{"id":"62b19d30-2197-49a5-8ff8-95115ebdd9ee","arxiv_id":"2607.17505","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On fractal sets, the Turán–Nazarov inequality holds only with a sharp frequency-bandwidth-dependent constant, and this forces failures of observability and unique continuation for Schrödinger equations.","lead":"Researchers generalize a classical inequality about trigonometric sums to fractal observation sets, showing the constant must grow with the frequency range and can no longer be uniform. This leads to explicit counterexamples where Schrödinger equations cannot be observed or uniquely continued from fractal sets, unlike heat equations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.1's unit-circle measure estimate is imported from [26] without proof; the q̃/q case inconsistency in §2.2 Step 2 further weakens the contraction.","rationale":"The reader correctly identified Lemma 2.1 as the load-bearing step. I agree that its proof is the most delicate point, but I locate the risk not merely in the claimed exponent −α but in the unsupported unit-circle measure estimate and the unresolved q̃/q case in Step 2. These are internal-consistency issues: the proof as written does not fully justify the transition to the logarithmic derivative of g in the q case, and the citation to [26] is to a real-line result that may not directly cover T. If the measure estimate fails or has a weaker exponent, the contraction (2.40) breaks and Theorem 1.2(ii) is unsupported. Since these are repairable with additional arguments or a correct citation, I recommend CONDITIONAL rather than REJECT: the central claim may be true, but the current proof is incomplete at its key lemma.","tokens_in":17703,"tokens_out":23173,"duration_ms":190859,"concrete_test":"Verify the unit-circle analog of [26, Lemma 1.2] by consulting the original reference or deriving it via the Möbius transform z=(i−x)/(i+x). Check whether the measure bound holds with a constant independent of the zeros z_j; if it does not, compute the sharp exponent for P(z)=z^N−1 on T and compare with Lemma 2.1. Separately, re-derive the contradiction chain in (2.40) for the case where Step 1 selects q (the z^{−r_1} derivative); if the expression involves g′/g plus a constant term, show how Lemma 2.1 still applies after reflection or another reduction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Lemma 2.1 hinges on the estimate μ({z∈T: |Σ 1/(z−z_j)|>0.8H}) ≤ (10/π)mH^{−1}, cited to [26, Lemma 1.2]. That lemma is a real-line result; no proof or reference is given for the unit-circle version. Transferring by a Möbius map z=(i−x)/(i+x) produces a rational function in x whose partial-fraction residues grow like 1+|x_j|^2, so a uniform constant independent of the zeros is not immediate. If the correct measure bound on T were weaker (e.g., only H^{−α/2}), the subsequent covering argument would give C_H^α ≤ C m H^{−α/2}, not the H^{−α} used in the contraction (2.40), and Theorem 1.2(ii) would lose its stated bandwidth dependence. Additionally, Step 1 of §2.2 defines p_{k−1} from either q (the z^{−r_1} derivative) or q̃ (the z^{−r_k} derivative), claiming WLOG the first case. But the equality chain in (2.40) is valid only for the q̃ case: |p_{k−1}/p_k| = |g′(1/z)/(r^{(k)}g(1/z))|. In the q case one obtains |1 − (g′(1/z)/(z r^{(k)}g(1/z)))|, which is not a pure level set of the logarithmic derivative of g, so Lemma 2.1 does not directly apply. The paper does not explain how to reduce the q case to the q̃ case, e.g., by reflecting z→1/z. This is a gap in the proof of the key inequality.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18130,"tokens_out":23360,"duration_ms":207827,"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":[{"comment":"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.","section":"§2.2, Step 1–2, Eqs. (2.34), (2.37)–(2.40)"},{"comment":"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).","section":"§2.2, Lemma 2.1, Eq. (2.25)"}],"minor_comments":[{"comment":"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).","section":"§2.2, Step 1"},{"comment":"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.","section":"§2.2, Step 4"},{"comment":"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.","section":"§2.2, Remark 2.2"},{"comment":"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.","section":"§3.2, Theorem 1.7(ii)"},{"comment":"There are numerous typographical errors ('whcih', 'uncertatinty', stray asterisk, inconsistent spacing in 'Schr¨odinger'). These should be corrected in revision.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for the journal and the results are significant if correct. The referee's accept is too optimistic: the Step 1–Step 2 inconsistency in the proof of (2.40) is a genuine gap in a load-bearing derivation, and the reliance on [26] for the unit-circle measure estimate needs explicit clarification. Neither issue looks fatal — both are local and likely repairable — so major revision rather than reject is appropriate. I would be willing to re-review after the authors reconcile the q/q̃ cases and state the exact external estimate used in Lemma 2.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nWhere it's soft.\n\nThe 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.\n\nSecond, 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.\n\nStep 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.\n\nBottom 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.","headline":"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.","tokens_in":18583,"tokens_out":16545,"would_cite":true,"duration_ms":132234,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B10","35A02"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Turán–Nazarov inequality","Hausdorff content","fractal sets","observability","Schrödinger equation","unique continuation","trigonometric polynomials","frequency bandwidth"],"falsifier":"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.","tokens_in":17598,"feed_emoji":"⚛️","tokens_out":11439,"duration_ms":103731,"temperature":0.7,"pith_summary":"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.","feed_headline":"Fractal sets defeat uniform Turán–Nazarov bounds","feed_subtitle":"The uniform bound is impossible; the bandwidth-dependent constant makes Schrödinger observability fail on fractals.","key_machinery":"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","core_discovery":"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} 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.","supporting_citations":[],"review_version":1}