ProofAtlas

Diophantine approximation · homogeneous dynamics · arithmetic lattices

Littlewood Conjecture

Collaboration beta

For every real pair alpha and beta, must n times the product of their two nearest-integer errors become arbitrarily small?

liminfn→∞n|nα||nβ|=0
Known results and sources
Three luminous coordinate axes compress a unimodular lattice while two approximation errors form a small unresolved product near the origin.
Littlewood asks whether every real pair produces arbitrarily small simultaneous multiplicative approximation.

Research problem

Exact mathematical statement

For every pair of real numbers α,β\alpha,\beta, the classical Littlewood conjecture asks whether

liminfn→∞n|nα||nβ|=0,\liminf_{n\to\infty} n\,\|n\alpha\|\,\|n\beta\|=0,

where |x|=minm∈Z|x-m|\|x\|=\min_{m\in\mathbf Z}|x-m|. The source reduces a possible counterexample to α,β∈Bad\alpha,\beta\in\mathrm{Bad} with 1,α,β1,\alpha,\beta linearly independent over Q\mathbf Q, and it does not prove that this hard core is empty.

Problem infographic

Problem at a glance

A problem-first diagram defines nearest-integer distance, displays the Littlewood liminf, isolates the badly approximable rationally independent hard core, and shows its equivalent positive-diagonal lattice-orbit question as open.
The universal approximation question reduces to a precise hard core and an equivalent compact-orbit problem in the space of unimodular lattices.

Current mathematical picture

Where work on Littlewood Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureTreating generic cubic-orbit shadowing as a contradiction

The audit says the compact orbit varies with n and uniform regulator or amplifier control is missing. Critical-corona generation, central-dominant integral rank defects, exceptional-face analysis, and uniform arithmetic repulsion for generic cubic orders remain live.

Route status · Narrowed route
Main reductionExact hard core

Any counterexample must have both coordinates badly approximable and 1, alpha, beta linearly independent over Q.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeGenerate incompatible affine dynamics on one critical root fiber or prove the exact uniqueness identity that contradicts the inherited regularity boundary.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Littlewood Conjecture in numbers

3.8kretained lines of mathematical investigation3,844 in the current working snapshot
Argument development
3,366 · 88%
Explored or eliminated routes
110 · 3%
Computational analysis
34 · 1%
Open obligations
163 · 4%
Definitions and setup
171 · 4%
8selected mapped statements1routes investigated4open questions4contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byJohn Edensor LittlewoodStatus checkedAug 14, 2026
Source recordedAug 14, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

13 selected steps

Scroll horizontally to explore the route

Working route overview for Littlewood ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Two simultaneous approximation errors become small enough that their n-weighted product has liminf zero. — Depends on missing premiseTwo simultaneousapproximation errors becomesmall…Current reduction — Depends on missing premiseCurrent reductionDiagonal-flow correspondence — Depends on missing premiseDiagonal-flow correspondenceExact hard core — Depends on missing premiseExact hard coreClosing target — Depends on missing premiseClosing targetLocal identity scope — Depends on missing premiseLocal identity scopeSymbolic ledger — Depends on missing premiseSymbolic ledgerVarying orbit limitation — Depends on missing premiseVarying orbit limitationTreating generic cubic-orbit shadowing as a contradiction — stoppedTreating generic cubic-orbitshadowing as a contradictionGenerate incompatible affine dynamics on one critical root fiber or prove the exact uniqueness identity that contradicts the inherited regularity boundary. — OpenGenerate incompatible affinedynamics on one criticalroot…Resolve central-dominant and exceptional large-cycle limits without using the local cocycle outside its identity chart. — OpenResolve central-dominant andexceptional large-cyclelimits…Obtain a quantitative unit or regulator bound strong enough for the generic varying-cubic-orbit branch. — OpenObtain a quantitative unitor regulator bound strongenough…Critical-corona closure — OpenCritical-corona closure
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeTreating generic cubic-orbit shadowing as a contradiction

The audit says the compact orbit varies with n and uniform regulator or amplifier control is missing. Critical-corona generation, central-dominant integral rank defects, exceptional-face analysis, and uniform arithmetic repulsion for generic cubic orders remain live.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Generate incompatible affine dynamics on one critical root fiber or prove the exact uniqueness identity that contradicts the inherited regularity boundary.Suggested move: Follow the six-step critical-corona protocol while recording common basepoints and composable arrows before every group-word identity.
Ready to work on
02
Resolve central-dominant and exceptional large-cycle limits without using the local cocycle outside its identity chart.Suggested move: Expand exact deck defects and adjugates, split by normalized rank, and seek a primal or dual coordinate-plane vector.
Ready to work on
03
Critical-corona closure

The primary live route seeks two distinct affine maps on a common root fiber, with exact basepoints and composability retained.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Obtain a quantitative unit or regulator bound strong enough for the generic varying-cubic-orbit branch.Suggested move: Compare an independently controlled unit displacement with the transverse approximation error before the latter is exhausted.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The universal Littlewood conjecture remains open. Einsiedler, Katok, and Lindenstrauss proved that its exceptional set has Hausdorff dimension zero; dimension zero does not imply that the set is empty.

[1]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedEinsiedler, Katok, and Lindenstrauss proved that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero using higher-rank measure rigidity.[1]
1 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusLittlewood conjecture
Dependency or reductionpositive diagonal action on SL(3,R)/SL(3,Z)

The modern dynamics approach relates exceptional pairs to bounded orbits and invariant measures for a higher-rank diagonal action.

[1]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA statement-aligned formalization must fix every ambient field, regularity, genericity, equivalence, and exceptional-set convention appearing in the external statement.
  • Formalization targetIt must formalize the exact prerequisite geometry, analysis, topology, or arithmetic library needed to express the theorem without replacing it by a weaker proxy.
  • Formalization targetNo problem-level formal statement or proof was identified in the bounded current Mathlib documentation and public Formal Conjectures repository search.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction2 of 82
  • lemma1 of 81
  • equivalence1 of 81
  • negative result2 of 82
  • computational claim1 of 81
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
  • retained route statementTwo simultaneous approximation errors become small enough that their n-weighted product has liminf zero.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact hard coreintermediate
  • retained route statementDiagonal-flow correspondenceintermediate
  • retained route statementLocal identity scopeintermediate
  • retained route statementSymbolic ledgerintermediate
  • retained route statementVarying orbit limitationintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureTreating generic cubic-orbit shadowing as a contradictionreported failure
  • Research targetGenerate incompatible affine dynamics on one critical root fiber or prove the exact uniqueness identity that contradicts the inherited regularity boundary.open
  • Research targetResolve central-dominant and exceptional large-cycle limits without using the local cocycle outside its identity chart.open
  • Research targetObtain a quantitative unit or regulator bound strong enough for the generic varying-cubic-orbit branch.open
  • Research targetCritical-corona closureopen
  • Narrowed routeTreating generic cubic-orbit shadowing as a contradictionThe audit says the compact orbit varies with n and uniform regulator or amplifier control is missing. Critical-corona generation, central-dominant integral rank defects, exceptional-face analysis, and uniform arithmetic repulsion for generic cubic orders remain live.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeGenerate incompatible affine dynamics on one critical root fiber or prove the exact uniqueness identity that contradicts the inherited regularity boundary.

The highlighted narrowed route is one recommended next task among several live source branches, including off-diagonal, critical-corona, central-dominant, exceptional-face, and varying-cubic work.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Read-only beta · actions unavailable
Prepared starting pointGenerate incompatible affine dynamics on one critical root fiber or prove the exact uniqueness identity that contradicts the inherited regularity boundary.

Littlewood Conjecture · ready to start

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Research contextPrepared context for any AI agent

For every real pair alpha and beta, must n times the product of their two nearest-integer errors become arbitrarily small?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

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Proof attempt or partial resultSupporting notes or data
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Sources and references1 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Invariant measures and the set of exceptions to Littlewood's conjecturepeer reviewed result · Manfred Einsiedler, Anatole Katok, Elon Lindenstrauss · Annals of Mathematics · 2006 · DOI 10.4007/annals.2006.164.513 · MR MR2247967 · accessed Aug 14, 2026

Important qualifications

  • External status was checked only against the listed primary, peer-reviewed, official-publisher, or author-hosted sources; omission is not a global nonexistence claim.
  • Special cases, reductions, preprints, and source-reported claims are kept at their exact scope and do not inherit proof, review, acceptance, publication, or priority authority.
  • The scoped formalization check found no problem-level statement or proof in current Mathlib documentation or the public Formal Conjectures repository; this is not evidence that no formalization exists elsewhere.
  • No source material, attachment, submitted URL, or packet computation was executed, rendered, fetched, or used as external authority.

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