Number theory · continued fractions · thin semigroups

Zaremba’s Conjecture

Collaboration beta

Is there one universal bound on continued-fraction digits that works for a reduced fraction with every nontrivial denominator?

\exists M_{0} \geq 1 \forall q \geq 2 \exists n \geq 1 \exists a \in {1, \dots, q - 1} \exists (a_{1,} \dots, a_{n}) \in {1, \dots, M_{0}}^{n} : (a, q) = 1, a_{n} \geq 2, \frac{a}{q} = [0; a_{1,} \dots, a_{n}]

Known results and sources

$A bounded-tile continued-fraction staircase approaches a ring of denominator states through an incomplete modular lattice seam.$

Zaremba’s conjecture asks whether one bounded digit alphabet reaches every nontrivial denominator; the current route has a conditional local-limit bridge rather than an accepted closure.

Research problem

Exact mathematical statement

There should be an absolute integer $M_0\ge 1$ such that, for every nontrivial denominator $q\ge 2$ , some reduced fraction

\frac aq=[0;a_1,a_2,…,a_n],\qquad 1\le a<q,\quad (a,q)=1,

has every partial quotient bounded by $M_0$ . Equivalently for $q\ge 2$ , every such denominator should be the continuant of a nonempty finite word over one fixed alphabet $\{1,2,…,M_0\}$ .

The governing source literally says $q\ge 1$ while also requiring $1\le a<q$ , which is empty at $q=1$ , and it supplies no empty-word continuant convention in the bound excerpt. This reader projection therefore states the nontrivial case $q\ge 2$ and makes no unsourced equivalence claim for denominator one.

The retained source does not close this target. Its live route is conditional on a positive-weight local limit theorem, with three controlling open obligations. It also records a recent full-conclusion source claim as noncontrolling until a corrected argument receives a new independent audit.

Problem infographic

Problem at a glance

A small fixed palette of geometric digit tiles builds several finite continued-fraction staircases, each ending at a different denominator marker along one extending sequence with an unresolved horizon. — The conjecture asks whether one finite bound on continued-fraction digits can realize a reduced fraction for every denominator at least two.

Current mathematical picture

Where work on Zaremba’s Conjecture stands

Recent proof claim under review

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

Useful failureSource-claimed deterministic Zhang–Shkredov closure

The current work's line-by-line attempts were falsified by a determinant kernel, and it classifies the outline as source-claimed-noncontrolling until a new independent checking addresses the kernel and all quantifiers. Use Zhang's finite-group flattening only as a bounded operator input inside the separate positive-weight FLLT architecture, subject to exact theorem matching.

Route status · Narrowed route

Main reductionConditional fiber local-limit closure

The current work proves that its displayed positive three-shell local limit formula, with positive singular integral and sufficiently large local series, would imply Zaremba’s conjecture.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeClose O6: obtain positive pointwise major arcs for the exact nonnegative even-word three-shell ensemble.Task status · Ready to work on

Work mapped so far

Zaremba’s Conjecture in numbers

890retained lines of mathematical investigation890 in the current working snapshot

Argument development: 773 · 87%
Explored or eliminated routes: 23 · 3%
Computational analysis: 17 · 2%
Open obligations: 43 · 5%
Definitions and setup: 34 · 4%

6selected mapped statements2routes investigated6open questions6contribution-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 byStanisław Krystyn ZarembaProposed1972Status checkedAug 26, 2026

Source recordedAug 26, 2026

Recommended next task

Close O6: obtain positive pointwise major arcs for the exact nonnegative even-word three-shell ensemble.

Suggested move: Write the exact shell-theorem specialization, fix the norm centers through P-NORMCONT, establish a positive singular integral, and match every modulus, parity, cylinder, and smoothing parameter to a pointwise one-over-q main term.

What would count as progress

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

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

14 selected steps

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

Scroll horizontally to explore the route

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

2 recorded

Narrowed routeSource-claimed deterministic Zhang–Shkredov closure

Route status · Narrowed route

Narrowed routeOld O5 rectangular transport-to-one-point route

The determinant-kernel identity destroys the required injectivity, so O5 and the old conditional closure are quarantined from every live dependency chain. Continue with the even-word three-shell local-limit route, which keeps the exact rank-one fiber and separates archimedean, oscillatory, and local-factor work.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks

Close O6: obtain positive pointwise major arcs for the exact nonnegative even-word three-shell ensemble.Suggested move: Write the exact shell-theorem specialization, fix the norm centers through P-NORMCONT, establish a positive singular integral, and match every modulus, parity, cylinder, and smoothing parameter to a pointwise one-over-q main term.

Ready to work on

Close O7: prove a pointwise minor-arc saving at the full main scale without spending the entropy budget.Suggested move: Formulate the bilinear oscillatory inverse theorem after splitting the middle word, distinguish rational frequency from exact-divisor or integral-hyperplane concentration, and use analytic convolution powers only as an operator bound.

Ready to work on

O6 pointwise major arcs

The exact positive three-shell ensemble still needs a pointwise major-arc formula with a positive singular integral and a power-saving rational tail.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

O7 pointwise minor arcs

The minor arcs require an oscillatory inverse theorem that forces rational frequency or prohibited hyperplane concentration while preserving the main-scale entropy budget.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

O8 uniform local factors

Fixed-good-modulus convergence must be upgraded to uniform prime-power convergence, a summable good-prime Euler tail, and explicit lower bounds at exceptional components.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Close O8: prove uniform shell local factors over every prime and prime power, including the Euler tail and exceptional components.Suggested move: Upgrade fixed-good-modulus convergence to uniform prime-power control, establish the good-prime Euler estimates, and handle each finite exceptional connector component with explicit lower bounds.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 26, 2026

Current statusRecent proof claim under review

Recent proof claim under review. Zhang's May 2026 preprint explicitly claims a full confirmation of Zaremba's conjecture using Shkredov's framework, but the bounded search found no peer-reviewed resolution or authoritative independent acceptance. Peer-reviewed sources establish density-one and related continuant results while the full-proof claim remains preprint-only.

[2][3][4]

External progress

What the literature has established

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

2026-05-08
PreprintZhang's preprint claims a full confirmation of Zaremba's conjecture using expansion in modular groups and Shkredov's framework; the claim remains preprint-only in this snapshot.[5]
2026-03-14
PreprintShkredov's preprint, initially submitted on 2026-03-14 and revised as v2 on 2026-06-24, develops a new framework and reports bounded-partial-quotient results for primes and integers under additional conditions.[4]
2014
Peer reviewedBourgain and Kontorovich prove that a fixed bounded alphabet realizes a density-one set of denominators, a major partial result that does not cover every denominator.[2]
1972
Historical sourceZaremba's work on good lattices introduced the bounded-partial-quotient denominator problem associated with the conjecture.[1]

Mathematical neighborhood

Claims, milestones, and routes in the current map

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

4 standing statements2 proposed statements6 open questions2 narrowed routes

Statements by mathematical role6 selected mapped statements

theorem candidate1 of 61
reduction2 of 62
lemma2 of 62
negative result1 of 61

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 2 routes included

retained route statementOne finite continued-fraction alphabet should realize every nontrivial denominator.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementConditional fiber local-limit closureintermediate
retained route statementNorm–continuant and exact local-fiber clusterintermediate
retained route statementDeterminant-kernel quarantineintermediate
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
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 failureSource-claimed deterministic Zhang–Shkredov closurereported failure
Useful failureOld O5 rectangular transport-to-one-point routereported failure
Research targetClose O6: obtain positive pointwise major arcs for the exact nonnegative even-word three-shell ensemble.open
Research targetClose O7: prove a pointwise minor-arc saving at the full main scale without spending the entropy budget.open
Research targetClose O8: prove uniform shell local factors over every prime and prime power, including the Euler tail and exceptional components.open
Research targetO6 pointwise major arcsopen
Research targetO7 pointwise minor arcsopen
Research targetO8 uniform local factorsopen
ComputationThe current work reports an exact full modular rank-one fiber calculation and machine-readable theorem statuses, but those submitted materials were not executed during intake.ProofAtlas has not reproduced the reported calculation. It remains narrative, source-reported evidence pending separately authorized bounded execution and independent review. · reported unreproduced
Narrowed routeSource-claimed deterministic Zhang–Shkredov closureThe current work's line-by-line attempts were falsified by a determinant kernel, and it classifies the outline as source-claimed-noncontrolling until a new independent checking addresses the kernel and all quantifiers. Use Zhang's finite-group flattening only as a bounded operator input inside the separate positive-weight FLLT architecture, subject to exact theorem matching.
Narrowed routeOld O5 rectangular transport-to-one-point routeThe determinant-kernel identity destroys the required injectivity, so O5 and the old conditional closure are quarantined from every live dependency chain. Continue with the even-word three-shell local-limit route, which keeps the exact rank-one fiber and separates archimedean, oscillatory, and local-factor work.

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 bridgeClose O6: obtain positive pointwise major arcs for the exact nonnegative even-word three-shell ensemble.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

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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Prepared starting pointClose O6: obtain positive pointwise major arcs for the exact nonnegative even-word three-shell ensemble.

Zaremba’s Conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Is there one universal bound on continued-fraction digits that works for a reduced fraction with every nontrivial denominator?

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data

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Sources and references5 cited works · next context review by Nov 26, 2026

The mathematical context was checked on Aug 26, 2026. Status can be refreshed sooner after a material result or claim.

1
La méthode des ‘bons treillis’ pour le calcul des intégrales multiplesoriginal source · Stanisław Krystyn Zaremba · Academic Press · 1972 · DOI 10.1016/B978-0-12-775950-0.50009-1 · accessed Aug 26, 2026
2
On Zaremba’s Conjecturepeer reviewed result · Jean Bourgain, Alex Kontorovich · Annals of Mathematics · 2014 · ARXIV 1107.3776 · DOI 10.4007/annals.2014.180.1.3 · accessed Aug 26, 2026
3
Modular values of continuants with fixed prefixes and endingspeer reviewed result · I. D. Kan · Sbornik: Mathematics · 2026 · accessed Aug 26, 2026
4
On some results of Korobov and Larcher and Zaremba's conjecturepreprint · Ilya D. Shkredov · arXiv · 2026-03-14; v2 revised 2026-06-24 · ARXIV 2603.14116 · accessed Aug 26, 2026
5
Expansion in SL_2(Z/qZ) and Zaremba's conjecturepreprint · Xin Zhang · arXiv · 2026-05-08 · ARXIV 2605.02518 · accessed Aug 26, 2026

Important qualifications

The bounded primary/direct-source review establishes representative historical and current status evidence, not an exhaustive literature history.
Zhang's 2026 full-proof claim was assessed at direct preprint-record and abstract scope; this collection is not a mathematical audit of the proof.
No peer-reviewed publication or authoritative independent acceptance of the claimed full resolution was located in the bounded search.
No packet attachment, submitted URL, or source-reported computation was used as independent external status authority.
No statement-aligned formalization or independently reproduced computation was established by this scoped search.

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Zaremba’s Conjecture

Exact mathematical statement

Problem at a glance

Where work on Zaremba’s Conjecture stands

Zaremba’s Conjecture in numbers

Close O6: obtain positive pointwise major arcs for the exact nonnegative even-word three-shell ensemble.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

Compare another research frontier