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HRT counterexamples with exponential tails

T0 review · 2 major / 2 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The HRT conjecture is false for functions with exponential tails.

desk verdict A genuinely useful exposition and a plausible near-optimal strengthening, but the load-bearing holomorphy claim in Step 1 is false, so the main theorem is not proved as written. read the letter →

arxiv 2608.13539 v1 pith:2U32QDWU submitted 2026-08-13 math.FA

John Jasper , Dustin G. Mixon This is my paper · ORCID
classification math.FA MSC 42C1546E1011J25
keywords HRTconjecturetime-frequencyshiftsGelfand-ShilovclassexponentialdecayZaktransformDiophantineapproximationlineardependenceGaborsystem
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 claims that the Heil–Ramanathan–Topiwala (HRT) conjecture—that finitely many distinct time–frequency shifts of any nonzero square-integrable function are linearly independent—fails even under very strong decay. For every Diophantine pair $\alpha,\beta$, the authors construct a nonzero function $f$ in the Roumieu Gelfand–Shilov class $\mathcal{S}^1_1(\mathbb{R})$ and finitely many half-integer frequency shifts such that $\pi(\alpha,\beta)f$ is a finite linear combination of the $\pi(m,n/2)f$. Functions in $\mathcal{S}^1_1(\mathbb{R})$ have exponential or faster decay in both $f$ and its Fourier transform, so this improves on the earlier Schwartz-function counterexamples. Because Bownik and Speegle showed that faster-than-exponential decay (up to a logarithmic factor) forces HRT to hold, the exponential-tail decay achieved here is essentially the fastest possible for a counterexample. The paper does not, however, bound the number of shifts in the dependence.

What carries the argument

The load-bearing object is the two-by-two vector-Zak transform $Z_2f(x,\omega)=2^{-1/2}(Zf(x,\omega/2),Zf(x,(\omega+1)/2))$, whose quasi-periodicity forces the half-lattice time-frequency shifts to act as pointwise multiplication operators. It converts the desired HRT dependence into an equation between strip-holomorphic sections of a vector bundle. The proof then runs on three levers: rank-one multipliers built from a pointwise unit-norm section, Fourier approximation of strip-holomorphic multipliers by finite sums of $L_{m,n}$, and the Diophantine cocycle estimate $|1-e^{-2\pi i k\cdot\tau}|\geq 4C(1+\|k\|_1)^{-s}$, which keeps the Fourier solution of $q(z)h(z-\tau)=ch(z)$ exponentially decaying and hence strip-holomorphic.

What would settle it

Compute $\partial_{\bar z_1}\big(\chi_\varepsilon(z)^*\chi_\varepsilon(z)\big)$ for the explicitly defined Gaussian-smoothed seed of Step 1; if it is nonzero anywhere, $\|\chi_\varepsilon(z)\|$ is not holomorphic and the proof of Theorem 6 as written breaks at the normalization step.

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Extended reading notes

Core claim

The paper's central claim is that HRT counterexamples can be built with exponential tails via a transform-domain construction. Using the vector-Zak transform $Z_2$, the half-lattice shifts $\pi(m,n/2)$ become pointwise multiplication by the $2\times2$ matrix multipliers $L_{m,n}(x,\omega)=e^{\pi i(nx-m\omega+mn)}Z^mX^n$, while the shift $\pi(\alpha,\beta)$ becomes $e^{2\pi i\beta x}F(x-\alpha,\omega-2\beta)$. The paper proves that for a Diophantine pair there exists a nonzero strip-holomorphic vector-Zak section $F$ and a finite multiplier $A=\sum a_{m,n}L_{m,n}$ satisfying $e^{2\pi i\beta x}F(x-\alpha,\omega-2\beta)=A(x,\omega)F(x,\omega)$; pulling this identity back through $Z_2$ yields the time-domain dependence $\pi(\alpha,\beta)f=\sum a_{m,n}\pi(m,n/2)f$. The construction proceeds by normalizing a Gaussian-smoothed compactly supported seed to unit norm, approximating the resulting rank-one multiplier by a finite Fourier polynomial in the $L_{m,n}$, and using a Banach fixed-point argument to pass to an exact nearby solution; the Diophantine condition is then used, through a Fourier series solution of the cocycle equation $q(z)h(z-\tau)=ch(z)$, to absorb the residual multiplier into a constant.

Load-bearing premise

The construction depends on having a holomorphic, length-one two-component function with the right periodicity that satisfies the shifted identity up to a rank-one factor; the proof gets this by asserting the holomorphy of the norm of its Gaussian-smoothed seed, and if that fails, the approximation and contraction steps have no input.

Editorial extensions

If this is right

  • For every Diophantine pair $(\alpha,\beta)$, the HRT conjecture fails in the class $\mathcal{S}^1_1(\mathbb{R})$, so exponential decay is not enough to restore linear independence of time-frequency shifts.
  • The fastest possible decay of an HRT counterexample lies between exponential and the Bownik–Speegle threshold $\exp(-cx\log x)$; in particular the decay barrier is tight up to a logarithmic factor in the exponent.
  • The counterexample functions can be taken to have exponential decay in both time and frequency, since $\mathcal{S}^1_1$ is characterized by that double decay.
  • The construction yields no control on the number of time-frequency shifts in the dependence, in contrast with the 12-point and 4-point counterexamples.

Reading between the lines

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

  • The mechanism is modular: any nowhere-zero strip-holomorphic section with a rank-one multiplier would seed the same Fourier-approximation and fixed-point machinery, so the construction may transfer to other decay classes or to more general irrational shifts.
  • The cocycle equation $q(z)h(z-\tau)=ch(z)$ is solved by a Fourier series whose coefficient decay is preserved because the Diophantine denominator is polynomial; this suggests the same construction could yield counterexamples for any shift whose small-denominator obstruction is subexponential.
  • A natural stress test is numerical: build $\chi_\varepsilon$ for the Gaussian-smoothed compactly supported seed and check whether $\|\chi_\varepsilon(z)\|$ admits a holomorphic square root; if not, the proof of Theorem 6 as written needs an alternate normalization before the later steps can run.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript claims that for every Diophantine pair (α,β) there exist a nonzero function f in the Roumieu Gelfand–Shilov class S_1^1(R), a finite set I ⊆ Z², and coefficients a_{m,n} such that π(α,β)f = Σ a_{m,n}π(m,n/2)f. The proof works in a vector-Zak transform domain, aiming to construct a strip-holomorphic section F and a multiplier A satisfying e(z)F(z−τ) = A(z)F(z). Step 1 selects a pointwise unit-norm F♯ by convolving a compactly supported function with a Gaussian and normalizing; Step 2 replaces the resulting rank-one multiplier A♯ by a finite combination of the L_{m,n} multipliers via Fourier approximation and a contraction-mapping argument; Step 3 uses the Diophantine property to make the resulting multiplier constant. If valid, Theorem 2 would give HRT counterexamples with exponential decay, which is optimal up to a logarithmic factor by the Bownik–Speegle result.

Significance. The claimed result is significant: it would push the decay of HRT counterexample functions to essentially the fastest possible while also providing a more transparent account of the mechanism behind the recent breakthrough. The paper is clearly written, and the three-step strategy is intelligible and well motivated. However, the central analytic step of the construction is invalid: the pointwise normalization of a holomorphic section is not holomorphic, so the strip-holomorphic objects F♯ and A♯ that feed the Fourier approximation and the fixed-point argument are not actually constructed. The significance is therefore conditional on a repair that the manuscript does not provide.

major comments (2)
  1. [§3.1.1] The claim “The map z↦χε(z)^* is holomorphic” is false: for a holomorphic vector-valued function χε, the conjugate transpose is anti-holomorphic, not holomorphic. Consequently sε(z)=χε(z)^*χε(z) is real-analytic but not entire, and the asserted holomorphic square root of sε on Ωρ is not established. Since F♯ is defined as χε/∥χε∥ using this denominator, F♯ is not shown to be strip-holomorphic. This is exactly the analytic regularity needed for the rest of the construction, so Steps 2 and 3 lose their required input.
  2. [§3.1.2] The proof repeats the same error when it states that both F♯ and z↦F♯(z)^* are holomorphic over Ωρ. The latter map is anti-holomorphic, so the identity-theorem argument that F♯(z)^*F♯(z) is identically 1 over Ωρ is invalid, and A♯(z)=e(z)F♯(z−τ)F♯(z)^* in (10) is not a strip-holomorphic vector-Zak multiplier. Thus A♯ does not satisfy the hypothesis of Proposition 7, and the rank-one fixed-point construction in Lemma 8 has no valid holomorphic starting point.
minor comments (2)
  1. [§3.1.1] The outline's phrase “z↦∥χε(z)∥ has a strip-holomorphic extension” is misleading: the Euclidean norm of a holomorphic vector is not a holomorphic function in several complex variables.
  2. [§2.4 / §3.3] The statement that τ=(α,2β) is Diophantine whenever (α,β) is could be expanded with the concrete inequality, since the current one-line justification is terse.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the construction derives the counterexample from external lemmas and standard analysis rather than from the target identity.

full rationale

The paper's central claim is Theorem 2, which asserts the existence of a nonzero $f\in S^1_1(\mathbb{R})$ and coefficients such that $\pi(\alpha,\beta)f=\sum a_{m,n}\pi(m,n/2)f$. The proof does not fit any parameter to the target identity; instead, it constructs a vector-Zak section $F^\sharp$ of unit norm, defines a rank-one multiplier $A^\sharp$ by equation (10), approximates $A^\sharp$ by a finite linear combination $A$ of basic multipliers $L_{m,n}$ via the Fourier approximation in Proposition 7, obtains a nearby $F^\flat$ via the Banach fixed point argument in Lemma 8, and then removes the scalar multiplier $q$ using the Diophantine property in Lemma 9. Each step is an existence/approximation result whose statement does not presuppose the final linear dependence (7). The external citations, chiefly [4], supply the vector-Zak transform formalism, but the present paper redevelops the relevant correspondences in Lemmas 3-5 rather than merely renaming them. The claim about fastest possible decay is imported from Bownik and Speegle [1], an independent external result, and is used only to contextualize optimality. There is no self-citation chain, no fitted parameter renamed as a prediction, and no input definition that already contains the output. The possible mathematical error in Section 3.1.1, where the paper states that $z\mapsto\chi_\varepsilon(z)^*$ is holomorphic (it is anti-holomorphic), is a correctness concern about whether the constructed $F^\sharp$ is strip-holomorphic; it is not a circularity because the false assertion does not make the theorem equivalent to its assumptions. Therefore the derivation is self-contained relative to its cited external machinery and receives a circularity score of 0.

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

The proof rests on external lemmas from Faulhuber et al. and standard functional analysis. No free parameters or invented entities are introduced. The only nontrivial analytic input is the construction of a holomorphic unit-norm vector-Zak section, and the paper's Step 1 fails to justify that construction because of the false holomorphy claim about χε^*χε.

assumptions (4)
  • domain assumption There exists a compactly supported f0 whose vector-Zak transform has pointwise unit norm (Lemma 3.6 of [4]).
    Used in Step 1 to seed the construction of F#; imported from [4] without proof.
  • domain assumption The vector-Zak transform inversion lemma (Lemma 3.3 of [4]) and the conjugation rules for time-frequency shifts (Lemma 4.1 and Subsection 4.2 of [4]).
    Used to pass between the time-domain and transform-domain statements of the main theorem.
  • standard math Diophantine pairs exist, e.g., (2^{1/3}, 2^{2/3}) is Diophantine with s=2.
    The theorem's scope covers all Diophantine pairs; the concrete example uses algebraic norm estimates in the field Q(2^{1/3}).
  • standard math Standard Banach fixed-point theorem and Fourier series facts for holomorphic periodic functions.
    Used in Step 2 for the contraction argument and in Step 3 for the cohomological equation.

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Cite this review

Pith. "Pith review of HRT counterexamples with exponential tails." pith.science (2026). https://pith.science/paper/2U32QDWU

@misc{pith2026260813539,
  author       = {Pith},
  title        = {Pith review of: HRT counterexamples with exponential tails},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2U32QDWU}},
  note         = {Machine review of arXiv:2608.13539}
}
abstract

We build on the recent breakthrough of Faulhuber, Petersen, van Velthoven, and Voigtlaender that disproved the HRT conjecture with a Schwartz function and a $12$-point configuration. We give a human-readable treatment of their mechanism and find HRT counterexample functions with exponential (or faster) decay. By a result of Bownik and Speegle, this is the fastest possible decay for an HRT counterexample, up to a logarithmic factor in the exponent.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

  1. [1]

    Bownik and D

    M. Bownik and D. Speegle, Linear independence of time–frequency translates of functions with faster than exponential decay, Bull. Lond. Math. Soc. 45 (2013) 554–566

    work page 2013

  2. [2]

    Chung, S.-Y

    J. Chung, S.-Y. Chung, D. Kim, Characterizations of the Gelfand–Shilov spaces via Fourier transforms, Proc. Amer. Math. Soc. 124 (1996) 2101–2108

    work page 1996

  3. [3]

    X. Dai, W. Deng, Y. Shi, T. Wu, Y. Yang, The minimum cardinality of a dependent finite Gabor system is four, arXiv:2608.08190 (2026)

    work page Pith review arXiv 2026

  4. [4]

    Faulhuber, P

    M. Faulhuber, P. Petersen, J. T. van Velthoven, F. Voigtlaender, Linear dependence of time-frequency shifts of a Schwartz function, arXiv:2608.05044 (2026)

    arXiv 2026

  5. [5]

    Gr¨ ochenig,Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkh¨ auser, Boston, 2001

    K. Gr¨ ochenig,Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkh¨ auser, Boston, 2001

    work page 2001

  6. [6]

    C. Heil, J. Ramanathan, P. Topiwala, Linear independence of time-frequency translates, Proc. Amer. Math. Soc. 124 (1996) 2787–2795

    work page 1996

  7. [7]

    A. J. E. M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys. 23 (1982) 720–731

    work page 1982

  8. [8]

    An intrinsically subcritical four-point counterexample

    V. Oussa, An intrinsically subcritical four-point counterexample, arXiv:2608.07604 (2026)

    work page Pith review arXiv 2026

Show all 9 references
  1. [9]

    Wahlberg, Semigroups for quadratic evolution equations acting on Shubin–Sobolev and Gelfand–Shilov spaces, Ann

    P. Wahlberg, Semigroups for quadratic evolution equations acting on Shubin–Sobolev and Gelfand–Shilov spaces, Ann. Fenn. Math. 47 (2022) 821–853. 12

    2022

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