Linear dependence of time-frequency shifts of a Schwartz function

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.

• The paper constructs an explicit twelve-term linear dependence among distinct time-frequency shifts of a nonzero Schwartz function, disproving the HRT conjecture for both L² and Schwartz windows.
• The authors combine a vector-valued Zak transform, an eleven-term Weyl polynomial, validated operator-norm estimates below 1/3, and a contraction argument to construct an invariant line for a matrix cocycle.
• A Diophantine cohomological reduction converts the variable cocycle multiplier into a constant eigenvalue, while certified numerics verify the exact construction’s stability rather than merely measuring a small residual.

A Constructive Schwartz-Class Counterexample to the HRT Conjecture

Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven, and Felix Voigtlaender present an explicit counterexample to the Heil–Ramanathan–Topiwala (HRT) conjecture in “Linear dependence of time-frequency shifts of a Schwartz function” (2608.05044). The paper proves that twelve pairwise distinct time-frequency shifts of a nonzero Schwartz function can be linearly dependent. This contradicts both the general $L^2(\mathbb{R})$ formulation of the HRT conjecture and its restriction to Schwartz windows.

The result is formulated constructively rather than existentially. The authors specify eleven phase-space points, exact dyadic coefficients, an irrational translation, and a nonzero $f_* \in \mathcal{S}(\mathbb{R})$ for which a twelve-term relation holds identically. The construction combines Weyl operators, a two-component vector-valued Zak transform, a computer-assisted operator-norm estimate, a contraction argument, and a Fourier-analytic cohomological reduction.

The HRT Conjecture and the Counterexample

For $z=(x,\omega)\in\mathbb{R}^2$, the time-frequency shift is

$\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$

The HRT conjecture asserts that if $f\neq 0$ and $z_1,\ldots,z_n$ are pairwise distinct, then the finite system

$\{\pi(z_1)f,\ldots,\pi(z_n)f\}$

is linearly independent over $\mathbb{C}$. Equivalently, no nonzero finite time-frequency polynomial should annihilate a nonzero function.

The paper disproves this assertion by constructing coefficients $\alpha_1,\ldots,\alpha_{12}\neq 0$, pairwise distinct $z_1,\ldots,z_{12}\in\mathbb{R}^2$, and $f_* \in \mathcal{S}(\mathbb{R})$0 such that

$f_* \in \mathcal{S}(\mathbb{R})$1

The configuration has a specific arithmetic structure. Eleven points lie in a translated half-integer lattice,

$f_* \in \mathcal{S}(\mathbb{R})$2

while the twelfth point is the origin. The inclusion of the origin is essential: Linnell-type results preserve linear independence for finite subsets contained in a discrete subgroup, so the counterexample must use a translated lattice together with a point outside that translated lattice.

The authors use the symmetric Weyl operators

$f_* \in \mathcal{S}(\mathbb{R})$3

which differ from the standard time-frequency shifts only by phase factors. They reduce the desired dependence relation to an eigenvalue problem for an eleven-term Weyl polynomial

$f_* \in \mathcal{S}(\mathbb{R})$4

Specifically, the central objective is to produce a nonzero Schwartz function satisfying

$f_* \in \mathcal{S}(\mathbb{R})$5

Then

$f_* \in \mathcal{S}(\mathbb{R})$6

gives the required twelve-term dependence relation.

Fixed Arithmetic Data

The construction fixes

$f_* \in \mathcal{S}(\mathbb{R})$7

and introduces the irrational phase-space translation

$f_* \in \mathcal{S}(\mathbb{R})$8

The eleven lattice indices are selected from a finite subset of $f_* \in \mathcal{S}(\mathbb{R})$9 involving only $z=(x,\omega)\in\mathbb{R}^2$0 and a bounded range of $z=(x,\omega)\in\mathbb{R}^2$1. The coefficients are exact dyadic complex numbers of the form

$z=(x,\omega)\in\mathbb{R}^2$2

with integer numerators. This exact arithmetic is important: the analytical argument does not depend on rounded decimal coefficients. The corresponding Weyl polynomial is

$z=(x,\omega)\in\mathbb{R}^2$3

The coefficients were selected so that the induced matrix-valued cocycle is close to an explicitly constructed rank-one model. The numerical coefficient magnitudes are moderate, but the cancellation in the final relation is highly nontrivial: the individual summands have sizes on the order of unity, whereas their sum vanishes.

Why the Scalar Zak Transform Is Insufficient

The classical Zak transform is

$z=(x,\omega)\in\mathbb{R}^2$4

It converts lattice time-frequency shifts into multiplication by simple phase factors and is therefore a natural framework for finite Gabor dependence problems. However, continuous scalar Zak functions obey the quasiperiodic relations

$z=(x,\omega)\in\mathbb{R}^2$5

Every continuous function satisfying these relations must vanish somewhere in each fundamental domain. This topological obstruction is incompatible with the construction required in the paper. A scalar eigenvalue equation transported along a dense irrational orbit would propagate any zero to a dense set, forcing the function to vanish identically.

The authors overcome this obstruction by folding the frequency variable into two components:

$z=(x,\omega)\in\mathbb{R}^2$6

The resulting vector-Zak functions satisfy matrix-valued sewing relations. This two-dimensional fiber is not merely a technical convenience. The paper argues that dimension two is both necessary and minimal for avoiding the scalar zero obstruction while retaining enough covariance to represent the half-integer lattice shifts.

The vector-Zak transform is unitary from $z=(x,\omega)\in\mathbb{R}^2$7 onto the Hilbert space of measurable vector-Zak functions. Under this transform, the lattice portion of the Weyl polynomial becomes multiplication by an explicit $z=(x,\omega)\in\mathbb{R}^2$8 matrix field, while the irrational translation becomes a shift of the base point.

The Matrix-Cocycle Reformulation

For each lattice index $z=(x,\omega)\in\mathbb{R}^2$9, the inverse Weyl shift $\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$0 acts on vector-Zak functions by multiplication with a unitary matrix $\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$1. The eleven-term lattice operator therefore produces

$\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$2

The irrational Weyl shift contributes a base transformation

$\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$3

and a scalar phase factor. Consequently, the eigenvalue equation is converted into a matrix cocycle equation

$\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$4

where $\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$5 is a smooth $\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$6 matrix field satisfying the appropriate sewing relations.

The authors construct an explicit smooth unit vector $\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$7 and use it to define the rank-one model

$\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$8

This model has the exact invariant relation

$\pi(z)f(t)=e^{2\pi i\omega t}f(t-x).$9

Thus, if $f\neq 0$0 were equal to $f\neq 0$1, the desired eigenfunction would be immediate. The central analytical task is to show that the fixed matrix field $f\neq 0$2 is sufficiently close to $f\neq 0$3 in operator norm.

Certified Numerical Estimate

The paper establishes the global bound

$f\neq 0$4

More precisely, the certified estimate is

$f\neq 0$5

This is not presented as an empirical floating-point observation. The authors reduce the two-variable bound analytically to a one-dimensional estimate in the spatial coordinate. They then use a grid of $f\neq 0$6 points and 256-bit Arb ball arithmetic to enclose all relevant quantities rigorously. The derivative estimate controlling interpolation between grid points is

$f\neq 0$7

and the interpolation contribution is certified to be less than $f\neq 0$8. At the grid centers, the bound is less than $f\neq 0$9, yielding the stated global estimate.

The role of this threshold is structural. The inequality $z_1,\ldots,z_n$0 supplies a sufficiently strong perturbative regime for a nonlinear graph transform. In particular, the rank-one invariant line of $z_1,\ldots,z_n$1 persists under the perturbation $z_1,\ldots,z_n$2.

The use of validated numerics is one of the paper’s most consequential methodological features. The numerical calculation does not approximate the final function and infer exact dependence from small residuals; rather, it certifies the operator inequality on which the existence proof rests. The final floating-point calculations are used only as visualization and consistency checks.

Construction of the Invariant Vector Field

Let $z_1,\ldots,z_n$3 and $z_1,\ldots,z_n$4. The authors seek an invariant vector field of the form

$z_1,\ldots,z_n$5

where $z_1,\ldots,z_n$6 lies in the orthogonal complement of $z_1,\ldots,z_n$7. A nonlinear correction map $z_1,\ldots,z_n$8 is defined by projecting the action of $z_1,\ldots,z_n$9 onto this orthogonal complement and normalizing by the scalar component along $\{\pi(z_1)f,\ldots,\pi(z_n)f\}$0.

The perturbative bound implies that the denominator in this normalization remains separated from zero. On a ball of radius $\{\pi(z_1)f,\ldots,\pi(z_n)f\}$1, the map is shown to be a contraction with Lipschitz constant strictly below

$\{\pi(z_1)f,\ldots,\pi(z_n)f\}$2

The Banach fixed-point theorem therefore produces a unique correction $\{\pi(z_1)f,\ldots,\pi(z_n)f\}$3 satisfying

$\{\pi(z_1)f,\ldots,\pi(z_n)f\}$4

The associated field $\{\pi(z_1)f,\ldots,\pi(z_n)f\}$5 is nowhere zero and obeys

$\{\pi(z_1)f,\ldots,\pi(z_n)f\}$6

where the scalar multiplier satisfies

$\{\pi(z_1)f,\ldots,\pi(z_n)f\}$7

The contraction argument initially gives only continuity. The authors then prove smoothness by expressing the correction in a scalar projective coordinate and iterating a holomorphic fiber map. The derivative of this fiber map is uniformly bounded by a constant below one. A carefully organized induction, including a mixed real/complex Faà di Bruno estimate, yields locally uniform bounds for derivatives of every order. Arzelà–Ascoli then upgrades the uniform fixed point to a $\{\pi(z_1)f,\ldots,\pi(z_n)f\}$8 vector-Zak function.

Cohomological Reduction to a Constant Eigenvalue

The invariant-line equation contains the nonconstant scalar multiplier $\{\pi(z_1)f,\ldots,\pi(z_n)f\}$9. To obtain an actual eigenfunction of the Weyl polynomial, the multiplier must be reduced to a constant.

Since $\mathbb{C}$0 lies in the disk $\mathbb{C}$1, the principal logarithm is well-defined:

$\mathbb{C}$2

The irrational translation vector $\mathbb{C}$3 has a quantitative Diophantine property. For every nonzero $\mathbb{C}$4,

$\mathbb{C}$5

This estimate follows from an algebraic identity associated with the cubic irrational $\mathbb{C}$6. It prevents the Fourier denominators

$\mathbb{C}$7

from becoming too small too rapidly.

Writing $\mathbb{C}$8 for the Fourier coefficients of $\mathbb{C}$9, the authors define

$\alpha_1,\ldots,\alpha_{12}\neq 0$0

The Diophantine lower bound and the rapid decay of $\alpha_1,\ldots,\alpha_{12}\neq 0$1 imply that $\alpha_1,\ldots,\alpha_{12}\neq 0$2. It solves the additive cohomological equation

$\alpha_1,\ldots,\alpha_{12}\neq 0$3

Setting

$\alpha_1,\ldots,\alpha_{12}\neq 0$4

gives the multiplicative relation

$\alpha_1,\ldots,\alpha_{12}\neq 0$5

Thus, with

$\alpha_1,\ldots,\alpha_{12}\neq 0$6

one obtains

$\alpha_1,\ldots,\alpha_{12}\neq 0$7

The construction separates the problem into two conceptually distinct components: a perturbative invariant-line problem and a scalar cohomological equation. The first is controlled by the certified operator bound; the second is controlled by the arithmetic of the irrational translation.

Lifting Back to a Schwartz Function

The inverse vector-Zak transform produces

$\alpha_1,\ldots,\alpha_{12}\neq 0$8

Since $\alpha_1,\ldots,\alpha_{12}\neq 0$9 is smooth and satisfies the vector-Zak sewing relations, the ordinary scalar Zak transform of $z_1,\ldots,z_{12}\in\mathbb{R}^2$0 is also smooth. Standard characterization results for the Zak transform then imply

$z_1,\ldots,z_{12}\in\mathbb{R}^2$1

The eigenvalue relation

$z_1,\ldots,z_{12}\in\mathbb{R}^2$2

is therefore exact. Moving the eigenvalue term to the left yields a twelve-term dependence relation involving the origin and eleven translated half-lattice points.

This conclusion is stronger than a counterexample in $z_1,\ldots,z_{12}\in\mathbb{R}^2$3. The window is Schwartz, the coefficients are explicitly specified, the phase-space points are pairwise distinct, and the dependence holds pointwise after choosing the smooth representative.

Numerical Reconstruction and Cancellation

The paper reconstructs a discretized approximation $z_1,\ldots,z_{12}\in\mathbb{R}^2$4 using a $z_1,\ldots,z_{12}\in\mathbb{R}^2$5 grid and $z_1,\ldots,z_{12}\in\mathbb{R}^2$6 iterations of the invariant-line iteration. The estimated eigenvalue is

$z_1,\ldots,z_{12}\in\mathbb{R}^2$7

The reported discretization residuals are at approximately double-precision scale:

• invariant-line residual: $z_1,\ldots,z_{12}\in\mathbb{R}^2$8;
• normalization residual: $z_1,\ldots,z_{12}\in\mathbb{R}^2$9;
• cohomological residual: $f_* \in \mathcal{S}(\mathbb{R})$00.

These numbers are not proofs of the exact result, but they verify numerical consistency with the analytical construction.

Figure 1: Floating-point reconstruction of the Schwartz window, displaying its modulus, real part, and imaginary part.

The reconstructed window is concentrated primarily on a bounded interval in the displayed numerical approximation. With the chosen normalization, its largest sampled value is approximately $f_* \in \mathcal{S}(\mathbb{R})$01 at $f_* \in \mathcal{S}(\mathbb{R})$02, and the sampled $f_* \in \mathcal{S}(\mathbb{R})$03 mass on $f_* \in \mathcal{S}(\mathbb{R})$04 is approximately $f_* \in \mathcal{S}(\mathbb{R})$05.

The direct physical-space evaluation is particularly informative. The largest individual summand has magnitude about $f_* \in \mathcal{S}(\mathbb{R})$06, while the maximum sum of the magnitudes of all twelve terms is approximately $f_* \in \mathcal{S}(\mathbb{R})$07. Nevertheless, the residual of their complex sum is only

$f_* \in \mathcal{S}(\mathbb{R})$08

and the ratio between the residual $f_* \in \mathcal{S}(\mathbb{R})$09 norm and the sum of the individual $f_* \in \mathcal{S}(\mathbb{R})$10 norms is

$f_* \in \mathcal{S}(\mathbb{R})$11

Thus, the relation is not explained by the individual terms being small. It is a finely structured cancellation among twelve nontrivial time-frequency translates.

Figure 2: Direct floating-point evaluation showing cancellation among the twelve summands and a residual at approximately double-precision rounding level.

The Role of LLMs

The paper explicitly reports that the counterexample and much of the initial proof strategy were developed through interaction with a LLM, identified by the authors as ChatGPT GPT-5.6 Pro. The authors describe a workflow in which the model proposed configurations and proof strategies, while the researchers selected the final construction, replaced several arguments, supplied missing details, corrected an erroneous lemma, and independently verified the mathematical content.

This disclosure is significant for the methodology of formal mathematics, but it does not alter the logical status of the theorem. The proof depends on exact symbolic identities, a reproducible validated-numerics certificate, and a sequence of analytical arguments that the authors claim to have independently checked. At the same time, the paper documents substantial gaps in the initial machine-generated reasoning, including an underdeveloped regularity argument and a nontrivial incorrect proof that required repair.

The example illustrates both the utility and the limitations of LLM-assisted mathematical discovery. LLMs can support combinatorial exploration, suggest operator factorizations, and generate candidate proof architectures. They remain unreliable as autonomous sources of proof, particularly in arguments involving regularity propagation, boundary sewing, noncommutative phases, and quantified numerical estimates. The effective workflow here is therefore not automated theorem proving but machine-assisted conjecture generation followed by expert mathematical validation.

Future AI systems for mathematical research would need stronger support for formal verification, exact symbolic computation, dependency tracking, and certified numerical reasoning. In this setting, an especially valuable capability would be an integrated system that could generate candidate Weyl polynomials, formally verify the associated Zak-transform identities, and produce machine-checkable interval certificates for the global perturbation bound. The paper’s workflow provides a concrete benchmark for such systems, because success requires coordination between algebra, harmonic analysis, dynamical systems, and validated computation.

Implications for Time-Frequency Analysis

The result changes the status of the HRT problem from a universal linear-independence conjecture to a classification problem. Several substantial positive results remain intact: independence for lattice configurations, for many small configurations, and for various special classes of windows and point sets. Linnell’s theorem is not contradicted because the twelve-point support is not contained in a single discrete subgroup in the relevant way.

The counterexample also identifies several mechanisms that may be essential for dependence:

1. An irrational translation: the phase-space configuration is not purely lattice-based.
2. A translated half-lattice plus the origin: the origin lies outside the translated lattice component.
3. A matrix-valued Zak representation: scalar Zak methods are topologically obstructed.
4. A rank-one cocycle approximation: the finite Weyl polynomial is engineered to approximate a solvable invariant-line model.
5. Diophantine control: the irrational translation must have sufficiently controlled resonances to solve the cohomological equation smoothly.

The construction is not generic evidence that arbitrary finite time-frequency systems are dependent. It is an explicit engineered example. Important questions remain open concerning the minimum number of shifts, the possible geometry of minimal counterexamples, dependence for other function classes, and the stability of dependence under perturbations of the phase-space points or coefficients.

The numerical architecture also suggests a broader constructive program. One could search for further counterexamples by designing matrix cocycles close to rank-one transport models and then enforcing the necessary sewing and arithmetic constraints. Conversely, one might seek geometric criteria that rule out such cocycle approximations for configurations with fewer points or more restrictive phase-space arrangements.

Conclusion

“Linear dependence of time-frequency shifts of a Schwartz function” (2608.05044) gives an explicit twelve-term Schwartz-class counterexample to the HRT conjecture. Its proof combines a vector-Zak reformulation, an exact finite Weyl polynomial, a certified global estimate

$f_* \in \mathcal{S}(\mathbb{R})$12

a contraction-based invariant-line construction, and a Diophantine Fourier solution of the resulting scalar cohomological equation. The paper’s central contribution is both negative and constructive: finite time-frequency shifts are not universally linearly independent, even for Schwartz windows, but the dependence can be realized through a rigorously specified and reproducible analytic mechanism.