Harmonic analysis · additive combinatorics · cosine polynomials

Chowla’s Cosine Conjecture

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For every finite set of positive integer frequencies, must the corresponding cosine sum have a negative value whose magnitude is at least a constant times the square root of the set size?

\exists c > 0 \forall A \subset ℤ_{> 0} finite, K (A) = - \min_{x} \sum_{a \in A} \cos (ax) \geq c \sqrt{| A |}

Erdős Problems, Problem 510Listed inBen Green's 100 Open Problems, Problem 81

Known results and sources

Positive integer frequencies, a negative cosine trough, residue cells, and open orbit motifs give the Chowla workspace a mathematical identity without claiming that the conjectured square-root bound is proved.

Research problem

Exact mathematical statement

For a finite set $A\subset\mathbb Z_{>0}$ , let $K(A)=-\min_x\sum_{a\in A}\cos(ax)$ . Chowla’s conjecture asks whether there is one absolute constant $c>0$ such that

\exists c>0\;\forall A\subset\mathbb Z_{>0}\text{ finite},\qquad K(A)\ge c\sqrt{|A|}.

the source reports substantial exact internal algebra and a narrowed proof route, but no complete proof or counterexample.

Problem infographic

Problem at a glance

A forest-green scientific plate begins with a finite comb of positive integer frequencies and overlays their cosine waves into one large sum curve. The curve crosses a zero line and reaches a marked negative trough whose depth is K(A); the central open question asks whether K(A) is always at least an absolute positive constant times the square root of the number of frequencies. Small insets note square-root-order examples and dilation invariance. — For a finite set A of positive integer frequencies, K(A) measures the depth of the most negative value of the cosine sum. Chowla’s conjecture asks whether K(A) is always bounded below by c√|A| for one absolute c>0. Square-root-order examples show that the exponent would be best possible; the general conjecture remains open.

Current mathematical picture

Where work on Chowla’s Cosine Conjecture stands

Open conjecture

This curated overview retains the current work's collision-to-quotient route, the revised-v10 oriented-fiber collapse of every nonempty near-transversal row to h=1, the exact cell-square, Fejer, entropy, prime-factor, and quotient-Haar charges, and the current large-loss arithmetic frontier. The centered-moment period module remains available only as an explicitly superseded consistency tool inside the pair branch. The source reports no complete proof or counterexample, and no claim here has proof, review, acceptance, or publication effect.

Strongest supported footholdEvery filtered odd pair stage receives a two-step payment

The second audit combines singleton-or-wrap with entropy to produce an exhaustive Fejer, Schur, or entropy alternative and parity-telescoping budgets.

Evidence posture · Reported result

Leading routeCollision-to-primitive-quotient route

Active collision energy is terminalized into teeth, then split between nonprimitive divisor ledgers and a popular primitive quotient row.

Route status · Active route

Useful failureNontrivial-period near-transversal branch

Eliminated as a current pair branch: oriented-fiber divisibility forces h=1 whenever the near-transversal row is nonempty. Nontrivial periods remain only in paid wrap branches.

Route status · Eliminated route

Main reductionBoundary paths split into Haar, long-path, and orbit branches

Exact boundary holes, filtered sections, spectral slack, and common orbit length give the current arithmetic trichotomy for odd rough rows.

Evidence posture · Reported reduction

Priority open bridgeNormalize quotient-Haar payments globally

Construct a scale-aware Bellman potential that makes local Delta_d payments and strict descendants telescope alongside entropy, Fejer, and Schur charges.

Task status · Prerequisites still open

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

Chowla’s Cosine Conjecture in numbers

5.9kretained lines of mathematical investigation5,898 in the current working snapshot

Argument development: 5,146 · 87%
Explored or eliminated routes: 98 · 2%
Computational analysis: 48 · 1%
Open obligations: 308 · 5%
Definitions and setup: 298 · 5%

22selected mapped statements12routes investigated9reported milestones6open questions

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 bySarvadaman ChowlaProposed1965Status checkedAug 2, 2026

Source recordedAug 2, 2026

Recommended next task

Normalize quotient-Haar payments globally

Construct a scale-aware Bellman potential that makes local Delta_d payments and strict descendants telescope alongside entropy, Fejer, and Schur charges.

Suggested move: Test a potential combining log(n/T^2), normalized entropy, scale-weighted quotient defects, and normalized Schur-triple mass.

What would count as progress

Prove every local Haar payment or retained descendant decreases one common bounded potential at the popular-row scale.

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

26 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 routeCollision-to-primitive-quotient route

Active collision energy is terminalized into teeth, then split between nonprimitive divisor ledgers and a popular primitive quotient row.

Route status · Active route

Active routeLarge singleton/interface fringe

Active current route for exceptional rows where C is much larger than tooth mass W; it must retain aperiodic teeth and missing self-sums.

Route status · Active route

Active routeSuperpopular perfect-cell row

Active exceptional-row route combining signed-residue distinctness, boundary holes, Haar slack, long paths, and complete orbit cosets.

Route status · Active route

Active routeLong signed-carry path

Active arithmetic route using filtered 2-3 sections and the exact affine recurrence while preserving signs, carries, and missing self-sums.

Route status · Active route

Active routeComplete signed 2-orbit closure

Active arithmetic route on complete cosets of the signed subgroup generated by two, with common orbit length and affine lifts retained.

Route status · Active route

Active routeNormalized Haar-entropy Bellman potential

Active global route intended to combine entropy, density, quotient-Haar, Fejer, Schur, and strict-descendant payments at compatible scales.

Route status · Active route

Explored alternatives

Other routes

6 recorded

Eliminated routeNontrivial-period near-transversal branch

Eliminated as a current pair branch: oriented-fiber divisibility forces h=1 whenever the near-transversal row is nonempty. Nontrivial periods remain only in paid wrap branches.

Route status · Eliminated route

Narrowed routeCentered-moment period module

recorded as an exact consistency and non-pair external-wrap audit module, but explicitly superseded as the main route inside the near-transversal pair branch.

Route status · Narrowed route

Narrowed routeNormal retentive entropy chain

Closed in its stated regime: rows with W comparable to n/T and pair defect O(W) cannot persist while normalized density is unbounded.

Route status · Narrowed route

Browse 3 more explored routes

Narrowed routeTwo-step Fejer-Schur-entropy ledger

Every odd h=1 pair stage with a following filter is assigned to one of three explicit parity-telescoping charges.

Route status · Narrowed route

Useful but insufficientCardinality payment alone

Too weak as a termination proof; retained only as one exact account inside a larger Bellman argument.

Route status · Useful but insufficient

Useful but insufficientGeneric polynomial-loss additive reductions

Rejected for the square-root target because polynomial losses in E overwhelm the relevant 1/E densities.

Route status · Useful but insufficient

Route statements and reductions

Statements the next route can inspect and build on

Route statementCollision forcing to terminal teeth

Under an unbounded normalized-density counterexample regime, the inherited collision argument supplies on the order of n^2/E terminal teeth d<b with d,b,b+d in A and 2b+d outside A, then concentrates primitive tooth mass W on a quotient row or produces a nonprimitive divisor ledger.

Source-reported route statement · dependencies incomplete

Route statementExact quotient interface split

For a complete primitive row d=hq, the quotient defect and external wrap satisfy one exact interface identity; substantial active or inactive wrap is assigned to existing telescoping payments, leaving the near-transversal inequality as the locally uncharged branch.

Source-reported route statement · dependencies incomplete

Route statementGenuine boundary-hole count

Small pair defect converts almost every low-occupancy signed-doubling boundary image into an exact quotient hole attached injectively to a missing self-sum 2b+d.

Source-reported route statement · dependencies incomplete

Route statementLong paths force a filtered 3-section

A sufficiently long no-wrap signed-doubling path in the odd h=1 branch forces a weighted occupied 3-section, a proper divisor descendant when 3 divides d, or an exact filtered zero.

Source-reported route statement · dependencies incomplete

Route statementQuotient-Haar payment

Low-occupancy boundary images and short signed-doubling paths are bounded by quotient spectral slack, which in turn is bounded by the ordinary quotient-Haar defect Delta_d.

Source-reported route statement · dependencies incomplete

Route statementExact signed 2-orbit cycles

Every directed cycle of perfect signed unit classes has the common length k_d=min{k>=1:2^k congruent to plus or minus 1 mod d} and is a complete coset of the signed subgroup generated by 2.

Source-reported route statement · dependencies incomplete

Route statementLarge-loss trivial-period closure

An exceptional h=1 row outside the normal retentive entropy regime must yield a square-root bound, a sharp quotient block, a summably controlled strict descendant, a no-wrap recurrence, a missing-self-sum payment, signed-carry closure, orbit closure, or normalized quotient-Haar payment.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks

Normalize quotient-Haar payments globally

Construct a scale-aware Bellman potential that makes local Delta_d payments and strict descendants telescope alongside entropy, Fejer, and Schur charges.

Suggested move: Test a potential combining log(n/T^2), normalized entropy, scale-weighted quotient defects, and normalized Schur-triple mass.

Prerequisites still open

Close superpopular perfect-cell rows

When TW/n is large and pair defect is moderate, combine boundary holes, Haar slack, filtered long paths, complete signed 2-orbits, and exact missing self-sums to obtain a one-row contradiction or strict descendant.

Suggested move: Optimize the path cutoff against spectral slack while keeping all perfect-cell multiplicities and missing values.

Prerequisites still open

Close the large singleton/interface fringe

When C is much larger than W, use the aperiodic lift, terminal teeth, missing self-sums, and quotient positivity to force a sharp quotient block, no-wrap interval, strict descendant, or Bellman-paid module.

Suggested move: Separate occupancy-one signed classes from structurally malformed cells and retain their exact missing self-sums through the quotient argument.

Prerequisites still open

Close complete signed 2-orbits

Use common orbit length, full orbit cosets, affine lifts, missing values 2b+d, and singleton fringe to force a common divisor, arithmetic spine, short quotient block, or autocorrelation contradiction.

Suggested move: Classify the complete orbit cosets together with their actual lifted missing self-sums and affine carries.

Prerequisites still open

Close long signed-carry paths

Extend the filtered 2-3 section output into a lossless recurrence, arithmetic relation, internal interval, or proper divisor descendant while preserving edge signs, affine carries, multiplicity, and missing self-sums.

Suggested move: Iterate the exact g=3 filter only while no-wrap hypotheses remain valid and record the signed affine state at every stage.

Prerequisites still open

Terminate even and small-prime descents

Integrate the exact almost-half-retentive and small-prime descendant bounds into a global descent that consumes prime factors, reduces maximum frequency, and preserves a uniform square-root constant.

Suggested move: Track consumed prime factors and maximum-frequency reduction together with entropy, Fejer, and quotient-defect charges.

Prerequisites still open

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026

Current statusOpen problem

For K(n), the least possible magnitude of the negative minimum of an n-frequency cosine sum, the conjectural and construction-sharp order is sqrt(n). The current bounds are n^(1/5-o(1)) << K(n) << n^(1/2), so the exponent gap remains open.

[2][10]

External progress

What the literature has established

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

2026
PreprintBedert's v3 improved the lower-bound exponent to 1/5-o(1).[2]
2025
PreprintJin, Milojević, Tomon, and Zhang obtained an independent polynomial lower bound through spectral graph theory.[1]
2010
Peer reviewedSanders proved structural results for symmetric subsets of discrete abelian groups and a polynomial bound in a restricted dense-frequency regime.[4]
2004
Peer reviewedRuzsa proved K(n) >= exp(c sqrt(log n)), the record lower bound until 2025.[6]

Mathematical neighborhood

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 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. 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.

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 1Collision forcing reaches a primitive quotient row
stage 2External-interface split isolates near transversality
stage 3Oriented fibers eliminate nontrivial near-transversal periods
stage 4Exact h=1 normal form and prime descent
stage 5Cell squares yield singleton-or-wrap payments
stage 6Entropy closes the normal retentive regime
stage 7Odd rough rows split into Haar, paths, and orbits
stage 8Centered-moment period work recorded as an archive
stage 9Two-step ledger assigns every filtered odd pair stage
stage 10Frontier refined to four exceptional arithmetic mechanisms

Collision forcing reaches a primitive quotient rowThe inherited route converts active-difference collisions into terminal teeth and then a primitive quotient row or a nonprimitive divisor ledger.