The Minimum Cardinality of a Dependent Finite Gabor System Is Four

Tailin Wu; Wenhao Deng; Xinan Dai; Yingdong Shi; Yuchen Yang

REVIEW 2 major objections 3 minor 16 references

The Minimum Cardinality of a Dependent Finite Gabor System Is Four

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Four time–frequency shifts of a nonzero Schwartz function can be linearly dependent, and no configuration of three shifts can be.

desk verdict A sharp resolution of the minimal Gabor dependence question, with a proof that rests on a large interval certificate that should be independently reproducible before the result is treated as settled. read the letter →

arxiv 2608.08190 v1 pith:BCZRSOCA submitted 2026-08-08 math.FA math.CAmath.DS

Xinan Dai , Wenhao Deng , Yingdong Shi , Tailin Wu , Yuchen Yang This is my paper · ORCID

classification math.FAmath.CAmath.DS MSC 42C1542A3847A16

keywords HRTconjectureGaborsystemstime-frequencyshiftsZaktransformdominatedcocyclesDiophantinecohomologyintervalarithmeticlineardependence

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 establishes that the smallest possible number of time–frequency shifts of a nonzero square-integrable function that can be linearly dependent is four. The previous state of the art had shown only that twelve shifts suffice and that three never do. The paper constructs an explicit nonzero Schwartz function and an explicit set of four points in the time–frequency plane at which the shifted functions satisfy a nontrivial linear relation. If the proof is correct, this closes the finite-cardinality version of the HRT question exactly. The construction is complex-valued, in a way that is forced by known four-point independence results for real-valued windows.

What carries the argument

The central mechanism is the rank-two Zak bundle over the torus associated with the lattice generated by $(1,0)$ and $(0,1/2)$, together with the matrix field $A(x,\omega)=I+\tfrac12 e^{-\pi i\omega}Z_0+\tfrac12 e^{\pi i x}X_0$, whose uniform invertibility makes the three-term lattice operator a bundle automorphism. The argument concentrates on the three-step projective return $P$ at the rational translation $\theta=(1/3,1/3)$, whose eigenline is shown by a $1{,}043{,}952$-box interval certificate to be a uniformly contracting graph over a trivial line. Quantitative perturbation carries this dominated line to the explicit algebraic translation, and a winding calculation plus a Diophantine cohomological equation flatten the scalar multiplier to a constant $\lambda$. Inverse Zak folding turns the resulting section into the Schwartz window $f$.

What would settle it

Re-evaluate the bounds (32) with an independent rigorous interval package or a formal proof checker, evaluating the explicit formulas (13), (19), (26), and (27) on a fine dyadic grid; any certified leaf box violating $d \ge 15.4044994528652104323$, $b \le 0.2499996482177990124$, or $c \le 0.7725133354867000155$ would disprove Proposition 3.2. Alternatively, symbolically verify the Laurent identities (20)–(21); if they fail, the winding and domination arguments collapse.

Watch

Extended reading notes

Core claim

The paper's central claim is Corollary 1.2: the minimum cardinality $N^*$ of a dependent finite Gabor system generated by a nonzero $L^2(\mathbb R)$ function is $4$. Theorem 1.1 provides the witness: with $\alpha=\frac13+10^{-12}\sqrt2$ and $\beta=\frac13+10^{-12}\sqrt3$, there are a nonzero $f\in\mathcal S(\mathbb R)$ and a nonzero scalar $\lambda$ such that $\bigl(I+\tfrac12 W(1,0)+\tfrac12 W(0,1/2)\bigr)W(\alpha,\beta/2)f=\lambda f$. Expanding by the Weyl commutation relations turns this into a nontrivial linear dependence among four distinct time–frequency shifts. Since a result of the paper's predecessors shows every three-point system is independent, the four-point example is sharp.

Load-bearing premise

The whole construction rests on the appendix's interval-arithmetic certificate that the rational three-step return has a uniformly contracting line; a bug in that million-box computation would undo the proof.

Editorial extensions

If this is right

The cardinality threshold $N^*=4$ holds for windows in $L^2(\mathbb R)$ and for windows in the Schwartz class.
Any attempt to build a dependent system with three time–frequency shifts is provably futile; four is the exact boundary.
The explicit example must be complex-valued, because real-valued windows retain four-point independence.
The near-rational, algebraically irrational parameters are essential: the rational model is lattice-contained and therefore independent by known results, while the irrational perturbation destroys the lattice obstruction.

Reading between the lines

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

The chosen $10^{-12}$ perturbation size is a convenience, so the same construction is likely to work for a wide band of nearby parameters, possibly with simpler explicit values.
The method plausibly extends to other subcritical covolumes where the natural Zak bundle has rank higher than two, potentially yielding sharp thresholds for larger minimal cardinalities.
A machine-checkable formalization of the appendix's interval certificate would remove any doubt about the computer-assisted step, since the paper provides the bounds but not the verifying code.

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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

The result is the real thing: N*=4, sharp, with an explicit Schwartz function and explicit parameters. The proof architecture is genuinely new—rank-two Zak bundle, rational three-step return with a dominated line, quantitative continuation, winding plus Diophantine cohomology to flatten the multiplier, and inverse unfolding. I read through the analytic steps and found no internal inconsistency; the perturbation estimates in §4 are coarse but the margins are large, and the winding/cohomology section is sound. The reliance on [5] for the Zak reduction is legitimate, and the lower bound from HRT is external. This would be a major within-field result if the certificate holds up.

The soft spot is exactly the one the stress-test names: Proposition 3.2 depends on a 1,043,952-box interval arithmetic computation, and no code or machine-checkable log is shipped. The description is detailed—Arb/Acb, 160-bit precision, outward rounding, refinement scheme—and the stated bounds have a real margin (e.g., b is just under 1/4, d is ~15.4), which makes a systematic bug less likely. But 'less likely' is not the same as verified. A referee can't re-run the computation without reimplementing it from scratch. The paper also asserts the Laurent identities (20)–(21) without derivation; the authors note they are exact over Q(r) and can be checked, but a four-line verification would remove an annoyance. A second issue: the claim that replacing the holonomy fails as a negative control is a nice sanity check, but it doesn't substitute for a certificate that someone else can execute.

I would not block the paper on these points, but I would insist that the authors upload the certificate-generating code or a machine-checked certificate before publication. The rest of the proof is contingent on those bounds, so the reproducibility burden is high. If the bounds check, the result is almost certainly correct; the analytic argument around them is careful and the margins are comfortable.

Bottom line: send this to referees, but ask for the computational artifact as a condition of acceptance. I'd bring it to reading group, and I'd cite it after the certificate is independently confirmed.

Referee Report

2 major / 3 minor

Summary. The paper proves that the minimum cardinality of a dependent finite Gabor system generated by a nonzero L2(R) function is N* = 4. The main theorem, Theorem 1.1, constructs an explicit Schwartz function f and a scalar λ such that (I + (1/2)W(1,0) + (1/2)W(0,1/2))W(α,β/2)f = λf for α = 1/3 + 10^(-12)√2 and β = 1/3 + 10^(-12)√3. Since the Heil–Ramanathan–Topiwala theorem already excludes dependence for three or fewer shifts, this establishes the sharp threshold. The proof uses a rank-two Zak bundle reduction, analyzes a rational model at θ = (1/3,1/3) whose three-step return has a dominated contracting line, proves the existence of this line by a large interval-arithmetic certificate (Proposition 3.2), then carries the invariant line to the explicit irrational parameter by a quantitative perturbation argument. A winding calculation and a Diophantine cohomology equation flatten the corresponding scalar multiplier, and inverse Zak folding yields the desired Schwartz function.

Significance. If the proof is correct, the result is a substantial advance: it reduces the known dependent Gabor system from twelve time–frequency shifts to four, and it identifies the sharp cardinality threshold, resolving a natural question left open by the recent disproof of the HRT conjecture. The architecture of the proof is original and broadly interesting: a rational finite model supplies a dominated line that cannot itself violate HRT because of Linnell's theorem, and an explicit algebraic perturbation plus Diophantine cohomology turns that geometric object into a genuine counterexample. The paper also gives concrete parameters and a directly checkable linear relation, so the central claim is falsifiable. The main strength is the combination of a clean analytic reduction with quantitative estimates; the main weakness is that the load-bearing interval certificate is not independently reproducible from the manuscript as written.

major comments (2)

[Appendix A, Proposition 3.2 and bounds (32)] The entire construction rests on the finite interval certificate that establishes Proposition 3.2, but the manuscript does not supply the code, the exact input formulas, or a machine-checkable certificate that produced the bounds in (32). The description of the 128x128 grid, dyadic splitting, and 160-bit Arb/Acb precision is not enough for an independent referee to verify the global bounds d>=15.404..., b<=0.249999..., c<=0.7725... within reasonable effort. These bounds are load-bearing: Proposition 4.1, the continuation argument, and hence the existence of the invariant line for the explicit parameters all depend on them. Please provide the verifying program and its exact output, or a formal certificate (for example, a proof-producing interval log), so that the computation can be independently rerun.
[Section 3, Eqs. (20)-(21)] The identities for trP and detP are stated only as the result of exact multiplication, with the note that they 'may be checked exactly over Q(r)[u^{±1},v^{±1}]' deferred to Appendix A. These identities are used directly in Lemma 3.1 to obtain |trP|^2 = 27/4 and |detP| <= 39/32, which is the basis for the uniform domination gap. An algebraic identity of this kind is easy to verify with a computer algebra system, but it is still a load-bearing step in the proof; please include a derivation or a short script that verifies (20)-(21).

minor comments (3)

[Appendix A] The box counting '1,043,952 evaluated boxes and 787,060 certified leaf boxes' is not fully transparent: it is unclear whether a box that fails at one level and is split is counted once or several times in the evaluated count. Please clarify the counting convention.
[Proof of Lemma 5.1] The nonvanishing of ζ is justified by the linear independence of 1, √2, √3 over Q, but the argument would be clearer if it explicitly stated that ζ = 0 forces m = n = 0 and hence (m,n) = 0, contradicting the choice of (m,n).
[Section 4, Eq. (44)] The displayed inequality '>5.3·10^(-6) > 0' contains a redundant nested inequality; a single lower bound, for example '>= 5.3·10^(-6)', would be read more cleanly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the four-shift construction is carried out from explicit algebraic formulas and an interval certificate, while the lower bound is cited from an external three-point independence theorem.

full rationale

The paper's central claim, N* = 4, splits into a lower bound and an upper bound. The lower bound N* >= 4 is taken from the Heil-Ramanathan-Topiwala three-point independence theorem [11, Theorem 1], which is an external result and is not derived from or equivalent to the present construction. The upper bound is established by an explicit construction: the paper defines alpha and beta, sets up the rank-two Zak bundle, proves the algebraic identities (20)-(21), invokes a finite interval certificate for Proposition 3.2, and then carries the invariant line through quantitative continuation, winding, cohomology, and inverse Zak folding. None of these steps fits a parameter to the target four-term dependence; the final eigen-equation (4) is the conclusion of the derivation, not an input. The rank-two Zak representation is borrowed from [5], but the paper states the transform and its sewing matrices explicitly and derives the covariance formulas (11)-(15) directly, so this is not a black-box reliance on the authors' own prior work. The interval certificate in Appendix A is a computational proof aid and not circular: it evaluates explicit rational formulas and yields the bounds (32); its correctness is a reproducibility concern, not a self-referential one. There are no load-bearing self-citations, no definitions that presuppose the target result, and no fitted quantity renamed as a prediction. The derivation is self-contained against external benchmarks, and any reservations about the interval certificate belong to correctness risk rather than circularity.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The proof introduces no new physical entities and no fitted parameters; the rank-two Zak bundle and the invariant graph are mathematical constructions built from standard objects. The main hand-chosen free parameter is the perturbation scale epsilon = 10^(-12). The proof relies on the three-point lower bound, standard Zak transform theory, and a computer-assisted interval certificate. The algebraic trace/determinant identities and the interval certificate are the entries with the least independent support.

free parameters (1)

perturbation scale epsilon = 10^(-12) = 10^(-12)
The parameters alpha and beta differ from the rational point theta = (1/3, 1/3) by epsilon*sqrt(2) and epsilon*sqrt(3). The scale epsilon is chosen by hand to be small enough to satisfy the continuation bound (35) with a comfortable margin; it is not fitted to data.

assumptions (3)

standard math Heil-Ramanathan-Topiwala three-point independence theorem
Used to establish the lower bound N* >= 4 in Corollary 1.2; cited as [11, Theorem 1].
domain assumption Correctness of the Arb/Acb interval arithmetic implementation
Appendix A relies on outward-rounded interval arithmetic to certify the graph inequalities; the paper does not ship code or machine-checked certificates, so correctness depends on the library and the authors' implementation.
ad hoc to paper Exactness of the symbolic identities (20)-(21) for trace and determinant of the rational return
These identities are asserted as results of 'exact multiplication' but no derivation or companion output is provided; the rest of Section 3 depends on them.

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Pith. "Pith review of The Minimum Cardinality of a Dependent Finite Gabor System Is Four." pith.science (2026). https://pith.science/paper/BCZRSOCA

@misc{pith2026260808190,
  author       = {Pith},
  title        = {Pith review of: The Minimum Cardinality of a Dependent Finite Gabor System Is Four},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BCZRSOCA}},
  note         = {Machine review of arXiv:2608.08190}
}

abstract

Recent work produced a linearly dependent system of twelve time--frequency shifts of a Schwartz function, disproving the HRT conjecture. We show that four shifts already suffice, and hence that four is the smallest possible cardinality of a dependent finite Gabor system. More precisely, set $\alpha=\frac13+10^{-12}\sqrt2$ and $\beta=\frac13+10^{-12}\sqrt3$. We construct a nonzero complex-valued function $f\in\mathcal S(\mathbb R)$ and $\lambda\ne0$ such that $\left(I+\frac12W(1,0)+\frac12W(0,1/2)\right)W(\alpha,\beta/2)f=\lambda f$, where $W$ denotes the Weyl time--frequency shift. Since every system of at most three shifts of a nonzero $L^2(\mathbb R)$ function is linearly independent, this gives the sharp cardinality threshold. The construction uses the rank-two Zak bundle naturally associated with the covolume-$1/2$ lattice generated by $(1,0)$ and $(0,1/2)$. At the rational translation $(1/3,1/3)$, the three-step return has a uniformly dominated contracting line. A finite outward-rounded interval certificate proves that this line is topologically trivial. A quantitative perturbation argument carries the dominated line to the explicit algebraic translation above. A winding calculation and a Diophantine cohomological equation then flatten its scalar multiplier, and inverse Zak folding produces the required Schwartz function.

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Reference graph

Works this paper leans on

16 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [5]

Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven, and Felix Voigtlaender, Linear dependence of time-frequency shifts of a Schwartz function, arXiv:2608.05044v1, 5 August 2026

arXiv 2026

[2] [1]

Benedetto and Abdelkrim Bourouihiya, Linear independence of finite Gabor systems determined by behavior at infinity,J

John J. Benedetto and Abdelkrim Bourouihiya, Linear independence of finite Gabor systems determined by behavior at infinity,J. Geom. Anal.25(2015), no. 1, 226–254. DOI: 10.1007/s12220-013-9423-8

work page doi:10.1007/s12220-013-9423-8 2015

[3] [2]

Marcin Bownik and Darrin Speegle, Linear independence of time–frequency translates of functions with faster than exponential decay,Bull. Lond. Math. Soc.45(2013), no. 3, 554–566. DOI: 10.1112/blms/bds119

work page doi:10.1112/blms/bds119 2013

[4] [3]

Ciprian Demeter, Linear independence of time frequency translates for special configurations,Math. Res. Lett.17 (2010), no. 4, 761–779. DOI: 10.4310/MRL.2010.v17.n4.a14

work page doi:10.4310/mrl.2010.v17.n4.a14 2010

[5] [4]

Ciprian Demeter and Alexandru Zaharescu, Proof of the HRT conjecture for(2, 2)configurations,J. Math. Anal. Appl.388(2012), no. 1, 151–159. DOI: 10.1016/j.jmaa.2011.11.030

work page doi:10.1016/j.jmaa.2011.11.030 2012

[6] [6]

Folland,Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol

Gerald B. Folland,Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989

work page 1989

[7] [7]

DOI: 10.1007/978-1-4612-0003-1

Karlheinz Gröchenig,Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Boston, MA, 2001. DOI: 10.1007/978-1-4612-0003-1

doi:10.1007/978-1-4612-0003-1 2001

[8] [8]

Math.177(2015), no

Karlheinz Gröchenig, Linear independence of time-frequency shifts?,Monatsh. Math.177(2015), no. 1, 67–77. DOI: 10.1007/s00605-014-0637-z

work page doi:10.1007/s00605-014-0637-z 2015