{"id":"c7df0023-6855-4f89-b084-3a1d330217dc","arxiv_id":"1908.03053","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Coherent frames from integrable vectors on homogeneous nilpotent Lie groups have lower density strictly larger than the formal dimension, and Riesz sequences have upper density strictly smaller.","lead":"The paper proves that in a large family of non-flat spaces with self-similar scaling, a complete set of shifted copies of one wave must use strictly more than a critical density of shifts. It is an extension of a fundamental 'uncertainty' theorem from signal processing to new geometric settings.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The strict density proof is internally sound; the least internally supported link is Theorem B.1's borrowed inverse-closedness for relatively separated index sets, which powers the universality theorem.","rationale":"After reading the paper through the contradiction argument, I find no internal flaw. The deformation of Lambda by dilations D_{r_n} with r_n tending to 1 is handled by weak-limit techniques; Lemma 3.4 correctly identifies the weak limits as elements of W(Lambda); Theorems 3.5 and 3.6 then transfer frame and Riesz properties to nearby dilated sets. Proposition 4.4 legitimately upgrades an integrable vector to a smooth vector while preserving the frame or Riesz property, so the later application of Theorem 4.1 to the deformed sets is legitimate. The only place where the proof leans on a nontrivial result not demonstrated in the paper is the spectral invariance of the weighted Schur class on relatively separated subsets of a homogeneous group, stated in Theorem B.1 and used in Proposition B.4. This is exactly the reader's weakest assumption, and I agree with that identification. The concern does not amount to a demonstrated error: the packing argument covers large-scale polynomial growth, and the admissible-weight condition is cited to a published example. However, because the entire universality theorem and hence the dilation stability results depend on the localized pseudo-inverse, a specialist should verify the hypotheses of Sun's theorem line-by-line, especially the small-radius behavior for non-separated relatively separated sets. No adjustment to the reader's ACCEPT verdict is warranted.","tokens_in":27825,"tokens_out":16942,"duration_ms":182909,"concrete_test":"Check Sun [45, Section 2] against the triple (Gamma, d_G|_Gamma, mu_c) for a merely relatively separated Gamma. Verify exactly whether the polynomial-growth condition is required for all radii r > 0 or only for large r, and whether the weight v_alpha(gamma, gamma') = (1 + |gamma^{-1} gamma'|_G)^alpha is admissible for all alpha > 0. If the polynomial-growth condition is only verified for r >= 1, either prove the missing small-radius bound using the finiteness of Gamma cap B_1(x) and the exact form of Sun's growth condition, or exhibit a non-separated relatively separated set with pairs at distance epsilon_n tending to 0 and a weighted-Schur-invertible Gramian whose pseudo-inverse fails to lie in A^1_{v_alpha}(Gamma).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The contradiction in Theorem 1.1 rests on a chain: smooth approximation (Prop. 4.4), dilation stability (Thms. 3.5 and 3.6), weak-limit stability (Thms. 3.2 and 3.3), universality (Thm. 2.2), the localized pseudo-inverse of the reference Gramian (Prop. B.4), and finally inverse-closedness of the weighted Schur algebra A^1_{v_alpha}(Gamma), stated as Theorem B.1 and imported from Sun [45, Thms. 4.1 and 5.1]. This is the only step in the main argument that is not proved in the paper. The paper verifies the polynomial-growth and admissible-weight hypotheses by a packing estimate for r at least 1, but it does not re-derive spectral invariance in the non-abelian homogeneous-group setting. If Sun's theorem requires stronger separation than 'relatively separated', or if its pseudo-inverse conclusion fails for sets with pairs clustered at arbitrarily small distances, then the Gramian pseudo-inverse G^dagger need not have the off-diagonal decay asserted in Proposition B.4, and the universality theorem collapses. This is a genuine external dependency rather than an observed contradiction; the cited hypotheses appear plausible, but the step remains the least secure load-bearing assumption in the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":28096,"tokens_out":29115,"duration_ms":274421,"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.","major_comments":[],"minor_comments":[{"comment":"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":"Appendix B.1, Theorem B.1"},{"comment":"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.","section":"Section 3.1, proof of Theorem 3.6"},{"comment":"The symbol '/greaterorsimilar' appears twice in the displayed inequalities; these should be the relation '≳'.","section":"Appendix B, proof of Theorem B.5"},{"comment":"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.","section":"Section 2.6, proof of Theorem 2.2 (ii)"},{"comment":"The notation d_pi is written as both 'd_pi' and 'dpi' in several places; please standardize the typesetting.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a strong contribution to abstract harmonic analysis and frame theory. The main theorem is significant and the proof is careful. The reliance on Sun's inverse-closedness theorem is not a concern for correctness, but the authors could improve the paper by adding a short remark that quotes the exact statements from [45] being used and notes explicitly that they apply verbatim to relatively separated subsets of homogeneous groups. No novelty concerns: the work builds naturally on earlier papers by the same group and others, and the extension to homogeneous groups is nontrivial."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nEverything 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.\n\nWho 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.\n\nRecommendation: send to peer review with a referee who knows Sun's Wiener-lemma paper, not just frame theory.","headline":"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.","tokens_in":830,"tokens_out":988,"would_cite":true,"duration_ms":36159,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E25","22E27","42C15","42C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Balian-Low theorem","homogeneous groups","coherent frames","Riesz sequences","Beurling density","formal dimension","deformation theory","spectral invariance"],"falsifier":"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.","tokens_in":27648,"feed_emoji":"📏","tokens_out":9092,"duration_ms":87925,"temperature":0.7,"pith_summary":"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 $D^+(\\Lambda)<d_\\pi$. This upgrades the previously known non-strict inequalities to strict ones and rules out orthonormal bases and Riesz bases at the critical density, in particular for smooth vectors. A sympathetic reader should care because it extends the Balian-Low phenomenon, originally a no-go statement for Gabor systems on the Heisenberg group, to the whole class of homogeneous groups.","feed_headline":"Frames need supercritical density on homogeneous groups","feed_subtitle":"A Balian-Low-type theorem shows integrable generators cannot form frames or Riesz bases at the critical density.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the non-strict density inequalities $D^-(\\Lambda)\\ge d_\\pi$ and $D^+(\\Lambda)\\le d_\\pi$ that the proof contradicts.","marker":"[21]"},{"why":"Supplies the deformation-of-frames strategy (dilating the index set near critical density) that the proof adapts to homogeneous groups.","marker":"[25]"},{"why":"Proves inverse-closedness and pseudo-inverse closedness of the weighted Schur algebra, the spectral-invariance fact the universality theorem rests on.","marker":"[45]"},{"why":"Supplies the weak-limit stability technique and the linear-algebra extension (Theorem B.5) used to propagate frame and Riesz properties to dilations.","marker":"[28]"},{"why":"Foundational coorbit theory and atomic decompositions used to construct localized reference frames and dual frames.","marker":"[14]"},{"why":"Sjöstrand's Wiener-type lemma, the template for the off-diagonal decay and $\\ell^p$-stability results (Proposition B.3) on homogeneous groups.","marker":"[44]"},{"why":"Introduces the two-sided amalgam spaces used in the envelope estimates for Gramian matrices over relatively separated sets.","marker":"[40]"},{"why":"Establishes the homogeneous approximation property used to verify the hypotheses of the non-strict density theorem.","marker":"[23]"}],"fun_headline_variants":["Strict density needed for coherent frames on homogeneous groups","Balian-Low on homogeneous groups: no critical density frames","Integrable vectors can't hit critical density for frames or Riesz","Homogeneous groups force strict density for coherent systems","No Riesz bases from integrable vectors at critical density"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Strict density needed for coherent frames on homogeneous groups","Balian-Low on homogeneous groups: no critical density frames","Integrable vectors can't hit critical density for frames or Riesz","Homogeneous groups force strict density for coherent systems","No Riesz bases from integrable vectors at critical density"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1455,"prompt_tokens":976,"completion_tokens":479,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":398}},"tokens_in":592,"tokens_out":479,"duration_ms":5104,"temperature":1.0,"reasoning_tokens":398,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:26:11.812461+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the non-strict density inequalities $D^-(\\Lambda)\\ge d_\\pi$ and $D^+(\\Lambda)\\le d_\\pi$ that the proof contradicts."},{"cited_title":"Gröchenig, J","cited_arxiv_id":null,"evidence_quote":"Supplies the deformation-of-frames strategy (dilating the index set near critical density) that the proof adapts to homogeneous groups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves inverse-closedness and pseudo-inverse closedness of the weighted Schur algebra, the spectral-invariance fact the universality theorem rests on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the weak-limit stability technique and the linear-algebra extension (Theorem B.5) used to propagate frame and Riesz properties to dilations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Foundational coorbit theory and atomic decompositions used to construct localized reference frames and dual frames."},{"cited_title":"Sjöstrand","cited_arxiv_id":null,"evidence_quote":"Sjöstrand's Wiener-type lemma, the template for the off-diagonal decay and $\\ell^p$-stability results (Proposition B.3) on homogeneous groups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the two-sided amalgam spaces used in the envelope estimates for Gramian matrices over relatively separated sets."},{"cited_title":"Gröchenig","cited_arxiv_id":null,"evidence_quote":"Establishes the homogeneous approximation property used to verify the hypotheses of the non-strict density theorem."}],"review_version":1}