Claims

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19 results

The branch of analysis concerned with Fourier transforms, oscillatory integrals, restriction estimates, maximal operators, and their open problems. Claims about conjectures, theorems, or techniques in harmonic analysis belong here; claims in unrelated branches of mathematics do not.

The restriction, Bochner–Riesz, and local smoothing conjectures each imply the Kakeya conjecture.

VerifiedThe evidence, examined directly, establishes the claim.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Kakeya conjectureimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The Fourier coefficients of the Syracuse random variable mod 3^n at frequencies not divisible by 3 decay faster than any power of n.

VerifiedThe evidence, examined directly, establishes the claim.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theory Probabilistic Collatz dynamicsimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The proof of the three-dimensional Kakeya conjecture is a once-in-a-century mathematical result.

SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Kakeya conjecture Wang–Zahl methodimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The jump from two to three dimensions is the hardest step in resolving the Kakeya conjecture in all dimensions.

SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Kakeya conjectureimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

Any counterexample to the three-dimensional Kakeya conjecture must be a grainy set.

SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Kakeya conjecture Wang–Zahl methodimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

Every Kakeya set in R^n has Hausdorff dimension at least (n+2)/2 (Wolff's 1995 bound).

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjectureimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The Wang–Zahl proof of the three-dimensional Kakeya conjecture can be adapted to prove the conjecture in higher dimensions.

SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjectureimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

Extending the Wang–Zahl method to Kakeya in dimensions four and higher faces primarily technical rather than fundamental obstacles.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjecture Wang–Zahl methodimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The three-dimensional Kakeya conjecture was a central open problem in harmonic analysis for about fifty years.

The Kakeya set conjecture remains open in dimensions four and higher.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjectureimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

Every set of delta-tubes pointing in delta-separated directions satisfies the polynomial Wolff axioms in every dimension.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjecture Polynomial Wolff axioms Convex Wolff axiomsimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The convex Wolff axioms version of the Kakeya conjecture fails in dimensions four and above.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjecture Convex Wolff axiomsimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The three-dimensional Kakeya conjecture has been proved: every three-dimensional Kakeya set has dimension three.

VerifiedThe evidence, examined directly, establishes the claim.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjectureimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

A small-volume union of tubes in R^3 whose points lie in tubes pointing in three quantitatively different directions clusters into rectangular slabs (grains) at an intermediate scale.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Wang–Zahl method Kakeya conjectureimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The local smoothing conjecture for the wave equation implies the Bochner–Riesz conjecture.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Local smoothing conjecture Bochner–Riesz conjectureimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The Bochner–Riesz conjecture implies the Fourier restriction conjecture.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Fourier restriction conjecture Bochner–Riesz conjectureimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The Fourier restriction conjecture implies the Kakeya conjecture.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjecture Fourier restriction conjectureimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The mathematical community accepts the Wang–Zahl proof of the three-dimensional Kakeya conjecture as correct.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjecture Wang–Zahl method Mathematicsimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

Hong Wang and Joshua Zahl posted an arXiv preprint in February 2025 claiming a proof of the three-dimensional Kakeya set conjecture.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Kakeya conjecture Wang–Zahl methodimportance · settledImportance 0.12, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

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