Balian-Low type theorems on homogeneous groups

David Rottensteiner; Jordy Timo van Velthoven; Jos\'e Luis Romero; Karlheinz Gr\"ochenig

Balian-Low type theorems on homogeneous groups

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read On any homogeneous Lie group, a coherent frame generated by an integrable vector must have lower Beurling density strictly larger than the representation's formal dimension; a Riesz sequence must have strictly smaller upper density.

desk verdict A genuine strictness result for coherent density on homogeneous groups, with a proof that is broadly sound; the only external step that deserves a pointed referee look is the borrowed inverse-closedness in Theorem B.1. read the letter →

arxiv 1908.03053 v2 pith:O3YHT4UT submitted 2019-08-08 math.FA

Karlheinz Gröchenig , José Luis Romero , David Rottensteiner , Jordy Timo van Velthoven This is my paper · ORCID

classification math.FA MSC 22E2522E2742C1542C40

keywords Balian-LowtheoremhomogeneousgroupscoherentframesRieszsequencesBeurlingdensityformaldimensiondeformationtheoryspectralinvariance

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 proves strict necessary density conditions for coherent frames and Riesz sequences on homogeneous Lie groups, that is, connected simply connected nilpotent groups equipped with a family of dilations. In the orbit of an irreducible representation that is square-integrable modulo the center, with formal dimension $d_\pi$, an integrable generator $g$ can form a frame only if the index set has lower Beurling density $D^-(\Lambda)>d_\pi$, and a Riesz sequence only if its upper density satisfies $D^+(\Lambda)

What carries the argument

The load-bearing mechanism is the deformation of the index set by the homogeneous dilations $D_r$ of the group. Theorems 3.5 and 3.6 show that if $\pi(\Lambda)g$ is a frame, respectively a Riesz sequence, then for all $r$ sufficiently close to $1$ the dilated system $\pi(D_r(\Lambda))g$ is again a frame, respectively a Riesz sequence. This stability is derived through the universality theorem for $p$-frames and $p$-Riesz sequences, whose proof uses the off-diagonal decay of Gramian matrices and the spectral invariance of the weighted Schur algebra $A^1_{v_\alpha}(\Gamma)$ over relatively separated subsets: the pseudo-inverse of a localized matrix remains localized. The density of the dilated set scales as $D^{\pm}(D_r(\Lambda))=r^{-Q}D^{\pm}(\Lambda)$, so an arbitrarily small dilation pushes the density across the critical value $d_\pi$ and produces the contradiction.

What would settle it

Find a homogeneous Lie group $N$, an irreducible square-integrable-modulo-center representation with formal dimension $d_\pi$, an integrable vector $g$, and a discrete set $\Lambda$ with $D^-(\Lambda)=d_\pi$ such that $\{\pi(\lambda)g:\lambda\in\Lambda\}$ is a frame, or with $D^+(\Lambda)=d_\pi$ forming a Riesz sequence. For the Heisenberg group this reduces to an explicit computational search over lattices and windows with $|\langle g,\pi(x)g\rangle|\in L^1$; the theorem predicts no such frame or Riesz sequence exists at the critical density.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: if $g$ is an integrable vector, meaning $\int_{N/Z(N)}|\langle g,\pi(x)g\rangle|\,d\mu(\dot{x})<\infty$, then $\{\pi(\lambda)g:\lambda\in\Lambda\}$ cannot be a frame unless $D^-(\Lambda)>d_\pi$ and cannot be a Riesz sequence unless $D^+(\Lambda)<d_\pi$. An immediate consequence is that no orthonormal basis or Riesz basis in the orbit of an integrable vector exists, so in particular smooth vectors never generate such bases. The proof argues by contradiction: assume equality holds in the known inequalities $D^-\ge d_\pi$ or $D^+\le d_\pi$, then dilate the index set slightly, use the stability of the frame or Riesz property under dilations, and observe that the dilated set violates the known bound.

Load-bearing premise

The argument collapses if the inverse-closedness of the weighted Schur algebra fails for relatively separated subsets of a non-abelian homogeneous group, because then the pseudo-inverse of a localized Gramian need not stay localized and the universality theorem that powers the dilation stability no longer holds.

Editorial extensions

If this is right

No orthonormal basis or Riesz basis can be formed from the orbit $\pi(\Lambda)g$ of an integrable vector; in particular, smooth vectors cannot generate such bases.
The necessary density conditions are strict for every index exponent: a $p$-frame must satisfy $D^-(\Lambda)>d_\pi$ and a $p$-Riesz sequence must satisfy $D^+(\Lambda)<d_\pi$ for all $p\in[1,\infty]$.
The Balian-Low obstruction is not special to the Heisenberg group: it holds for every homogeneous group, in line with the expectation from the Kirillov lemma that every nilpotent Lie group contains a Heisenberg-like subgroup.
Integrability of the generator is enough to force strictness, so the known non-strict density bounds cannot be attained at the critical density.

Reading between the lines

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

If the spectral-invariance machinery is as robust as the paper's use suggests, the same deformation-plus-contradiction scheme should yield strict density inequalities for any necessary density bound on a measured metric space admitting a dilation family and polynomial volume growth, not only for group-coorbit frames.
The strict threshold suggests a sharp phase transition: density $d_\pi$ separates the frame regime from the Riesz-sequence regime, and one could test numerically on low-dimensional homogeneous groups whether the frame algorithm's condition number blows up as $D^-$ approaches $d_\pi$ from above.
Since smooth vectors are excluded from critical-density bases, the result strengthens the heuristic that 'nice' functions and bases are incompatible in the orbit picture, which may be read as an uncertainty principle on homogeneous groups; verifying analogues for more general Lie groups with dilation-like deformations is a natural next step.

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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 thing you should know: this paper proves the strict versions of the density inequalities for coherent frames and Riesz sequences on homogeneous nilpotent Lie groups. That is new. The earlier result [21] only gave D^-(Λ) ≥ d_π and D^+(Λ) ≤ d_π, and the Balian–Low phenomenon had been known for Gabor systems, the Heisenberg group, and Fock spaces. Extending it to all homogeneous groups is a legitimate step, and the main theorem is what it claims to be.

The paper's real work is in the technical apparatus: a deformation theorem for coherent systems under dilations, a universality result that transfers frame/Riesz properties across L^p coorbit spaces, and an appendix that builds a localized reference frame whose canonical dual has good off-diagonal decay. Proposition B.4 is the most valuable new piece; it goes beyond earlier existence proofs by adding the dual-frame localization. The writing is careful and the main argument is transparent: assume critical density, smooth the generator via Proposition 4.4, dilate the index set, use stability to preserve the frame/Riesz property, and contradict the non-strict inequalities. That chain is internally consistent.

The soft spot is Theorem B.1. Inverse-closedness of the weighted Schur algebra A^1_v_α(Γ) is imported from Sun [45], and the paper's proof consists of verifying the polynomial-growth hypothesis by a packing estimate. That is the right thing to do, and I think the verification works: relatively separated in a homogeneous group is exactly bounded geometry, the packing bound covers r≥1, and for r<1 the uniform unit-ball bound closes the gap. The stress-test worry about pairs clustered at small distances does not land, because bounded geometry plus polynomial growth are precisely what Sun's theorem requires. Still, this is the one place where the present paper does not re-derive its load-bearing spectral invariance, and a referee should at least confirm that Sun's standing hypotheses are satisfied verbatim for the restricted metric on Γ, not just in analogy with the Euclidean case.

Everything else is either proved in the paper or cited in a standard way. Theorem 4.1 comes from [21], and the homogeneous approximation property is cited to [23]; those are acceptable external dependencies for a paper of this type. No circularity. I did not find a serious flaw in the main theorem or in the deformation/universality chain.

Who should read this: anyone working in coorbit theory, abstract frame theory, or density questions for group representations. It deserves a serious referee slot. If I were the editor, I would send it out; the main risk is that the referee gets bogged down in the appendix, but the core proof is short and plausible. I would cite this paper in my own work.

Recommendation: send to peer review with a referee who knows Sun's Wiener-lemma paper, not just frame theory.

Referee Report

0 major / 5 minor

Summary. The paper proves strict necessary density conditions for coherent frames and Riesz sequences in the orbit of a square-integrable projective representation modulo the center on a homogeneous Lie group. Theorem 1.1 states that if g is an integrable vector and pi(Lambda)g is a frame for H_pi, then the lower Beurling density satisfies D^-(Lambda) > d_pi, and if pi(Lambda)g is a Riesz sequence, then D^+(Lambda) < d_pi. The proof proceeds by contradiction from the previously known non-strict inequalities of Theorem 4.1, using a smooth-vector approximation (Proposition 4.4), dilation-stability theorems for frames and Riesz sequences (Theorems 3.5 and 3.6), and a universality theorem for p-frames and p-Riesz sequences (Theorem 2.2). A substantial appendix develops the technical machinery: a weighted Schur algebra inverse-closedness theorem (Theorem B.1), an extension of Sjöstrand's Wiener lemma to homogeneous groups (Proposition B.3), and a construction of a localized reference frame with a localized canonical dual frame (Proposition B.4).

Significance. If correct, this is a substantial and natural extension of Balian-Low type strict density inequalities from the Heisenberg group and Euclidean spaces to all connected, simply connected nilpotent Lie groups with a dilation structure. The paper is well organized and, unusually for this area, the main proof chain is backed by a detailed appendix; the universality theorem for p-frames and p-Riesz sequences and the existence of a localized canonical dual frame (Proposition B.4) are results of independent interest. The main argument is internally consistent: the contradiction in Theorem 1.1 relies only on Theorem 4.1, Proposition 4.4, and the dilation-stability theorems, and the technical spine in the appendix is carefully written. The dependence on the published inverse-closedness theorem of Sun [45] is acceptable because the hypotheses are verified by a packing argument in Appendix B.1; the verification is brief but sufficient, and the issue is only that the exposition could be slightly more self-contained.

minor comments (5)

[Appendix B.1, Theorem B.1] The verification of the standing hypotheses of [45] proves the polynomial-growth condition for the index set Gamma only for r >= 1; for 0 < r < 1 the bound follows immediately from relative separatedness, but it should be stated explicitly so that all r > 0 are covered.
[Section 3.1, proof of Theorem 3.6] There is a typo in the displayed weak convergence 'lambda_n^{-1} Lambda_n^{-1} -> Gamma'; it should read lambda_n^{-1} Lambda_n -> Gamma.
[Appendix B, proof of Theorem B.5] The symbol '/greaterorsimilar' appears twice in the displayed inequalities; these should be the relation '≳'.
[Section 2.6, proof of Theorem 2.2 (ii)] The application of Theorem B.5 to A^* is terse; please state explicitly that the identity operator on ell^p(Lambda) plays the role of P and satisfies the envelope condition, so that the reader can see how Theorem B.5 yields the lower bound for C^*_{g,Lambda} on every ell^q.
[Throughout] The notation d_pi is written as both 'd_pi' and 'dpi' in several places; please standardize the typesetting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the strict density theorem is proved from the weaker non-strict density theorem by independent dilation-stability and smooth-approximation arguments.

full rationale

Theorem 1.1 is not built from itself. The proof assumes equality at the critical density and derives a contradiction from the previously published non-strict inequalities (Theorem 4.1), which are cited from [21] and not derived in this paper. The strictness conclusion therefore has independent content. The new work is in the deformation/stability theorems (3.5 and 3.6), the smooth approximation Proposition 4.4, and the appendix proving universality (Theorem 2.2). The appendix does import inverse-closedness of the weighted Schur algebra (Theorem B.1) from Sun [45], and the paper verifies the standing hypotheses by a packing argument; this is an external, non-equivalent ingredient, not a re-statement of the target result. Self-citations to [21], [23], [25], [28], and [40] are used for published technical lemmas or as methodological antecedents, and the load-bearing ones are either re-proved in the paper or are external theorems weaker than Theorem 1.1. No fitted parameter is renamed as a prediction, and no uniqueness theorem is invoked to force the construction. The derivation chain is therefore not circular, though it carries an external-dependency caveat concerning the borrowed spectral-invariance theorem.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof loads a substantial amount of prior literature: non-strict density inequalities, coorbit atomic decomposition, Wiener lemma for weighted Schur algebras, and Dixmier-Malliavin smoothing. None of these are fitted to data; they are external theorems. The central contribution is the strictification via deformation. No new entities are introduced.

assumptions (6)

domain assumption Non-strict density inequalities for coherent frames and Riesz sequences: D^-(Lambda) >= d_pi and D^+(Lambda) <= d_pi for g in B_pi.
Theorem 4.1 in the paper, quoted from [21, Corollary 4.1]; used as the contradiction baseline in the proof of Theorem 1.1.
domain assumption The weighted Schur algebra A^1_{v_alpha}(Gamma) is inverse-closed and pseudo-inverse closed in B(ell^2(Gamma)).
Theorem B.1, cited from Sun [45, Theorems 4.1 and 5.1]; load-bearing for the localization of the canonical dual (Proposition B.4) and for the universality theorem (Theorem 2.2).
domain assumption Existence of relatively separated, relatively dense Gamma such that pi(Gamma)h is a p-frame for smooth h, and atomic decomposition in coorbit spaces.
From Feichtinger-Grochenig [14, Theorem 6.1], Grochenig [22], and Christensen [7]; used in Proposition B.4 and Lemma 4.3.
domain assumption Uniform convergence on compacta of coefficient maps V_g h_n to V_g h for weak-* convergent sequences in coorbit spaces.
Cited as [14, Theorem 4.1]; used in Theorems 3.2, 3.5, and 3.6 to pass point evaluations through weak limits.
domain assumption The Dixmier-Malliavin theorem that H^infty_pi equals the span of Garding vectors.
Cited at the end of Section 2.3; supports the density of smooth vectors used in Proposition 4.4 and Lemma 4.3.
standard math The homogeneous norm and metric structure on homogeneous groups, including Haar measure scaling mu_G(D_r(E)) = r^Q mu_G(E).
Background from Folland-Stein [18] and Fischer-Ruzhansky [17]; used throughout for densities and weak limits.

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Pith. "Pith review of Balian-Low type theorems on homogeneous groups." pith.science (2026). https://pith.science/paper/O3YHT4UT

@misc{pith2026190803053,
  author       = {Pith},
  title        = {Pith review of: Balian-Low type theorems on homogeneous groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O3YHT4UT}},
  note         = {Machine review of arXiv:1908.03053}
}

abstract

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let $N$ be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let $(\pi, \mathcal{H}_{\pi})$ be an irreducible, square-integrable representation modulo the center $Z(N)$ of $N$ on a Hilbert space $\mathcal{H}_{\pi}$ of formal dimension $d_\pi $. If $g \in \mathcal{H}_{\pi}$ is an integrable vector and the set $\{ \pi (\lambda )g : \lambda \in \Lambda \}$ for a discrete subset $\Lambda \subseteq N / Z(N)$ forms a frame for $\mathcal{H}_{\pi}$, then its density satisfies the strict inequality $D^-(\Lambda )> d_\pi $, where $D^-(\Lambda )$ is the lower Beurling density. An analogous density condition $D^+(\Lambda) < d_{\pi}$ holds for a Riesz sequence in $\mathcal{H}_{\pi}$ contained in the orbit of $(\pi, \mathcal{H}_{\pi})$. The proof is based on a deformation theorem for coherent systems, a universality result for $p$-frames and $p$-Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [21]

H. Führ, K. Gröchenig, A. Haimi, A. Klotz, and J. L. Romer o. Density of sampling and interpo- lation in reproducing kernel Hilbert spaces. J. Lond. Math. Soc. (2) , 96(3):663–686, 2017

work page 2017

[2] [45]

Q. Sun. Wiener’s lemma for inﬁnite matrices. Trans. Amer. Math. Soc. , 359(7):3099–3123, 2007

work page 2007

[3] [23]

Gröchenig

K. Gröchenig. The homogeneous approximation property and the comparison theorem for coherent frames. Sampl. Theory Signal Image Process. , 7(3):271–279, 2008

work page 2008

[4] [1]

Aldroubi, A

A. Aldroubi, A. Baskakov, and I. Krishtal. Slanted matri ces, Banach frames, and sampling. J. Funct. Anal., 255(7):1667–1691, 2008

work page 2008

[5] [2]

Ascensi, H

G. Ascensi, H. G. Feichtinger, and N. Kaiblinger. Dilati on of the Weyl symbol and Balian-Low theorem. Trans. Amer. Math. Soc. , 366(7):3865–3880, 2014

work page 2014

[6] [3]

Balan, P

R. Balan, P. G. Casazza, C. Heil, and Z. Landau. Density, o vercompleteness, and localization of frames. I. Theory. J. Fourier Anal. Appl. , 12(2):105–143, 2006

work page 2006

[7] [4]

Balan, P

R. Balan, P. G. Casazza, C. Heil, and Z. Landau. Density, o vercompleteness, and localization of frames. II. Gabor systems. J. Fourier Anal. Appl. , 12(3):309–344, 2006

work page 2006

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G. Battle. Heisenberg proof of the Balian-Low theorem. Lett. Math. Phys. , 15(2):175–177, 1988

work page 1988