Fock-space HRT Conjecture

• The Fock-space HRT Conjecture is defined through the linear independence of finite Weyl translates in the Bargmann–Fock space, connecting to deep zero uniqueness phenomena in entire functions.
• The research employs rotation-eigenspace decomposition and reproducing kernel Hilbert space techniques to establish affirmative results for cases where d is in {2,3,4,6}.
• The study bridges time-frequency analysis and complex function theory, with implications for extending the HRT conjecture to new settings and addressing open cases for other d values.

The Fock-space HRT Conjecture explores the linear independence of finite Weyl (time–frequency) translates in the Bargmann–Fock space, drawing a profound connection with “deep zero” uniqueness phenomena for entire functions. Motivated by Hedenmalm’s deep zero problems, this conjecture bridges classical questions in time-frequency analysis (the HRT conjecture) and the theory of reproducing kernel Hilbert spaces through the geometry of regular polygons and lattices in the complex plane (Li et al., 14 Jan 2026).

1. Structural Foundations: The Bargmann–Fock Space, Kernels, and Weyl Operators

Let $dA(z)$ denote Lebesgue area measure on $\mathbb{C}$, normalized as $dA = \pi^{-1} dx\,dy$. The Bargmann–Fock space $F^2$ is the Hilbert space of entire functions $f: \mathbb{C}\rightarrow\mathbb{C}$ with finite norm under the Gaussian measure $d\mu_G(z) = e^{-|z|^2} dA(z)$:

$$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$

The inner product (after possible rescaling) takes the form $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$. $F^2$ is a reproducing‐kernel Hilbert space with kernel $K_w(z) = e^{z\bar{w}}$ or, in the $\mathbb{C}$0-normalized convention, $\mathbb{C}$1.

The Bargmann transform $\mathbb{C}$2 is the unitary operator:

$\mathbb{C}$3

Under $\mathbb{C}$4, Gabor shifts $\mathbb{C}$5 transfer to Weyl (Heisenberg) unitary operators $\mathbb{C}$6 in Fock space:

$\mathbb{C}$7

with $\mathbb{C}$8. Finite linear relations among Gabor shifts in $\mathbb{C}$9 thus correspond exactly to relations among the Weyl translates in $dA = \pi^{-1} dx\,dy$0.

2. Hedenmalm's Deep Zero Problems and Problem 5.2

The general deep zero problem in $dA = \pi^{-1} dx\,dy$1 fixes $dA = \pi^{-1} dx\,dy$2, decomposing $dA = \pi^{-1} dx\,dy$3 into $dA = \pi^{-1} dx\,dy$4 arithmetic progressions: $dA = \pi^{-1} dx\,dy$5, $dA = \pi^{-1} dx\,dy$6. Bounded operators $dA = \pi^{-1} dx\,dy$7 act on $dA = \pi^{-1} dx\,dy$8, and one considers:

$dA = \pi^{-1} dx\,dy$9

For Problem 5.2 (d = 4), with $F^2$0, $F^2$1 for $F^2$2 and $F^2$3, this becomes:

• $F^2$4 for $F^2$5,
• $F^2$6 for $F^2$7.

By Taylor expansion, these are equivalent to the functional system:

• $F^2$8,
• $F^2$9.

The question becomes: must $f: \mathbb{C}\rightarrow\mathbb{C}$0 if $f: \mathbb{C}\rightarrow\mathbb{C}$1 satisfies this system?

3. Connection to the HRT Conjecture

The classical HRT (Heil–Ramanathan–Topiwala) conjecture posits that if $f: \mathbb{C}\rightarrow\mathbb{C}$2 and $f: \mathbb{C}\rightarrow\mathbb{C}$3 are distinct, then

$f: \mathbb{C}\rightarrow\mathbb{C}$4

are linearly independent in $f: \mathbb{C}\rightarrow\mathbb{C}$5. In Fock space, this equivalently asserts that for $f: \mathbb{C}\rightarrow\mathbb{C}$6 and distinct $f: \mathbb{C}\rightarrow\mathbb{C}$7, the Weyl translates $f: \mathbb{C}\rightarrow\mathbb{C}$8 are linearly independent.

A deep zero vanishing condition translates into a finite linear relation among Weyl translates. Thus, an affirmative answer to the HRT conjecture for the corresponding point configuration rules out nontrivial solutions to the deep zero uniqueness problem.

Established results (collected as Theorem D in (Li et al., 14 Jan 2026)) give special cases of HRT:

• $f: \mathbb{C}\rightarrow\mathbb{C}$9,
• $d\mu_G(z) = e^{-|z|^2} dA(z)$0 where points lie on two parallel lines,
• $d\mu_G(z) = e^{-|z|^2} dA(z)$1 forming a regular lattice (up to an affine shift).

4. Principal Results and Proof Strategies

4.1 Solutions for Special Values of $d\mu_G(z) = e^{-|z|^2} dA(z)$2

Theorem A (for $d\mu_G(z) = e^{-|z|^2} dA(z)$3) affirms: for even/odd progressions and any $d\mu_G(z) = e^{-|z|^2} dA(z)$4,

$d\mu_G(z) = e^{-|z|^2} dA(z)$5

The main theorem (Theorems 1, 6; Corollary 7) establishes for $d\mu_G(z) = e^{-|z|^2} dA(z)$6: If $d\mu_G(z) = e^{-|z|^2} dA(z)$7, $d\mu_G(z) = e^{-|z|^2} dA(z)$8, and $d\mu_G(z) = e^{-|z|^2} dA(z)$9 for $$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$0, then $$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$1. This solves Hedenmalm’s Problem 5.2 ($$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$2, $$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$3), and extends to new analytic cases for $$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$4.

4.2 Proof Structure for General $$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$5

• $$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$6 decomposes into $$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$7 rotation-eigenspaces: $$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$8, $$\|f\|^2_{F^2} := \int_{\mathbb{C}} |f(z)|^2\, d\mu_G(z) < \infty.$$9.
• Let $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$0, $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$1. The rotation operator $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$2.
• Lemma 5: $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$3.
• The vanishing hypothesis propagates as constraints on the projections $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$4 for $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$5, forcing a linear relation of the form $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$6.
• If the HRT conjecture is known for the configuration $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$7, these translates are linearly independent, forcing $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$8 and thus $$\langle f, g \rangle = \int_{\mathbb{C}} f(z)\, \overline{g(z)}\, e^{-\pi|z|^2}\, dz$$9.

4.3 Geometric Aspects

$F^2$0th roots of unity form regular polygons; for $F^2$1 they comprise a subset of a lattice. Theorems prove that outside these cases ($F^2$2), the $F^2$3th roots of unity are not subsets of any regular lattice, restricting the solvable cases to those covered by lattice-based HRT results.

5. Consequences, Open Cases, and Analytic Continuation

The deep zero problem analysis yields new affirmative HRT cases: for $F^2$4, $F^2$5, and $F^2$6, the Weyl translates $F^2$7 are linearly independent in $F^2$8. Thus, the Fock-space HRT conjecture holds for all regular $F^2$9-gons of order $K_w(z) = e^{z\bar{w}}$0.

The problem remains open for other $K_w(z) = e^{z\bar{w}}$1, particularly $K_w(z) = e^{z\bar{w}}$2, where the associated roots of unity do not lie on a regular lattice. Problem 8 in (Li et al., 14 Jan 2026) asks whether the HRT conjecture holds for the configuration $K_w(z) = e^{z\bar{w}}$3 for all $K_w(z) = e^{z\bar{w}}$4; an affirmative resolution here would generalize deep zero uniqueness to all $K_w(z) = e^{z\bar{w}}$5.

6. Significance and Interrelation of Deep-Zero and HRT Uniqueness

This research reveals a deep structural correspondence between two seemingly disparate areas:

• The “deep zero” phenomena (uniqueness given vanishing of many derivatives in $K_w(z) = e^{z\bar{w}}$6),
• Finite linear independence of Weyl time-frequency systems (HRT).

Progress on one front immediately impacts the other; new lattice/regular polygon HRT cases generate new deep zero theorems, and vice versa. This synergy illustrates a previously underexplored mathematical unity at the interface of time-frequency analysis, complex function theory, and operator theory (Li et al., 14 Jan 2026).