Fractal Tur\'{a}n-Nazarov Inequality and Observability for Schr\"{o}dinger Equations

Jiaqi Yu; Shanlin Huang

REVIEW 2 major objections 5 minor 1 cited by

Fractal Tur\'{a}n-Nazarov Inequality and Observability for Schr\"{o}dinger Equations

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For sets of positive α-Hausdorff content with α<1, the Turán–Nazarov constant must depend on the frequency spread with sharp exponent, and this forces Schrödinger observability and unique continuation to fail on fractal sets.

desk verdict First genuine fractal Turán–Nazarov bound with sharp bandwidth dependence, and the applications to Schrödinger observability are clean; two proof gaps in Lemma 2.1 need referee attention before I'd trust the contraction. read the letter →

arxiv 2607.17505 v1 pith:5J4M3PQ2 submitted 2026-07-20 math.AP math.CA

Jiaqi Yu , Shanlin Huang This is my paper · ORCID

classification math.APmath.CA MSC 42B1035A02

keywords Turán–NazarovinequalityHausdorffcontentfractalsetsobservabilitySchrödingerequationuniquecontinuationtrigonometricpolynomialsfrequencybandwidth

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's target is a fractal version of the Turán–Nazarov inequality, which in its classical form says that a trigonometric polynomial with n frequencies, measured on any subset of the circle of positive length, is controlled everywhere by a constant that depends only on n and that subset—not on which frequencies appear. The authors prove that this uniform control is impossible once the subset is fractal: for every 0<α<1 and every set with positive α-Hausdorff content, a uniform bound fails, and the best possible control constant must grow with the frequency bandwidth m_n−m_1, raised to the sharp power 1/α−1. This refinement is then used to build explicit counterexamples showing that the sup-type observability inequality and the two-time unique continuation inequality for Schrödinger equations fail on fractal observation sets of Hausdorff dimension less than 1, even for smooth initial data. The heart of the proof is a new estimate on the α-Hausdorff content of superlevel sets of the logarithmic derivative of polynomials, which carries the fractal dimension into the inequality. If the paper is right, control theory for Schrödinger equations on rough sets is fundamentally different from the heat equation, where such fractal sets still allow observability.

What carries the argument

The load-bearing mechanism is Lemma 2.1: for an algebraic polynomial P of degree m and H>0, the α-Hausdorff content of the superlevel set {z∈T: |d/dz log P(z)| > H} is bounded by C0 m H^{−α}. This replaces the measure-based level-set estimate used in the classical case and is precisely where the fractal dimension α enters. Around it, the proof runs a descent argument: from a trigonometric polynomial p with n frequencies, form p_{k−1} by differentiating z^{−r_1}p_k (or z^{−r_k}p_k) and normalising, so the degree drops by one while at least half the ℓ1-norm of the coefficients is preserved. The product of the ratios |p_{k−1}/p_k| over the observation set is then controlled by Lemma 2.1 through

What would settle it

Compute the α-Hausdorff content of the superlevel set {e^{it}: |(z^m−1)'/(z^m−1)|>m} for P_m(z)=z^m−1 with H=m. Lemma 2.1 predicts the content is at most C0 m^{1−α}; if a numerical covering estimate shows growth like m^{1−α/2} instead, the lemma is false and Theorem 1.2(ii) collapses. This is a direct test because the corresponding level-set rate in the complex plane is known to be exponent −α/2.

Watch

Extended reading notes

Core claim

Theorem 1.2 is the central discovery. It says (i) no inequality of the form sup_{[0,1]} |p| ≤ C sup_E |p| can hold uniformly over all trigonometric polynomials of degree n and all subsets E⊂[0,1] with C_H^α(E)>0; (ii) for every such E and every p(t)=Σ c_k e^{2π i m_k t} with m_1<...<m_n, one has sup_{[0,1]} |p| ≤ [ (C0(n−1)/C_H^α(E))^{1/α} (m_n−m_1)^{1/α−1} ]^{n−1} sup_E |p|, and the exponent 1/α−1 on the bandwidth cannot be improved. The positive estimate is proved via a chain of polynomial differentiations together with a logarithmic-derivative level-set bound on the circle; the negative part uses lacunary frequencies N_j and a Cantor-type set E on which sin(2πN_j t) decays to zero. These

Load-bearing premise

The entire positive half of the paper rests on Lemma 2.1: that for every degree-m polynomial, the α-Hausdorff content of the superlevel set of its logarithmic derivative on the circle is at most C0 m H^{−α}; if the true exponent were only −α/2, the contraction in the proof of (1.7) would break.

Editorial extensions

If this is right

Taking α→1 in (1.7) recovers the classical Turán–Nazarov inequality up to an absolute constant, so the fractal result is an extension of the classical one rather than a replacement.
Because the constant in (1.7) must grow like (m_n−m_1)^{(n−1)(1/α−1)}, any control or observability statement for band-limited functions on a fractal set must pay a cost that increases with the frequency bandwidth—something the classical inequality avoids.
On the torus, Theorem 1.5 rules out sup-type observability (1.11) for all positive α-Hausdorff-content observation sets with α<1, even when the initial data are C∞; only sets of full dimension can be observable in this sense.
On the real line, Theorem 1.7(ii) constructs α-thick fractal pairs E_1,E_2 for which the two-time unique continuation inequality (1.17) fails, while Theorem 1.7(i) shows the same inequality holds when the complements are ε-thin.

Reading between the lines

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

A natural extension the authors do not spell out: the same descent-chain mechanism should transfer to any dispersive equation whose solution map is a Fourier multiplier with polynomial phase, so the failure of observability on fractals is likely generic for Schrödinger-type equations rather than special to the flat torus.
The sharp bandwidth exponent 1/α−1 suggests a quantitative rule of thumb—to resolve a fractal of dimension α from band-limited data one needs a frequency window of size roughly cost^{α/(1−α)}; this could be tested numerically by computing optimal constants for spectral projectors onto low frequencies on Cantor sets.
The circle analogue of the level-set estimate has exponent −α, whereas the plane analogue has sharp exponent −α/2; locating a circle-version counterexample family with exponent −α/2 would directly refute Lemma 2.1 and with it Theorem 1.2(ii), so the gap between C and T is the place to stress-test the proof.
The contrast with heat equations suggests a structural principle: observability from fractal sets depends on whether the evolution regularises initial data (heat, where codimension-one fractals are observable) or merely propagates phase (Schrödinger, where they are not); this would predict similar failures for wave equations with finite speed—a question the paper leaves open.

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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

Key take: this is the first genuine fractal Turán–Nazarov bound with an explicit, sharp bandwidth factor, and it delivers what it promises—negative observability and unique continuation for Schrödinger on α-dimensional sets. The Cantor-type construction for failure of the uniform inequality is sound, the dimension computation checks out, and the applications to observability are clean. This deserves a serious referee.

What's new: Theorem 1.2(i)-(ii), especially the sharp exponent 1/α−1 in the bandwidth dependence. The qualitative failure of sup-type observability for C∞ data on T (Theorem 1.5) and the α-thick counterexamples in R (Theorem 1.7(ii)) are new and follow from the construction.

Where it's soft.

The load-bearing Lemma 2.1 has two problems. First, its proof invokes [26, Lemma 1.2] for the measure bound on T. The paper's Remark 2.2 says the α=1 result on T is in [26] with constant π/8, but (2.25) uses 10/π mH^{-1} for level 0.8H. Those constants don't line up, and the exact statement of the T-version isn't quoted. The stress-test note says [26, Lemma 1.2] is a real-line result; I can't confirm without the reference, but the authors should state and prove the unit-circle version or give a precise pointer. This is checkable, not necessarily fatal.

Second, the q/q̃ WLOG is a genuine gap. Step 1 says we may assume the q case; Step 2 uses the q̃ identity |p_{k-1}/p_k| = |g'(1/z)/(r^{(k)}g(1/z))|. For q you get |1 − g'(1/z)/(z r^{(k)} g(1/z))|, which is not a level set of the logarithmic derivative, so Lemma 2.1 doesn't directly apply. This is central to the contraction in (2.40). I suspect it's patchable—for the large threshold used here, |1−A| > θ implies |A| > θ−1, which puts you back in a level set up to a harmless constant—but the paper doesn't do that, and the WLOG isn't WLOG because the norm inequality for q̃ may fail. A referee should ask for this to be rewritten.

Step 4 (sharpness) is also compressed; the parameter chases around (2.41) are hard to verify. Minor typos like the missing √(2π) in (3.17) don't affect the decay argument.

Bottom line: the central claim is likely correct and the applications are honest. The paper needs a careful revision of Lemma 2.1's proof and the q/q̃ step. I'd send it to a knowledgeable referee and expect it to come back after those fixes. For a reading group, it's a good example of how sharp constants in quantitative unique continuation change on irregular sets.

Referee Report

2 major / 5 minor

Summary. The paper studies fractal analogues of the Turán–Nazarov inequality for trigonometric polynomials p(t)=∑ c_k e^{2π i m_k t}. It proves that on sets E⊂[0,1] with positive α-Hausdorff content, no uniform bound sup_T |p| ≤ C sup_E |p| can hold with C depending only on E and the degree, but that a bound with an additional frequency-bandwidth factor (m_n−m_1)^{1/α−1} is valid and sharp. The proof is based on a Cartan-type lemma for logarithmic derivatives of algebraic polynomials on the unit circle. These estimates are then used to construct explicit counterexamples showing failure of sup-type observability for Schrödinger equations on the torus and failure of a unique-continuation inequality on R for initial data in C^∞, for observation sets of Hausdorff dimension less than 1.

Significance. If the main theorem stands, this is a valuable contribution: it is the first fractal version of the Turán–Nazarov inequality with sharp dependence on the frequency bandwidth, and it gives an explicit, constructive contrast with the heat equation, where observability holds for sets of sufficiently large fractal dimension. The test functions and Cantor-type sets are explicit, and the sharpness argument is concrete. The main risk is not the architecture of the proof but a specific gap in the key contraction argument for the logarithmic-derivative lemma; this gap is local and appears repairable, so the central claims are plausibly correct.

major comments (2)

[§2.2, Step 1–2, Eqs. (2.34), (2.37)–(2.40)] The proof of the key inequality is internally inconsistent at the point where p_{k-1} is defined. Step 1 chooses the first case q(z)=d/dz(z^{-r_1}p_k(z)) and sets p_{k-1}=q/r(k). But Step 2's equality chain uses z^{-r_k}p_k(z)=g(1/z), which is the identity for the second case q̃, and gives |p_{k-1}/p_k|=|g'(1/z)|/r(k)|g(1/z)|. For the q-case the correct ratio is |h'(z)|/r(k)|h(z)| with h(z)=z^{-r_1}p_k(z). Lemma 2.1 applies to both h and g, so the argument can be repaired, but as written the displayed derivation of (2.40) is not valid for the polynomial actually constructed. Please rewrite Step 1–Step 2 to treat both alternatives explicitly or justify a reduction from one to the other.
[§2.2, Lemma 2.1, Eq. (2.25)] The proof of Lemma 2.1 relies on the estimate μ({z∈T: |Σ 1/(z-z_j)|>0.8H}) ≤ (10/π)mH^{-1}, quoted from [26, Lemma 1.2]. Remark 2.2 indicates that [26] contains both a unit-circle and a real-line version, but the proof would be much clearer if the exact statement for T were stated and attributed. If [26, Lemma 1.2] is in fact only for R, the transfer to T is not automatic and needs a proof. Since this measure estimate is the only external input in Lemma 2.1, please clarify the precise source and domain of (2.25).

minor comments (5)

[§2.2, Step 1] The two auxiliary polynomials are both written as q(z); the second should be q̃(z) throughout, including in the sentence introducing them and in (2.37)–(2.40).
[§2.2, Step 4] The sharpness proof is under-specified: after fixing ε_0, the text imposes conditions on M and ε that depend on α̃ and ε_0, but does not give the order of choices. Since the theorem claims sharpness for every α∈(0,1), please state explicitly how to choose α̃∈(α,1) and c=1/M so that (2.41) holds.
[§2.2, Remark 2.2] The notation C1/H should presumably be C_H^1, and the phrase 'C_H^α' in the complex-plane display is confusing; use C_H^α consistently and specify the ambient dimension in the Hausdorff content.
[§3.2, Theorem 1.7(ii)] The assertion that dim_H(E_1∩[x,x+π])=α̃ uniformly in x and that inf_x C_H^α(E_i∩[x,x+L])>0 is plausible from the periodic construction but is stated without proof. A short justification, or a reference to the exact step in the proof of Theorem 1.2 that gives uniformity, should be added.
[General] There are numerous typographical errors ('whcih', 'uncertatinty', stray asterisk, inconsistent spacing in 'Schr¨odinger'). These should be corrected in revision.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central derivation rests on independent external estimates and explicit constructions; the only self-citation [19] is non-load-bearing background/contrast.

full rationale

The paper's central inequality (1.7) is proved from Lemma 2.1, whose proof invokes Nazarov's measure estimate [26, Lemma 1.2] and the Eiderman--Nazarov--Volberg result [11, Thm 2.4] -- both external and neither by the present authors -- plus standard covering and Besicovitch arguments. No fitted parameter is introduced and no prediction is a renamed input: Lemma 2.1 is a genuinely new unit-circle estimate, and the contraction in (2.40) applies it with an explicit H chosen from the claimed bound, which is a legitimate proof step rather than a re-use of the conclusion. The counterexamples in Theorem 1.2(i), Theorem 1.5, and Theorem 1.7(ii) are explicit lacunary constructions whose decay on the constructed sets is verified directly, so they do not assume their conclusions. The only self-citation [19] appears as background/contrast in the introduction, Definition 1.6, and Remark 1.8, where it is used to recall heat-equation notions; it is not load-bearing for the new Schrödinger negative results. The possible unit-circle transfer gap in Lemma 2.1 and the q̃/q notational WLOG issue are correctness concerns, not circular reductions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No empirical or fitted parameters appear; all constants in the construction (α̃, c, δ, β, M) are explicit proof variables chosen to satisfy displayed inequalities. The central claims import three external results: Falconer's dimension formulas, Nazarov's measure bound, and Kovrizhkin's uncertainty principle, plus the standard free-Schrödinger formula. No new physical entities or mediators are introduced.

assumptions (4)

standard math Falconer's dimension estimates for generalized Cantor sets ([13, Ex. 4.6/Prop. 4.1])
Used in Step 2 of Theorem 1.2(i) to compute dim_H E = α̃ from m_k and ε_k; if this computation failed, C_H^α(E) > 0 for α < α̃ would not follow.
standard math Nazarov's Lebesgue-measure bound for logarithmic derivatives on the unit circle ([26, Lemma 1.2])
Used inside the proof of Lemma 2.1 to upper-bound µ(G) by C m H^{-1}; the fractal α-content bound is derived from this one-dimensional measure estimate.
domain assumption Kovrizhkin's uncertainty principle for ε-thin sets ([22, Theorem 1.1])
Powers the positive half of Theorem 1.7(i); the paper verifies its scaling conditions but the principle itself is imported unchanged.
standard math Standard free-Schrödinger solution formula (3.8)
Used in Theorem 1.7 to relate the Fourier transform of the initial data to the solution at time T = 1/2.

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Cite this review

Pith. "Pith review of Fractal Tur\'{a}n-Nazarov Inequality and Observability for Schr\"{o}dinger Equations." pith.science (2026). https://pith.science/paper/5J4M3PQ2

@misc{pith2026260717505,
  author       = {Pith},
  title        = {Pith review of: Fractal Tur\'an-Nazarov Inequality and Observability for Schr\"odinger Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5J4M3PQ2}},
  note         = {Machine review of arXiv:2607.17505}
}

abstract

This paper establishes limitations on observability inequality and unique continuation for Schr\"{o}dinger equations on fractal sets. We prove that, in contrast to the heat equation, such properties can fail in fractal settings. To achieve this, we first extend the classical Tur\'{a}n--Nazarov inequality, which provides lower bounds of trigonometric polynomials of the form $\sum_{k=1}^nc_ke^{2\pi im_kt}$ on sets of positive measure, to the fractal setting. Unlike in the classical case, the constant in the inequality loses uniformity in the degree $n$, and we obtain sharp bounds depending on both $n$ and the frequency difference $m_n-m_1$. These refinements then enable us to construct explicit counterexamples, showing that observability and unique continuation may fail for Schr\"{o}dinger equations when the observation set is fractal.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [26]

F. L. Nazarov, Local estimates of exponential polynomials and their applications to inequalities of the uncertainty principle type,St. Petersbg. Math. J.5(4), 663-717 (1994)

1994

[2] [1]

J. M. Anderson, V . Ya. Eiderman, Estimates for the Cauchy transform of point masses (the logarithmic derivative of polynomials).Dokl. Ross. Akad. Nauk401:5 583–586; English transl. in Dokl. Math. 71:2, 241–244 (2005)

2005

[3] [2]

J. M. Anderson, V . Ya. Eiderman, Cauchy transforms of point masses: the logarithmic derivative of polynomials. Ann. of Math.(2) 163(3) , 1057–1076 (2006)

2006

[4] [3]

Apraiz, L

J. Apraiz, L. Escauriaza, G. Wang, and C. Zhang. Observability inequalities and measurable sets.J. Eur. Math. Soc. (JEMS), 16(11), 2433–2475 (2014)

2014

[5] [4]

Bourgain, N

J. Bourgain, N. Burq, M. Zworski, Control for Schr ¨odinger operators on 2-tori: rough potentials.J. Eur. Math. Soc. 15, 1597–1628 (2013)

2013

[6] [5]

Brudnyi, Bernstein type inequalities for quasipolynomials,J

A. Brudnyi, Bernstein type inequalities for quasipolynomials,J. Approx. Theory112, 28–43 (2001). 20 JIAQI YU SHANLIN HUANG

2001

[7] [6]

N. Burq, M. Zworski, Rough controls for Schr ¨odinger operators on 2-tori.Annales H. Lebesgue2, 331–347 (2019)

2019

[8] [7]

N. Burq, I. Moyano, Propagation of smallness and control for heat equations.J. Eur. Math. Soc. (JEMS)25(4), 1349–1377 (2023)

2023