Harmonic analysis · Wiener algebra · additive and spectral structure

Kahane’s Quantitative Beurling–Helson Conjecture

Collaboration beta

Must a continuous circle phase be affine whenever the Wiener norms of all its large integer powers grow more slowly than logarithmically?

| e^{in φ} |_{A (T)} = o (\log | n |) \Rightarrow φ (x) = mx + b (\mod 2 π)

Known results and sources

A circle phase, its integer powers, and their Fourier mass form the visual vocabulary of Kahane’s rigidity question.

Research problem

Exact mathematical statement

Let $A(\mathbb T)$ be the Wiener algebra. Kahane’s quantitative Beurling–Helson conjecture asks whether

\phi: \mathbb T\to\mathbb T\text{ continuous},\qquad \|e^{in\phi}\|_{A(\mathbb T)}=o(\log|n|)\quad\Longrightarrow\quad \phi(x)=mx+b\pmod{2\pi}

for some $m\in\mathbb Z$ and $b\in\mathbb R$ .

Problem infographic

Problem at a glance

Problem-centered plate for Kahane's quantitative Beurling–Helson conjecture. A continuous phase map on the circle is paired with the Wiener algebra norm, the hypothesis that the norm of e^{inφ} is o(log |n|), and the question whether φ must equal mx + b modulo 2π. An affine phase appears as a one-spike Fourier benchmark, while a general continuous phase is illustrated with spread Fourier mass; the rigidity question is marked open. — Let 𝕋 = ℝ/2πℤ and let φ:𝕋→𝕋 be continuous. Kahane's quantitative Beurling–Helson conjecture asks whether ‖e^{inφ}‖A = o(log |n|), where ‖f‖A is the sum of the absolute Fourier coefficients of f, forces φ(x) = mx + b modulo 2π with m ∈ ℤ. Affine phases provide the clean benchmark: every powered phase is a single Fourier mode and has Wiener norm 1. The general rigidity question remains open.

Current mathematical picture

Where work on Kahane’s Quantitative Beurling–Helson Conjecture stands

Open conjecture

Selected mathematical highlights through revision 12. The retained project now gives measurable global centered residuals, exact affine residual transport after block labels stabilize, lossless transfer from residual dictionaries to both endpoint dictionaries, and exact Toeplitz/translation formulas for canonical good-set concentration operators. It also corrects two overstatements: finite-subgroup recurrence yields residue-block diagonalization rather than a quotient by itself, and spectral transport must retain the set error, dimension, and spectral-gap parameter-and-error budget. The former arbitrary hidden-mode recycling frontier is superseded by a canonical moving-set transport-or-boundary theorem, with block stabilization, a positive-mass primitive core spine, and a large primitive Fourier packet as named subfrontiers. These are source-reported arguments, not independent review;…

Strongest supported footholdInnovation, power, and quotient budgets

Sequential trace innovation, powered information, and tensor visibility constrain transverse, balanced, and exactly invisible refinement branches.

Evidence posture · Reported result

Leading routeOverlapping-chart residual globalization

Retained overlapping-chart preprocessing, now governed by the revision-12 bridge audit and exact centered residual transport.

Route status · Active route

Useful failureLocalized hidden-mode packing

Historical revision-11 hidden-mode classification route; revision 12 supersedes it with canonical moving-set transport-or-boundary analysis.

Route status · Eliminated route

Main reductionCell-normalized residual lifting

The current work exposes and pays the p^{-1/2} normalization factor, then gives explicit exponential rank and row-trimming ledgers.

Evidence posture · Reported reduction

Priority open bridgeCertify the revision-12 bridge

Audit the measurable branch, centered residual transport, dictionary transfer, entropy aggregation, concentration-operator signs, perturbation constants, cyclic thresholding, tournament error, and localized core-anchor supply before further inverse work.

Task status · Prerequisites still open

Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Kahane’s Quantitative Beurling–Helson Conjecture in numbers

7.4kretained lines of mathematical investigation7,376 in the current working snapshot

Argument development: 6,355 · 86%
Explored or eliminated routes: 252 · 3%
Computational analysis: 57 · 1%
Open obligations: 211 · 3%
Definitions and setup: 501 · 7%

29selected mapped statements12routes investigated10reported milestones7open questions2contribution-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 byJean-Pierre KahaneProposed1962Status checkedAug 2, 2026

Recommended next task

Retain the core-halo geometric fallback

When hidden-mode abstraction stalls, turn ordered first exits into Ferrers structure, interlacing into crossings, dispersed transitions into jump payments, and one recurrent transition mode into rank two.

Suggested move: Use the synchronized two-probe family to test first-exit order before launching longer hidden-mode recursion.

What would count as progress

Each geometric branch is connected to a retained analytic exit or explicitly returned to hidden-mode recycling.

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

27 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.

Active routeOverlapping-chart residual globalization

Retained overlapping-chart preprocessing, now governed by the revision-12 bridge audit and exact centered residual transport.

Route status · Active route

Active routeConcentration-operator incidence

Track canonical concentration matrices and stable low spectral bands under exact set translation; seek bands rather than individual hidden eigenvectors.

Route status · Active route

Active routeCocycle transport and quotient preservation

Use exact affine residual transport and canonical set/band comparison before quotient or consecutive-power residue analysis.

Route status · Active route

Active routeSynchronized core-halo fallback

Use common core and halo probes to expose Ferrers order, crossing tensorization, dispersed transition payments, or rank-two recurrence.

Route status · Active route

Active routePrimitive-model stress tests

Use primitive finite-Fourier and finite-Blaschke examples to falsify overstrong revision-12 transport or stabilization claims and measure the current canonical-set, spectral-band, core, crossing, residue, and entropy observables.

Route status · Active route

Active routeCanonical moving-set transport or boundary

The active revision-12 route compares canonical good sets and their low spectral bands under exact affine residual transport, forcing stable block transport or a trace-visible moving-boundary certificate.

Route status · Active route

Explored alternatives

Other routes

6 recorded

Eliminated routeLocalized hidden-mode packing

Historical revision-11 hidden-mode classification route; revision 12 supersedes it with canonical moving-set transport-or-boundary analysis.

Route status · Eliminated route

Narrowed routeCommon hidden-subspace classification

Narrowed subroute: common-space recurrence is usable only after canonical block/band stabilization and must feed one revision-12 structural exit.

Route status · Narrowed route

Narrowed routeSparse-Fourier good-domain norming

Under the exact Turan-Nazarov normalization and an exponentially accurate O(L/log L)-dimensional Fourier packet, norming on partial good sets closes the branch.

Route status · Narrowed route

Browse 3 more explored routes

Refuted routeIndependent chart-phase quantization

A generic J-parameter phase mesh has the wrong state-count versus error rate and cannot serve as the full-profile endpoint.

Route status · Refuted route

Useful but insufficientGeneric low-rank net

Low rank e^{O(L)} without structure yields doubly exponential covering cost; translation, incidence, and localization are essential.

Route status · Useful but insufficient

Not yet justifiedFixed-anchor algebra closure

The exact product cocycle changes the anchor, so finite-dimensional algebra collapse cannot be invoked before proving valid base transport.

Route status · Not yet justified

Route statements and reductions

Statements the next route can inspect and build on

Route statementExact tensor invisibility equals quotient equality

Equality of every tensor-depth averaged translation state determines spatial measures after quotienting by the gcd of Fourier-support differences; in the primitive case the measures coincide.

Source-reported route statement · dependencies incomplete

Route statementSynchronized two-probe core-halo family

A noncharacter center class yields a positive-measure family of relative rows sharing one common core probe and one common visible halo sector at a controlled power.

Source-reported route statement · dependencies incomplete

Route statementJump-frame and Fourier-dual transition budgets

Operator-orthogonal transitions are bounded by the trace-class jump-frame inequality, while physically or frequency-dispersed transitions have a separate width-free Fourier-dual bound.

Source-reported route statement · dependencies incomplete

Route statementPolynomial fixed-tolerance overlapping-chart atlas

On a fixed connected cover of short overlapping shift intervals, all but exponentially small row mass has polynomially many coarse spatial states, exact row gauges, fixed chart centers, uniform fixed-tolerance control, and N^{-1+o(1)} control on retained chart good sets.

Source-reported route statement · dependencies incomplete

Route statementFinite-overlap gauge synchronization

If every chart residual in one coarse cell is globally small, positive-measure overlaps synchronize all chart phases into one row phase and fixed chart phases, producing one full projective profile state.

Source-reported route statement · dependencies incomplete

Route statementNormalized chartwise trace-class lifting

Each short chart-gauged residual splits into a trace-class kernel of inherited nuclear norm O_delta(L) plus arbitrarily polynomially small Hilbert-Schmidt error; normalization on a row cell of mass p costs p^{-1/2}, followed by explicit rank truncation and row trimming.

Source-reported route statement · dependencies incomplete

Route statementNorming-or-localized-hidden-mode dichotomy

For a residual near a finite-dimensional chart space, either its retained good set norms the space and globalizes the residual, or a low-concentration unit mode almost absent from the good set carries a visible part of the discrepancy.

Source-reported route statement · dependencies incomplete

Route statementExact affine residual transport

When two rows share stable source and target coarse-cell labels under translation, their centered residual difference is transported exactly by the translation operator; the remaining correction is fixed for the block transition.

Source-reported route statement

Route statementCanonical concentration operators transport with the good sets

The Fourier concentration operator of a translated canonical good set is unitarily translated exactly; approximate set transport controls a spectrally gapped band with an explicit square-root dimension and inverse-gap loss.

Source-reported route statement

Route statementCanonical moving-set transport-or-boundary frontier

Canonical moving good-set or level-set incidence together with exact affine residual transport must force stable set/block transport with a recurrent spectral band, or a trace-visible moving-boundary mixed-difference certificate, with the listed endpoint-compatible exits.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

7 featured tasks

Retain the core-halo geometric fallback

Suggested move: Use the synchronized two-probe family to test first-exit order before launching longer hidden-mode recursion.

Ready to work on

Stress-test revision-12 transport and stabilization claims

For primitive finite-Fourier and finite-Blaschke examples, measure the full revision-12 transport, concentration, entropy, core, crossing, residue, and residual-state observables before investing in an overstrong inverse theorem.

Suggested move: Run exact or high-precision primitive examples against the complete revision-12 observable list and retain reproducible inputs and outputs.

Ready to work on

Certify the revision-12 bridge

Suggested move: Check the ten revision-12 bridge items line by line against the retained exponential parameter-and-error budget.

Prerequisites still open

Prove canonical set transport or expose a moving boundary

From a positive-mass family of canonical good sets and visible low-concentration modes, force stable block/set transport with a useful spectral gap, or extract a quantitatively valid moving-boundary mixed-difference certificate.

Suggested move: Stabilize source/target block labels on a positive-mass shift family, then measure canonical-set symmetric differences and the spectral gap of the visible low band; otherwise convert label/set instability into one audited boundary certificate.

Prerequisites still open

Extract structure from the large primitive Fourier packet

Use Toeplitz concentration matrices, frequency amplitudes, difference multiplicities, residue classes, and moving low bands to derive norming, innovation, a sample-matrix lower bound, or independent crossings.

Suggested move: Measure low-band motion and residue structure in the actual primitive packet rather than applying a generic dimension argument.

Prerequisites still open

Prove block stabilization or extract a certificate

Construct globally consistent source/target blocks with controlled annuli, synchronized core phases, halo confinement, residue preservation, and O(L) refinement entropy, or immediately extract a stated boundary certificate.

Suggested move: Stabilize rowwise pair data into source/target blocks while preserving the shift-set mass and exact affine correction.

Prerequisites still open

Transport the positive-mass primitive core spine

Use the localized anchor supply to transport a positive-mass primitive core spine or force a visible phase, third-difference, stable-transition, spectral, rank-two, residue, or state-collapse exit.

Suggested move: Follow the large child from the localized anchor set and test every listed structural exit before recursion.

Prerequisites still open

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026

Current statusOpen conjecture

Kahane’s o(log |n|) threshold remains open in the located literature. The latest located theorem proves affine linearity under the stronger hypothesis O(log^(1/8−epsilon) |n|) for every epsilon>0.

[3][1]

External progress

What the literature has established

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

2026
PreprintSanders proved affine linearity under O(log^(1/8−epsilon)|n|) for every epsilon>0.[3]
2015
PreprintKonyagin and Shkredov improved the sufficient bound to o(log^(1/22)|n|/(log log |n|)^(3/11)).[2]
2012
PreprintLebedev proved affine linearity under growth o((log log |n| / log log log |n|)^(1/12)).[1]
1953
PreprintThe Beurling–Helson theorem gives affine linearity when the Wiener-algebra norms of the powers are uniformly bounded.[4][1]

Mathematical neighborhood

Related results and reusable starting points

Current focusKahane’s quantitative Beurling–Helson conjecture

Related problemBeurling–Helson theorem

Kahane asks whether the bounded-norm hypothesis can be relaxed all the way to o(log |n|) while retaining the same affine-linearity conclusion.

[1]

Related problemnonlinear piecewise-linear circle maps

Kahane observed that nonlinear piecewise-linear maps can have Wiener norm of order log |n|, so the conjectured little-o threshold would be sharp in scale.

[3]

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Canonical moving-set frontier isolatedThe former arbitrary hidden-mode classification target is superseded by a quantitative transport-or-boundary theorem for canonical good sets, stable block labels, and recurrent spectral bands.

Changed the research frontierLater mathematical revision

Research stage 13

Finite-subgroup conclusion narrowedThe retained cyclic-recurrence route now concludes only residue-block diagonalization; a quotient requires additional consecutive-power, tensor, or phase concentration.

Changed the research frontierLater mathematical revision

Research stage 12

Residual transport and endpoint bridges sharpenedRevision 12 adds exact global-centered-residual, affine-transport, residual-dictionary, and concentration-operator bridges that move the live obstruction from chartwise mode matching to canonical set and block transport.

Changed the research frontierLater mathematical revision

Research stage 11

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

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 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

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

Research-record correctionWe 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

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.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

10 mapped milestonesretained argument map

Browse all 10 mapped stages

stage 1Fixed-phase contradiction and partial-atlas model
stage 2Positive-mass profile rigidity
stage 3Unseparated endpoint extended to exceptional mass
stage 4Single recurrent mode routed to rank two
stage 5Innovation, power, and tensor budgets assembled
stage 6Anchored and geometric transition tools established
stage 7Overlapping-chart completion replaces independent phase quantization
stage 8Cell normalization and exceptional losses made explicit
stage 9Single-vector residual dichotomy removes the reference-row gap
stage 10Chart-indexed hidden-mode recycling isolated

Fixed-phase contradiction and partial-atlas modelThe current work reduces a hypothetical counterexample to one fixed zero-winding Wiener phase and supplies at most 2L^2 actual centers with quantitative row-dependent good domains on each shift interval.