth-theorem/theorem-poster-v1.png",
      "sha256": "sha256:2021e296679319145dcf79ab6299b3e9bd01a9572462a1ebe10e58cc6bf758ad",
      "title": "Roth’s Theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "ADDITIVE COMBINATORICS · THREE-TERM PROGRESSIONS\nROTH’S THEOREM\nrothNumberNat N = o(N)\nrothNumberNat N is the maximum size of a 3AP-free subset of {0,…,N−1}\nMAXIMUM 3AP-FREE DENSITY TENDS TO ZERO\nDENSE SET\nCORNERS CONSTRUCTION\nTRIANGLE REMOVAL + REGULARITY\nNONTRIVIAL 3-TERM PROGRESSION\nEXACT SCOPE\nQualitative asymptotic result. No explicit rate is claimed."
    },
    "visual": {
      "alt": "A long interval has many emerald selected positions and three larger gold selected positions with two visibly equal positive gaps.",
      "byteSize": 2348231,
      "caption": "For sufficiently large intervals, a subset of fixed positive density cannot avoid a nontrivial three-term arithmetic progression.",
      "description": "A discrete integer interval is rendered as evenly spaced ticks. A relatively dense selected subset rises above the interval, while exactly three central gold selections are joined by two equal-span bars to isolate the decisive nontrivial progression without asserting a quantitative threshold.",
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      "generatedAt": "2026-07-25T20:42:04Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/roth-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:a8d74b1c210e17f2e0cfd9f5639521d2c6f5d8879be1047d5619c68ce3a11a92",
      "title": "Equal spacing forced inside a dense integer interval",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Roth’s theorem is a foundational density result in additive combinatorics. Mathlib’s retained route connects one-dimensional three-term progressions to two-dimensional corners, triangle removal, and Szemerédi regularity."
  },
  "nonClaims": [
    "Proof Atlas did not originate Roth’s theorem or Mathlib’s declaration.",
    "The selected declaration gives no explicit rate or numerical threshold.",
    "The selected endpoint concerns three-term arithmetic progressions, not arbitrary progression lengths.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.roth-theorem.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "The maximum size of a three-term-arithmetic-progression-free subset of {0,…,N−1} is little-o of N.",
  "targetId": "target.library.mathlib.roth-theorem.v001",
  "title": "Roth’s Theorem",
  "upstreamOrigin": {
    "declarationName": "rothNumberNat_isLittleO_id",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Combinatorics/Additive/Corner/Roth.lean#L197",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
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    "sourceFile": "Mathlib/Combinatorics/Additive/Corner/Roth.lean",
    "sourceLine": 197,
    "verificationKind": "git_worktree"
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  "publicPresentationReview": {
    "status": "reviewed"
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}