Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform

Longben Wei; Xingyu Zhao; Zhiwen Duan

REVIEW 3 major objections 5 minor 32 references

Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform

T0 review · 3 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read The paper proves a fractal uncertainty principle for the Fourier–Bessel transform: if position and frequency supports are scale-regular fractals, mass decays like $h^\beta$, uniformly in the translation parameter.

desk verdict Theorem 1.3 is a genuine, mostly proved extension of the Jaye–Mitkovski uniqueness theory to the Fourier–Bessel transform; Theorem 1.4's fractal uncertainty principle is not proved as written because the centered damping functions in Remark 4.5 are asserted, not constructed, and the natural symmetrizations fail. read the letter →

arxiv 2608.15126 v1 pith:5YJMYMHA submitted 2026-08-15 math.CA

Xingyu Zhao , Longben Wei , Zhiwen Duan This is my paper · ORCID

classification math.CA MSC 43A3228A80

keywords fractaluncertaintyprincipleFourier–BesseltransformHankelquantitativeuniquecontinuationscale-regularsetsquasi-analyticweightdampingfunctionsBesselasymptotics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper extends the fractal uncertainty principle—the statement that no $L^2$ function can be substantially concentrated on fractal sets in both position and frequency—from the classical Fourier transform to the Fourier–Bessel (Hankel) transform. Its main one-dimensional result says that if $X\subset[0,1]$ and $Y\subset[a,a+h^{-1}]$ are both scale-regular with exponent $\delta$ on the relevant scales, then any $f$ whose Hankel transform is supported on $Y$ satisfies $\|1_X f\|_{L^2_\nu}\le C h^\beta\|f\|_{L^2_\nu}$ for all $a\ge a_0 h^{-1}$, with constants independent of the translation $a$. Because the Hankel transform is not translation-invariant, the classical Fourier proof cannot be copied; the paper builds damping functions for translated regular sets and inserts them into a multiscale iteration. In arbitrary dimension it also proves a quantitative unique continuation estimate on relatively dense sets for functions whose Hankel transforms decay with a quasi-analytic weight. These are the two claims the reader should carry away.

What carries the argument

The central object is a damping function for a regular set $Y$: a function $\psi$ supported near the origin whose Hankel transform is bounded below by a positive constant on a neighbourhood of $Y$, decays faster than any polynomial globally, and decays against a superpolynomial weight along $Y$ itself. The construction goes through a band-limited multiplier obtained from a plurisubharmonic potential, and the paper's key move is to apply the construction to the translated set $Y-\eta$ and its reflection, producing damping functions whose constants do not depend on the translation center $\eta$. These damping functions localize $f$ through the generalized Hankel convolution, after which the large-argument asymptotic expansion of Bessel functions turns the high-frequency part of the estimate into a classical Fourier uncertainty estimate for the localized pieces. A multiscale iteration, using scale-wise porosity of $X$ to insert sharp weight functions, compounds the $h^\beta$ loss scale by scale.

What would settle it

Pick a concrete pair: $X$ a middle-thirds Cantor set scaled into $[0,1]$ and $Y$ the same Cantor construction dilated to length $h^{-1}$ and translated by $a$; compute $\|1_X f\|_{L^2_\nu}/\|f\|_{L^2_\nu}$ for $f=\mathcal H_\nu^{-1}(1_Y)$ as $h\to 0$ with $a$ growing like $a_0 h^{-1}$. The theorem predicts $O(h^\beta)$ decay with constants independent of $a$; any ratio that fails to decay, or whose constant grows with $a$ at fixed $h$, would falsify the uniform-in-$a$ claim.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.4: for $\nu>-1/2$ and $0\le\delta<1$, let $X\subset[0,1]$ be $\delta$-regular with constant $C_R$ on scales $h$ to $1$, and let $Y\subset[a,a+h^{-1}]$ be $\delta$-regular with constant $C_R$ on scales $1$ to $h^{-1}$. Then there exist $a_0,\beta,C>0$, depending only on $\delta,C_R,\nu$, such that every $f\in L^2_\nu(\mathbb R_+)$ with $\operatorname{supp}\mathcal H_\nu f\subset Y$ obeys $$\|1_X f\|_{$L^{2}$_\nu}\le C h^\$\beta$ \|f\|_{$L^{2}$_\nu},\qquad a\ge a_0 $h^{{-1}}$.$$ The load-bearing feature is uniformity in $a$: the translated location of the spectral fractal does not enter the constants, even though the Hankel transform lacks the translation invariance that makes the classical Fourier argument work. The paper's second claim, Theorem 1.3, is a higher-dimensional quantitative uniqueness statement: if the Hankel transform of $f$ decays against a unique continuation weight $W$, then $\|f\|_{L^2_\nu}$ is controlled by $\|f\|_{L^2_\nu(E)}$ for any relatively dense set $E$. The one-dimensional FUP is obtained from these ingredients by splitting the spectrum into high frequencies, applying Bessel asymptotics to reduce to a Fourier-type estimate, and iterating a scale-by-scale gain.

Load-bearing premise

The argument rests on the existence of damping functions for translated copies of the frequency fractal whose constants are independent of the translation; the paper cites earlier constructions for this step rather than proving it here.

Editorial extensions

If this is right

Theorem 1.4 supplies the first fractal uncertainty principle for the Fourier–Bessel transform in one dimension, with the translation-independent constants that any application to translated spectral fractals would require.
Theorem 1.3 gives a quantitative unique continuation statement: an $L^2_\nu$ function whose Hankel transform decays with a quasi-analytic weight cannot be small on any relatively dense set; its global norm is controlled by the norm on that set.
The crude $L^1_\nu\to L^\infty$ estimate diverges for $\nu>-\delta$, so the $h^\beta$ decay in Theorem 1.4 is a genuine improvement rather than a repackaging of the trivial bound.
The multiscale proof gives an explicit exponent $\beta\approx \gamma_0/(2T\ln 2)$, depending on scale-separation parameters, so the principle is quantitative rather than purely existential.
The restriction $a\ge a_0 h^{-1}$ means the theorem covers high-frequency spectral fractals; the low-frequency and higher-dimensional cases are explicitly not settled by the present argument.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the authors leave implicit is the many-window analogue: replacing the single spectral interval $[a,a+h^{-1}]$ by a union of well-separated translated regular sets, where the same damping-function construction should compound the loss with the number of components.
The same obstruction—no translation invariance—appears for other integral transforms whose kernels are Bessel-type functions, so the damping-function strategy should transfer to Jacobi or Dunkl transforms if their Paley–Wiener theorems support it.
If the uniform damping estimate can be pushed below the high-frequency threshold $a\ge a_0 h^{-1}$, the multiscale iteration would extend to low-frequency fractals, which the paper explicitly leaves open.

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Formalized claims in Lean

Claim #1: The paper's central claim is Theorem 1.4: for $\nu>-1/2$ and $0\le\delta<1$, let $X\subset[0,1]$ be $\delta$-regular with constant $C_R$ on scales $h$ to $1$, and let $Y\subset[a,a+h^{-1}]$ be $\delta$-regular with constant $C_R$ on scales $1$ to $h^{-1}$. Then there exist $a_0,\beta,C>0$, depending only on $\delta,C_R,\nu$, such that every $f\in L^2_\nu(\mathbb R_+)$ with $\operatorname{supp}\mat

/-- @claim 1 The paper's central claim is Theorem 1.4: for $\nu>-1/2$ and $0\le\delta<1$, let $X\subset[0,1]$ be $\delta$-regular with constant $C_R$ on scales $h$ to $1$, and let $Y\subset[a,a+h^{-1}]$ be $\delta$-regular with constant $C_R$ on scales $1$ to $h^{-1}$. Then there exist $a_0,\beta,C>0$, depending only on $\delta,C_R,\nu$, such that every $f\in L^2_\nu(\mathbb R_+)$ with $\operatorname{supp}\mat -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. Theorem 1.3, the multidimensional quantitative uniqueness estimate under quasi-analytic decay, is a real extension of Jaye–Mitkovski and of the authors' own band-limited result, and the Bessel-adapted derivative D_i = (1/2x_i)∂_{x_i} is the right tool. The overall Beurling–Malliavin/Denjoy–Carleman architecture is coherent and, apart from one exponent slip in the necessity part of Proposition 3.1, it goes through. Theorem 1.4 is another story: the high-frequency FUP is not established as written, because the centered damping functions carry the whole weight and are never actually produced.

The exact spot is Remark 4.5. Lemma 4.4 gives a damper centered at 0 for a set in [0,h^{-1}]. For eta ≈ a ≥ a0 h^{-1} you need an even Hankel multiplier with a lower bound at eta and W-decay on Y. Translating the lemma's damper does not give an even multiplier. Symmetrizing does, but the two natural symmetrizations fail: the sum loses the exponential decay on Y because the G(xi+eta) term is only polynomially small in 1/a, while the product loses the lower bound at eta because G(2eta) is small. The references cited, [3, Lemma 3.1] and [8, Prop 3.5], are for centered dampers, not for this uniform-in-eta even construction. Proposition 4.1 and Lemma 4.7 both rely on these psi_eta dampers, so the proof of Theorem 1.4 is incomplete at its advertised key step: the place where translation invariance is supposed to be bypassed.

Two more soft spots. Proposition 4.2 is stated for translated reflected sets and dismissed as a small modification of [3] or [8, Prop 3.5]; that needs a real proof or an exact citation. And the relation to Jaye–Sethi [21] is left unclear: the title of [21] is essentially Theorem 1.4, and the paper never says whether its result is independent, an improvement, or already contained there. The necessity part of Prop 3.1 also contains a power-counting error in the display after "for n ≥ n0"; it looks fixable, but as written the inequality does not balance.

Credit where due: the paper does not fit parameters or argue circularly. The external imports (Bourgain–Dyatlov, Cohen's PSH-BM/A-BM lemmas, the translated Fourier FUP) are legitimate, and the self-citations to [30] concern a band-limited predecessor, not the target theorem. Theorem 1.3 deserves to be in the literature.

Recommendation: send this to peer review, but make acceptance conditional on a complete proof of the centered damping construction — or a reference that genuinely contains it — and on an explicit novelty statement relative to [21]. A referee should not be expected to reconstruct Remark 4.5.

Referee Report

3 major / 5 minor

Summary. The paper studies quantitative uniqueness and a fractal uncertainty principle (FUP) for the one- and multidimensional Fourier–Bessel (Hankel) transform. Theorem 1.3 is a Logvinenko–Sereda-type estimate: if E is relatively dense with respect to the measure dμν and Hνf decays according to a quasi-analytic weight W, then the L2ν norm of f on the whole space is controlled by its L2ν norm on E, with an explicit necessity argument for the logarithmic-integral condition. Theorem 1.4 is the main FUP: for X⊂[0,1] that is δ-regular on scales h to 1 and Y⊂[a,a+h^{-1}] that is δ-regular on scales 1 to h^{-1}, one has ‖1_X f‖_{L2ν} ≤ C h^β ‖f‖_{L2ν} whenever supp Hνf is contained in Y, with constants independent of the translation parameter a for a ≥ a0 h^{-1}. The proof strategy is to construct translated and centered Hankel damping functions via Beurling–Malliavin theory, reduce the high-frequency estimate to the Fourier FUP through Bessel asymptotics, and then run a Bourgain–Dyatlov multiscale iteration.

Significance. If Theorem 1.4 is fully established, it is a genuine fractal uncertainty principle for the Fourier–Bessel transform and, in particular, gives h^β decay uniformly in the translation parameter a, which is the natural substitute for the lack of translation invariance of the Hankel transform. The paper is also useful in identifying the precise role of the obstacle: the damping functions must be centered at each lattice point while remaining even, since Hankel multipliers are even. The quantitative uniqueness part, Theorem 1.3, appears mostly self-contained and is interesting in its own right, extending the band-limited results of reference [30] to quasi-analytic decay. The remarks comparing the trivial L1–L∞ bound with the obtained high-frequency estimate are informative. However, the central damping construction for translated regular sets is not proved, and the proof of Proposition 4.1 contains an internal mismatch between the stated weight in the good/bad summation and the weight required by Lemma 4.6; these gaps are load-bearing for Theorem 1.4.

major comments (3)

[Section 4.2, Lemma 4.4 and Remark 4.5] Lemma 4.4 supplies the damping functions on which the whole proof of Theorem 1.4 rests, but its proof is deferred to references [3, Lemma 3.1] and [8, Proposition 3.5], both of which are statements in the Fourier setting. More seriously, Remark 4.5 asserts, without proof, that for every η in the lattice Λ=c2N the same construction applied to Y−η and its reflection gives an even entire multiplier Fη with the uniform bounds |Fη(ξ)| ≥ c2 on B_{c2}(η) and |Fη(ξ)| ≤ 1/W(|ξ−η|) on Y. For the Hankel transform the multiplier must be even in ξ. The two natural symmetrizations fail: Fη(ξ)=G(ξ−η)+G(ξ+η) does not have the required exponential decay on the far part of Y, since G(ξ+η) is evaluated far outside the regular set and is only polynomially small there; while the product Fη(ξ)=G(ξ−η)G(ξ+η) restores decay but loses the uniform lower bound at ξ=η because G(2η) need not be bounded below independently of η. Thus the stated argument does not establish the uniform-in-η damping property, and both Proposition 4.1 and Lemma 4.7, which rely on it, are left unsupported. A different centered construction, or a proof that a symmetrized multiplier satisfies both bounds uniformly in η, is needed.
[Section 4.2, Proposition 4.1, equations (4.22) and (4.17)] The good/bad summation in the proof of Proposition 4.1 does not close as written. Good indices are defined in (4.22) by the condition ||W(|η−·|)^{1/2} Hν fη||_{L2ν} ≤ A ||fη||_{L2ν}, but Lemma 4.6 is invoked with hypothesis (4.17), which requires ||W(|η−·|) Hν fη||_{L2ν} ≤ A ||fη||_{L2ν}. Since W(t) ≥ 1, the condition (4.22) is weaker than (4.17), so the hypotheses of Lemma 4.6 are not satisfied. The proof of Lemma 4.6 also uses a Cauchy–Schwarz splitting with W and with the integral of W^{-2}; to be compatible with (4.22) it would need the splitting with W^{1/2} and the integral of W^{-1}. This is likely repairable by changing the weights consistently, but as written the argument is internally inconsistent and Proposition 4.1 does not follow.
[Section 4.2, Proposition 4.2] Proposition 4.2 is a key input in Lemma 4.6: the Fourier reduction requires inequality (4.15) for the two-component set Y=(Y0+a)∪(−Y0−a) with a constant independent of a. The proof is not given; the text only says that a slight modification of reference [3, Proposition 3.3] or an argument in reference [8, Proposition 3.5] gives it. Since [3, Proposition 3.3] is stated for a set on a fixed interval and the parameter a here may be as large as h^{-1}, the uniformity in a is not a formal consequence of translation invariance alone: the regularity conditions are imposed on absolute scales 1 to h^{-1}, and the two components of Y are separated by a distance that grows with a. The full statement and proof, or an exact reference containing precisely this statement with the required uniformity, should be supplied.

minor comments (5)

[Section 4.2, proof of Lemma 4.6, Step 4] In Step 4 the integral I6 is written as the integral over E without a prior definition of the set E; it should presumably be S, and the integration should be over S intersected with the positive half-line in the region where the Hankel inversion formula is applied.
[Section 4.2, Proposition 4.1 and Lemma 4.6] The definition I={[j,j+1] with j in Z} is used for sets S in the one-dimensional Hankel setting, where the underlying space is the positive half-line. Intersecting S with R+ would avoid ambiguity about negative intervals that play no role for functions on R+.
[Section 3, Proposition 3.1, necessity part] The final displayed estimate in the necessity part, W(|ξ|) ≤ inf_n |ξ|^{n+1}/M_n ≤ C' divided by |Hνf(ξ)| times the product of |ξ_i|^{-n0}, is not immediate from the preceding product bounds. The cancellation between the infimum and the product estimate should be spelled out, or a supporting lemma from reference [20] quoted explicitly.
[Section 2.2, Lemma 2.3] In the converse direction of the higher-dimensional Paley–Wiener theorem, the text asserts that the partial inverse h1(x1,z′) is entire in z′ and retains the same exponential type after applying the one-dimensional result for almost every z′. A short justification of the measurability in z′ and uniformity of the exponential type would improve the rigor of the iteration.
[Section 4.3, Lemma 4.7] After rescaling by 2^{-mT}, the spectral support is only known to lie in the intersection of 2^{-mT}Y+[−2,2] with the positive half-line. The text should specify at which scale this expanded set is δ-regular and which value of h is used in Proposition 4.1 at the m-th step; otherwise the application of Proposition 4.1 is not immediate.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the main FUP proof reduces to external Fourier FUP and BM-damping results, not to its own target, and only a minor non-load-bearing self-citation to [30] appears.

full rationale

The derivation of Theorem 1.4 is an external-lemma chain: PSH-BM and A-BM are imported from [7]; the translated Fourier FUP (Proposition 4.2) is asserted by modification of [3, Proposition 3.3] and [8, Proposition 3.5]; damping functions for regular sets (Lemma 4.4 and Remark 4.5) are deferred to [3, Lemma 3.1] and [8, Proposition 3.5]; and the Bessel-to-Fourier reduction in Lemma 4.6 follows [21]. None of these inputs contains the target Hankel FUP, and no parameter is fitted to the conclusion. The only self-citation is [30], which is used for background and in Remark 1.7 as an explicitly unsuccessful alternative; Theorem 1.3's proof does not depend on [30]'s theorem. The genuine weakness is that Remark 4.5 asserts, without proof, that translating and reflecting the centered construction gives uniform-in-eta centered dampers; if untrue, Proposition 4.1 and Lemma 4.7 fail. That is an omitted proof and a correctness gap, not a circular reduction: the asserted damper is an input from external sources, not the target inequality itself. Similarly, the mismatch between W in (4.17) and W^{1/2} in (4.22) is a technical error, not circularity. Hence score 2 reflects a minor, non-load-bearing self-citation and external-reliance concerns, with no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted parameters. All constants are universal existence constants depending on delta, C_R, nu, and the dimension. The load-bearing imported inputs are the Cohen BM lemmas, the damping-function existence asserted in Lemma 4.4, and the translated Fourier FUP of Proposition 4.2.

assumptions (5)

ad hoc to paper Existence of damping functions for delta-regular sets in the Fourier-Bessel setting (Lemma 4.4).
This is load-bearing for Proposition 4.1 and Theorem 1.4, but its proof is deferred to [3, Lemma 3.1] and [8, Proposition 3.5] rather than given.
domain assumption Translated sets Y-eta and their reflections remain delta-regular with the same constant C_R.
Used in Remark 4.5 to build damping functions centered at eta; this translation invariance of regularity is asserted, not proved.
domain assumption Fourier FUP for translated and reflected sets (Proposition 4.2).
Stated as a variant of Bourgain-Dyatlov with proof omitted; it controls the Fourier-type integral I_3 in Lemma 4.6.
standard math Cohen's plurisubharmonic and analytic Beurling-Malliavin lemmas in dimension d.
Imported from [7] without proof; used to construct exponential-type multipliers and damping functions.
standard math Denjoy-Carleman characterization of quasi-analytic classes.
Used in Section 3 to obtain quantitative unique continuation on good cubes.

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Pith. "Pith review of Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform." pith.science (2026). https://pith.science/paper/5YJMYMHA

@misc{pith2026260815126,
  author       = {Pith},
  title        = {Pith review of: Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YJMYMHA}},
  note         = {Machine review of arXiv:2608.15126}
}

abstract

We establish quantitative uniqueness and a fractal uncertainty principle for the Fourier Bessel transform. In arbitrary dimension, we prove a quantitative uniqueness estimate on relatively dense sets for functions whose Fourier Bessel transforms decay according to a quasi-analytic weight. In dimension one, if $X\subset[0,1]$ and $Y\subset[a,a+h^{-1}]$ are$\delta$-regular on the relevant scales, then, for $a\geq a_0h^{-1}$, \[ \operatorname{supp}\mathcal H_\nu f\subset Y \quad\Longrightarrow\quad \|\mathbf 1_Xf\|_{L^2_\nu} \leq Ch^\beta\|f\|_{L^2_\nu}. \] The lack of translation invariance prevents a direct application of the classical Fourier argument. We overcome this by constructing damping functions adapted to translated regular sets and combining Beurling Malliavin multipliers with large-argument Bessel asymptotics and a Bourgain Dyatlov multiscale iteration.

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [3]

and Dyatlov, S.: Spectral gaps without the pressure condition.Ann

Bourgain, J. and Dyatlov, S.: Spectral gaps without the pressure condition.Ann. of Math.187, 825–867 (2018)

work page 2018

[2] [21]

and Sethi, R.: A high-frequency uncertainty principle for the Fourier-Bessel transform.J

Jaye, B. and Sethi, R.: A high-frequency uncertainty principle for the Fourier-Bessel transform.J. Fourier Anal. Appl.32(3), Paper No. 48, 22 pp. (2026)

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