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Linear dependence of time-frequency shifts of a Schwartz function

T0 review · 1 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read HRT conjecture falls to a 12-term linear dependence

desk verdict The paper claims a counterexample to HRT, but the printed numerical certificate appears to bound a different matrix field due to a conjugation error in Step 1b; worth refereeing, not citable yet. read the letter →

arxiv 2608.05044 v1 pith:MTJUFUFB submitted 2026-08-05 math.FA math.CA

Markus Faulhuber , Philipp Petersen , Jordy Timo van Velthoven , Felix Voigtlaender This is my paper · ORCID
classification math.FAmath.CA MSC 42C1543A65
keywords HRTconjecturetime-frequencyshiftsWeylpolynomialvectorZaktransformSchwartzfunctionlineardependencecomputer-assistedproofcontractionfixedpoint
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

This paper claims to settle a 30-year-old open problem, the HRT conjecture, by constructing a nonzero Schwartz function whose twelve time-frequency shifts are linearly dependent. The conjecture predicted that any finite set of time-frequency shifts of a nonzero square-integrable function must be independent, and the paper produces an explicit counterexample with twelve distinct shift points. Doing so requires a genuinely new mechanism: a vector-valued Zak transform that escapes the boundary obstructions of the scalar transform, followed by a computer-assisted estimate and a contraction argument. If correct, the result would close the HRT question in the negative and show that even the Schwartz class does not restore independence.

What carries the argument

The central objects are the vector Zak transform $\mathcal{Z}_2$, mapping $L^2(\mathbb{R})$ to $\mathbb{C}^2$-valued functions satisfying sewing relations, and the certified bound $\delta \le \tfrac{41629}{125000} < \tfrac13$ on the operator-norm distance between the matrix field $B_*$ and the rank-one field $B_0$. The vector Zak transform is necessary because the scalar Zak transform forces any continuous quasiperiodic function to vanish somewhere, which would destroy the nowhere-vanishing solution needed for the eigenvalue problem. The smallness bound $\delta < 1/3$ makes the contraction argument work: it keeps the scalar multiplier $q_*$ bounded away from zero and gives a Lipschitz constant below 1 in the fixed-point iteration for the smooth vector-Zak function.

What would settle it

Reproduce the supplementary interval-arithmetic script in an independent certified-arithmetic implementation and check whether the two key enclosures still hold: the bound $B(x_j) < 0.328813$ at all 2048 grid centers and the derivative allowance $D_*/2N < 0.004219$. If either fails by more than the small margin, or if a point $z$ is found with $\|B_*(z)-B_0(z)\|_{\mathrm{op}} \ge 1/3$, then the contraction argument collapses and the claimed counterexample would no longer be established.

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Extended reading notes

Core claim

The central discovery is Theorem 1.1: there exist nonzero complex coefficients and twelve pairwise distinct points in the plane such that a nonzero Schwartz function satisfies an exact linear relation among its time-frequency shifts. The proof reformulates this as an eigenvalue problem for an explicit eleven-term Weyl polynomial, transforms the problem through the vector Zak transform into a matrix-valued difference equation, and then proves existence of a smooth, nowhere-vanishing solution by a contraction fixed-point argument near an explicit rank-one matrix field. The decisive ingredient is a certified uniform bound, less than 1/3, on the distance between the true matrix field and that rank-one field; the remaining steps reduce the scalar multiplier to a constant by a Fourier argument that uses arithmetic properties of the cube root of two.

Load-bearing premise

The construction breaks if the computer-certified bound that the matrix field stays within one-third of the rank-one field is wrong, since that strict inequality drives the contraction, the nonzero scalar multiplier, and the smoothness of the fixed point.

Editorial extensions

If this is right

  • The HRT conjecture is false in $L^2(\mathbb{R})$, since the constructed function is in fact Schwartz.
  • Even the Schwartz-space formulation of the conjecture, raised by several authors, is disproved.
  • Because Linnell's theorem guarantees independence for any finite subset of a lattice, the counterexample necessarily uses an irrational translation vector; the construction does exactly that, with the origin added to a translated lattice.
  • The proof is fully explicit: the eleven coefficients are fixed dyadic numbers, and the validity reduces to a finite computer-assisted certificate that can be independently audited.
  • The vector Zak transform of dimension two is shown to be both necessary and minimal for this construction, since a scalar version cannot work.

Reading between the lines

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

  • The same vector-Zak framework may be adaptable to produce counterexamples with fewer than twelve shifts or with other algebraic irrational translation parameters, since the proof only needs a strong diophantine separation property of the translation vector.
  • The certified gap between $\delta$ and $1/3$ is only about $3\times 10^{-4}$, so a purely analytic proof of the needed bound may be within reach, which would replace the computer-assisted component by a rigorous closed-form estimate.
  • A natural testable extension is to search numerically for eigenvalues of the associated Weyl polynomial for the same or slightly perturbed coefficients; a robust nonzero eigenvalue would support the existence mechanism beyond the single certified configuration.
  • The failure of the HRT conjecture for $\mathbb{R}^2$ does not by itself settle the discrete zero-divisor conjecture for the Heisenberg group, but it removes the primary analytic evidence that the two conjectures must share the same fate.
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Formalized claims in Lean

  1. Claim #1: The central objects are the vector Zak transform $\mathcal{Z}_2$, mapping $L^2(\mathbb{R})$ to $\mathbb{C}^2$-valued functions satisfying sewing relations, and the certified bound $\delta \le \tfrac{41629}{125000} &lt; \tfrac13$ on the operator-norm distance between the matrix field $B_*$ and the rank-one field $B_0$. The vector Zak transform is necessary because the scalar Zak transform forces an

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 3 minor

Summary. The paper claims a counterexample to the Heil–Ramanathan–Topiwala (HRT) conjecture in the Schwartz class: twelve time-frequency shifts of a nonzero Schwartz function are linearly dependent. The proof constructs a nonzero Schwartz eigenfunction f* for an eleven-term Weyl polynomial P*, then adds the constant term to obtain a twelve-term relation. The construction uses the vector Zak transform, reducing the eigenvalue equation to a matrix-valued difference equation B*(z)F(Tτz)=cF(z). The key quantitative step is a certified numerical estimate δ≤41629/125000<1/3 for the uniform distance between B* and a rank-one matrix field B0 built from an explicit smooth Zak function χ. With this estimate, a fixed-point argument produces a smooth nonzero vector-valued solution v*, a scalar cocycle argument reduces the multiplier q* to a constant c*, and a Fourier-series correction gives a smooth function h satisfying q*(z)h(Tτz)=c*h(z). The product h·v* is shown to be a smooth vector-Zak function whose inverse vector Zak transform lies in the Schwartz class. The paper also contains appendices giving the Faà di Bruno estimates, the sewing-relation verifications, and a description of the computer-assisted certificate.

Significance. If the main theorem is correct, it disproves a long-standing open conjecture of Heil, Ramanathan, and Topiwala, including its Schwartz-class formulation. The paper provides an explicit finite configuration, exact dyadic coefficients, and a reproducible certified numerical script, which are valuable strengths. The analytic architecture—vector Zak transform, fixed-point contraction, and Fourier cocycle reduction—is coherent and detailed. However, the decisive estimate δ<1/3 currently rests on an algebraic error in the expansion of A0, so the central claim is not yet established by the written proof. With a corrected certificate the result could be a major advance; as submitted, the key bound is not proven.

major comments (1)
  1. [§5.2, Step 1b, Eqs. (5.4)–(5.8)] The Laurent expansion of A0 is computed incorrectly. Since A0(z)=χ(z)w(z)^* with w(z)=η(x)χ(Tτz), the (r,s) entry equals χ_r(z) overline{w_s(z)}, which is overline{η(x)} χ_r(z) overline{χ_s(Tτz)}. For x≥α this yields d_{r,s,0}=overline{η}/2(a A_- + ϵ_r ϵ_s b B_- σ^{-1}), d_{r,s,-1}=overline{η}/2 ϵ_s a B_- σ^{-1}, and d_{r,s,1}=overline{η}/2 ϵ_r b A_-, with the analogous σ^{-2} correction in the x<α case. The printed formulas instead contain η, σ, and σ^2 and multiply by w_s rather than overline{w_s}. Consequently, the certified bound in (5.11) applies to ∥A*(x,ω)-Ã0(x,ω)∥ for a different matrix field Ã0(x,ω)=χ(x,ω)w(x,ω)^T, not to the A0 of (4.9). Since (4.13), Corollary 5.3, Lemma 6.1, and Proposition 6.3 all rely on δ<1/3 for A*-A0, the numerical certificate does not connect to the fixed-point argument. The expansion must be corrected and the certified computation rerun; the margin 0.333032 versus 1/3 is far too small for this to be dismissed as a purely cosmetic issue.
minor comments (3)
  1. [§4.2, after Eq. (4.6)] The displayed identity w(z)^* = η(x)χ(Tτz)^* is false; since η(x) is a unit-modulus phase, the correct expression is overline{η(x)}χ(Tτz)^*. The subsequent identity B0(z)=A0(z)η(x)I2 remains true because η overline{η}=1, but the typo should be fixed as it is the likely source of the Step 1b error.
  2. [§5.2, Step 1b, Case 1] The text says the outer product A0=χw^* is expanded entry by entry as (χ)_{r+1}(w)_{s+1}; it should be (χ)_{r+1} overline{(w)_{s+1}}. This is the concrete form of the conjugation error that invalidates the current certificate.
  3. [§5.1 and Remark 5.2] The proof of Theorem 5.1 delegates the decisive bound to a Python/Arb script. The description of the interval arithmetic is clear, but since the script implements the incorrect d-coefficients, the reproducibility statement alone cannot compensate for the missing corrected certificate.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the counterexample is constructed from explicit data and verified by independent certified estimates; only a minor non-essential self-citation is present.

full rationale

The derivation chain is self-contained. The paper fixes explicit arithmetic data (Section 2.2), defines the matrix fields A_* and A_0 (Section 4), and proves the certified bound delta < 1/3 in Theorem 5.1 by an analytic reduction to finitely many grid points followed by interval-arithmetic verification. That bound is then used as an input to a Banach fixed-point argument (Proposition 6.3), a smoothness argument (Lemma 6.5), and a Fourier-analytic scalar reduction (Proposition 7.2). None of these steps assumes the conclusion of Theorem 1.1; the final eigenvalue equation is derived, not postulated. The coefficients in Table 1 are explicit constants that make the verified inequality hold, and the inequality is checked independently of the existence statement. The skeptic's concern about the Step 1b Laurent expansion of A_0 is a possible correctness or transcription issue in the numerical certificate, not a circularity: it does not make the theorem equivalent to its inputs. The only self-citation with author overlap is reference [9], used for context on elementary proofs of Linnell's theorem; it is not load-bearing for the main claim. The vector-Zak facts that are cited are also proved in the text (e.g., Lemma 3.3). Therefore there is no significant circularity, and the score reflects only the presence of a minor, non-essential self-citation.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The construction rests on several chosen constants (the eleven dyadic coefficients, the algebraic translation parameters, the index set, and the numerical grid). These are part of an explicit existence proof rather than fitted data, but the certified bound is attainable only for these exact choices. The mathematical framework uses standard tools (Zak transforms, Weyl calculus, fixed point theory, Fourier series) plus a domain assumption about the correctness of FLINT/Arb interval arithmetic. No new entities with independent falsifiable handles are introduced.

free parameters (4)
  • Eleven dyadic coefficients a_{m,n} = (p_{m,n}+i q_{m,n})/260 (Table 1)
    Chosen by search so that the vector-Zak matrix field A* approximates A0 well enough for the bound δ < 1/3.
  • Translation pair (α, β) = α = 2^(1/3)-1, β = 2^(2/3)-1
    Algebraic irrational choice that is outside the lattice, used in Lemma 7.1 for the Diophantine lower bound and to evade Linnell's theorem.
  • Index set I of the Weyl polynomial = eleven points listed in Section 2.2
    The support of the Weyl polynomial; chosen together with the coefficients to make the Laurent expansion and numerical bound tractable.
  • Grid size N and Arb precision = N=2048, 256 bits
    Numerical resolution needed to certify δ < 1/3 with the tight margin; changing them could change the outcome.
assumptions (3)
  • domain assumption The Arb/FLINT interval arithmetic routines used in hrts_fixed_12_point_arb_1d_audit.py return valid enclosures for the expressions in Theorem 5.1.
    The proof of (5.2) rests on the correctness of the library and on the reduction to grid points in the script; this is not checked inside the paper.
  • standard math The vector Zak transform Z2 is a unitary map from L^2(R) to H_Z and the standard Zak transform unitarity, quasi-periodicity, and Schwartz characterization hold.
    Invoked in Lemmas 3.1-3.3 and at the end of Section 8 to convert a smooth vector-Zak function back to a Schwartz function.
  • standard math The numbers 1, 2^(1/3), 2^(2/3) are linearly independent over Q (x^3-2 is irreducible) and the resulting Diophantine estimate in Lemma 7.1 is valid.
    Needed for the lower bound on |1-e^{2πik·τ}| used in the Fourier series construction of u in Proposition 7.2.

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Cite this review

Pith. "Pith review of Linear dependence of time-frequency shifts of a Schwartz function." pith.science (2026). https://pith.science/paper/MTJUFUFB

@misc{pith2026260805044,
  author       = {Pith},
  title        = {Pith review of: Linear dependence of time-frequency shifts of a Schwartz function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MTJUFUFB}},
  note         = {Machine review of arXiv:2608.05044}
}
read the original abstract

We show that a finite number of time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the so-called HRT conjecture of Heil, Ramanathan, and Topiwala. In particular, we provide an example consisting of 12 time-frequency shifts.

Figures

Figures reproduced from arXiv: 2608.05044 by the authors.

Figure 1
Figure 1. Floating-point approximation fG of the final window. The three panels show its modulus, real part, and imaginary part. With the stated normalization, the largest displayed sample has |fG(t)| ≈ 1.061 at t ≈ 1.258. The sampled L 2 -mass in [−2, 3) is 0.9999999958. The second exponential in (D.2) is precisely the symmetric Weyl phase from (2.1), with translation parameter m+α and modulation parameter (n+β)/2. The resid… view at source ↗
Figure 2
Figure 2. Direct floating-point evaluation of the twelve summands in (D.2). In the first two panels, the gray curves are the real and imaginary parts of the individual summands, while the dark curve is their sum. In the bottom panel, the blue curve is the sum of the magnitudes of the individual terms P11 j=0 |Sj,G(t)|, and the orange curve is the residual |RG(t)|. The latter remains at approximately double-precision rounding … view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Minimum Cardinality of a Dependent Finite Gabor System Is Four

    math.FA 2026-08 conditional novelty 8.0 of 10

    Four time-frequency shifts of a nonzero Schwartz function can be linearly dependent, and no smaller number can, so the minimum cardinality of a dependent Gabor system is exactly 4.

  2. An intrinsically subcritical four-point counterexample

    math.CA 2026-08 conditional novelty 7.0 of 10

    Four distinct time-frequency shifts of a nonzero Schwartz function are linearly dependent while every symplectic triangle determinant is strictly below one.

Reference graph

Works this paper leans on

27 extracted references · 25 canonical work pages · cited by 2 Pith papers

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