{"id":"c51a19ec-6db9-4106-90e9-cb3bac84dd44","arxiv_id":"1908.00579","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Doubly sparse measures on second countable locally compact Abelian groups are shown to be supported on finitely many translates of a lattice with trigonometric polynomial amplitudes.","lead":"Mathematicians classified a large family of \"doubly sparse\" measures, where both a measure and its Fourier transform are concentrated on sparse point sets, on very general groups. The result shows such measures are highly ordered, living on finitely many lattice translates with trigonometric-polynomial weights, and answers a question of Meyer in a weaker form.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 4.10 is supported by the paper's proofs plus two published black-box theorems; only minor repairable typos appear.","rationale":"The reader's verdict was ACCEPT with moderate confidence. My pass confirms the central chain is valid. Prop 4.5's use of the two external theorems is consistent with their published statements; the QUP/analyticity argument is self-contained. Theorem 4.8's two minor flaws are typographical/choice-of-representatives issues with evident repairs; they do not undermine Theorem 4.10. The remaining dependency on prior work is normal mathematical practice and is already disclosed by the reader. No manufactured concern: the paper's central claim appears sound, so the verdict should remain unchanged.","tokens_in":30658,"tokens_out":29451,"duration_ms":297631,"concrete_test":"Independently verify the two external inputs in the exact generality used here: (i) [43, Thm. 5.5.2] applied to strongly almost periodic pure point measures on any second countable LCAG with Meyer support; (ii) [37, Thm. 5.3] for weighted model combs with h in C_c(H), H=Z^m×K. If either requires extra hypotheses (e.g., G compactly generated or h∈L1 only), check whether Prop. 4.5's application can be adapted; then test the corrected coset-representative proof of supp(µ)⊆Γ+F and the sign fix in Γ′=Γ^0. If the corrected computations go through, Theorem 4.10 stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central dichotomy (Theorem 4.10) rests on Prop. 4.5, whose only non-self-contained inputs are [43, Thm. 5.5.2] (strongly almost periodic pure point measures with Meyer support are weighted model combs) and [37, Thm. 5.3] (Poisson-type identity for such combs). Both are published peer-reviewed results; the internal analytic argument forcing the internal space to be Z^m×K is complete and correct. I checked the later chain Theorem 4.8 → Lemma 4.9 → Theorem 4.10 and found no invalid step. Two minor presentation issues do not affect the claim: in Theorem 4.8 the finite set F is written as ⋏(S×{0}), whereas the valid inclusion uses one chosen lattice representative per H0-coset; and in the proof of Γ′=Γ^0 the displayed computation should use the inverse character (χ_k(x)ψ(t)=1 should be χ_k(x)overline{ψ(t)}=1). Both are immediately repairable without changing the statements. Consequently, no load-bearing concern about the central claim was identified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies Fourier-transformable Radon measures on second-countable locally compact Abelian groups for which both the measure and its Fourier transform are pure point measures with sparse support ('doubly sparse measures'). The main structural results are: Proposition 4.5 and Theorem 4.8, which show that a nonzero translation-bounded measure supported in a Meyer set with B-sparse Fourier-Bohr support is a weighted model comb over a cut-and-project scheme with internal space Z^m×K, and that both supports lie in finitely many translates of a lattice and its annihilator; Theorem 4.10, which turns this into explicit representations of µ and \\hat µ by trigonometric polynomials on lattice translates; and Section 5, which extends the dichotomy to positive definite strongly almost periodic measures with uniformly discrete support, proving sup-almost periodicity and a Poisson-summation-type identity for weighted model combs. Section 6 specializes to R^d, connects the results with work of Lev and Olevskii, and resolves a weak form of Meyer's question for fully Euclidean model sets. The proofs are detailed and explicitly cite the needed published tools from earlier work of the same authors.","tokens_in":30773,"tokens_out":8093,"duration_ms":87435,"significance":"The paper provides a clean and useful extension of the Euclidean 'crystalline measure' theory to general second-countable LCAGs. If the results are correct, the crystallographic dichotomy in Theorem 4.8 and the trigonometric-polynomial representation in Theorem 4.10 are substantial; they give a precise answer to when Meyer-set-supported measures with sparse Fourier-Bohr spectrum are crystallographic in the generalized sense. The treatment of positive definite measures in Section 5, including sup-almost periodicity and the PSF identity, is a further valuable contribution. The proofs are unusually complete: delicate reductions such as the d=0 step in Proposition 4.5 and the lattice duality in Theorem 4.8 are carried out explicitly rather than hand-waved. The main caveat is that some of the deepest inputs, notably [43, Thm. 5.5.2], [37, Thm. 5.3], and [45, Rem. 5], come from earlier papers by the same authors; this is not circularity, since they are published with proofs, but it does mean the central theorem inherits the full strength of those results.","major_comments":[],"minor_comments":[{"comment":"The displayed inclusion supp(µ) = ⋏(supp(h)) ⊆ ⋏(H0) + ⋏(S×{0}) is not justified as written, because (t,0) need not lie in π_H(L), so ⋏(S×{0}) may not contain the required representatives of the cosets {t}×K. The valid argument chooses, for each t∈S, one element x_t∈⋏({t}×K) (nonempty by density of π_H(L)) and sets F={x_t}; with that choice the inclusion holds and F is finite.","section":"Theorem 4.8, proof"},{"comment":"The displayed computation at the end of the proof should use the inverse character: one needs χ_k(x) overline{ψ(t)} = 1, equivalently χ_k(x)χ_{−ℓ}(t) = 1, rather than χ_k(x)ψ(t) = 1. As printed, the equality is incorrect because χ_{−ℓ} is the conjugate of χ_ℓ.","section":"Theorem 4.8, proof of Γ′ = Γ^0"},{"comment":"There are numerous typographical and encoding artifacts in the provided text (for example 'sp arse' in the title and various garbled symbols); I assume these are artifacts of the version supplied and not present in the submitted manuscript, but the final version should be checked carefully.","section":"Throughout"},{"comment":"The proposed canonical choice of the trigonometric polynomials would be easier to verify if the characters χ_j were explicitly tied to the finite set F′ from Theorem 4.8 (they are elements of Γ^0+F′), rather than described only as 'all characters which appear in the polynomials Pi'.","section":"Remark 4.11"}],"recommendation":"minor_revision","confidential_remarks":"For the editor: the manuscript is well within the journal's scope and the central claims appear sound. The reliance on several published theorems from the same authors is legitimate and transparent, but because those results are load-bearing, it may be prudent to ensure the cited papers have been thoroughly refereed. The two mathematical typos in Theorem 4.8's proof are local and immediately repairable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers what it promises: the Lev–Olevskii/Kellendonk–Lenz/Favorov classification of doubly sparse measures is extended from Euclidean space to all second countable locally compact Abelian groups, and the main dichotomy (Theorem 4.8/4.10) is clean. If a measure with Meyer set support has sparse Fourier–Bohr spectrum, then both the measure and its transform are supported on finitely many translates of a lattice, with coefficients given by trigonometric polynomials. That is a genuinely sharp structural result for the aperiodic order community, and it also gives a weak answer to Meyer's question in R^d. The authors have done the work of finding the right general framework, cut and project schemes plus almost periodic measures, and the proof of Proposition 4.5, forcing the internal space to be Z^m × K, is the key step and it is done carefully.\n\nThe paper is well written and the line-by-line arguments are mostly clear. I checked the chain from Proposition 4.5 through Lemma 4.9 to Theorem 4.10 and did not find a gap. The two typos the stress-test flags (the finite set F in Theorem 4.8 and the character computation in the proof of Γ′ = Γ^0) are indeed minor and repairable; they do not affect the statements.\n\nThe real soft spot is the heavy dependence on previously published theorems by the same authors, especially [43, Thm. 5.5.2] and [37, Thm. 5.3]. Those results are published and peer-reviewed, so this is not circular, but the central dichotomy inherits all their assumptions. If any of those black boxes were wrong, Theorem 4.10 would collapse. That is worth keeping in mind for the referee report, but it is not a flaw in the present paper.\n\nWho is this for? Harmonic analysts and people working on quasicrystals, aperiodic order, and diffraction. It is not a broad-circulation paper; its significance is within-field, but within that field it answers a natural open question. I would send it to a serious referee, and I would expect acceptance after minor revisions.","headline":"A solid, correct extension of the sparse-measure dichotomy to all second countable LCAGs; the heavy lifting is outsourced to earlier published theorems, but the proof chain is sound.","tokens_in":31426,"tokens_out":1328,"would_cite":true,"duration_ms":16710,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["43A05","52C23"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sparse Fourier–Bohr spectrum forces a Meyer-supported measure onto finitely many lattice translates.","keywords":["doubly sparse measures","pure point measures","Fourier–Bohr spectrum","Meyer sets","cut and project schemes","weighted model combs","almost periodic measures","Poisson summation formula"],"falsifier":"On $G=\\mathbb{R}$, construct a non-zero translation-bounded, Fourier-transformable measure whose support is a Meyer set and whose Fourier transform is a pure point measure supported on a set with finite upper density with respect to some van Hove sequence, and check whether the support is contained in finitely many translates of a lattice; Theorem 4.8 predicts it always is, so a single Meyer-supported measure with sparse spectrum whose support is not lattice-translate-bounded refutes the claim.","tokens_in":30352,"feed_emoji":"📐","tokens_out":16173,"duration_ms":145809,"temperature":0.7,"pith_summary":"Doubly sparse measures are Fourier-transformable measures for which both the measure and its transform are pure point measures whose supports are locally finite and have finite density. The paper proves that, on any second-countable locally compact abelian group, a translation-bounded measure supported inside a Meyer set—a relatively dense, almost-lattice set with uniformly discrete differences—with a sparse Fourier–Bohr spectrum is crystallographic in a generalized sense: both its support and its spectrum lie in finitely many translates of a single lattice and of its annihilator. Consequently the measure is a finite sum of Dirac combs on lattice cosets weighted by trigonometric polynomials, and its Fourier transform has exactly the same form. The paper also shows that positive definite measures with uniformly discrete support and sparse spectrum are almost periodic in the strongest sense, fit into a cut-and-project scheme, and satisfy a Poisson-summation-type identity. In the Euclidean setting it answers a weaker version of an open question from the literature: no non-trivial fully Euclidean model set in $\\mathbb{R}^d$ can carry such a measure.","feed_headline":"Sparse spectrum pins measures to lattice translates","feed_subtitle":"Doubly sparse measures on abelian groups decompose into trigonometric polynomials over lattice translates.","key_machinery":"The load-bearing object is the weighted model comb: a Dirac comb $\\omega_g = \\sum_{x\\in\\pi_G(L)} g(x^\\star)\\,\\delta_x$ obtained by pulling a weight function $g$ back from the internal group $H$ of a cut-and-project scheme. The proof chains three facts. First, almost periodic pure point measures with Meyer-set support are weighted model combs with compactly supported weight. Second, sparse Fourier–Bohr spectrum forces the Fourier-side weight to have finite-measure support, and the qualitative uncertainty principle—which forbids a function and its Fourier transform from both having finite-measure support unless the identity component is compact—then rules out a Euclidean factor in the internal group, leaving $H\\cong\\mathbb{Z}^m\\times K$ with $K$ compact. Third, on such an internal group, compactness of the window and discreteness of the dual group imply that the weights are trigonometric polynomials, which yields the lattice-translate representation via the annihilator lattice $\\Gamma^0$.","core_discovery":"The central discovery is a rigidity dichotomy. For a nonzero translation-bounded, transformable measure with Meyer-set support, either its Fourier–Bohr spectrum meets the translates of any non-empty open set in unboundedly many points, or the spectrum is sparse—and then both the measure and its transform are crystallographic in a generalized sense. Theorem 4.8 shows that in the sparse case the support of $\\mu$ is contained in $\\Gamma+F$ and the support of $\\widehat{\\mu}$ is contained in $\\Gamma^0+F'$, where $\\Gamma$ is a lattice in $G$, $\\Gamma^0$ its annihilator, and $F,F'$ finite sets. Theorem 4.10 upgrades this to explicit formulas, $\\mu = \\sum_{i=1}^N \\sum_{x\\in\\Gamma+\\tau_i} P_i(x)\\,\\delta_x$ and the corresponding dual formula, with each $P_i$ and $Q_j$ a trigonometric polynomial. In other words, doubly sparse measures with Meyer support are finite superpositions of lattice combs with smoothly varying periodic weights, and the same structure is forced on the Fourier side.","pith_inferences":["The same lattice-translate rigidity should persist for sup-almost periodic measures with only weakly uniformly discrete support: Theorem 5.3 already supplies the cut-and-project frame and finite-measure windows, so the missing step—compactness of the window—is the only obstruction to a full analogue of Theorem 4.8 in that broader class.","The obstruction for fully Euclidean model sets suggests a stronger principle: in any cut-and-project scheme with non-compact connected internal space, sparse Fourier–Bohr spectrum can only occur when the window is finite, so the model set is actually a lattice. Testing this on partially Euclidean schemes with internal space $\\mathbb{R}^n\\times K$ would delimit the boundary of the phenomenon.","The trigonometric polynomials in Theorem 4.10 are determined by finitely many Fourier–Bohr coefficients, so one could extract quantitative bounds: the number of lattice cosets and the degree of the weights should be controlled by the sizes of the finite sets $F$ and $F'$, making the class of doubly sparse Meyer-supported measures explicitly parametrizable.","On the diffraction side, the theorem implies that the Bragg-peak set of a Meyer-supported measure with sparse Fourier–Bohr spectrum is a finite union of dual-lattice cosets, which means its pure point diffraction is finitely generated over the lattice; this could be tested numerically on candidate quasicrystal models."],"forward_implications":["On every second-countable locally compact abelian group, a nonzero translation-bounded measure with Meyer-set support and sparse Fourier–Bohr spectrum has both support and spectrum contained in finitely many lattice translates (Theorem 4.8).","Such measures admit the explicit trigonometric-polynomial representations of Theorem 4.10, simultaneously for the measure and its Fourier transform, so doubly sparse measures form a finite-dimensional family once the lattice and the finite translating sets are fixed.","Positive definite measures with uniformly discrete support and sparse spectrum are sup-almost periodic and norm-almost periodic, and satisfy the Poisson-type formula $\\widehat{\\omega_h}=\\operatorname{dens}(L)\\,\\omega_{\\widehat{h}}$ in a cut-and-project scheme (Theorem 5.3 and Corollary 5.8).","In $\\mathbb{R}^d$, no non-trivial fully Euclidean model set can support a tempered measure whose distributional Fourier transform is a translation-bounded measure with sparse support; the internal space must collapse to a point (Corollary 6.9).","For tempered measures supported in a Meyer set, the pure point part of the diffraction spectrum is either uniformly discrete inside finitely many dual-lattice translates or has a relatively dense set of accumulation points (Corollary 6.13)."],"supporting_citations":[{"why":"Supplies the characterization of almost periodic pure point measures with Meyer-set support as weighted model combs, the entry point for Theorem 4.8.","marker":"[43]"},{"why":"Supplies the Poisson-type identity $\\widehat{\\omega_h}=\\mathrm{dens}(L)\\,\\omega_{\\widehat{h}}$ for transformable weighted combs, used throughout Sections 4 and 5.","marker":"[37]"},{"why":"Supplies the fact that measures with pure point Fourier transform are strongly almost periodic, together with background on Fourier transforms of Radon measures on locally compact abelian groups.","marker":"[33]"},{"why":"Supplies the lemma used in Lemma 4.9 that turns finite lattice-translate support into trigonometric-polynomial weights.","marker":"[45]"},{"why":"Supplies Fourier–Bohr coefficient formulas and strong almost periodicity of compactly supported weighted combs used in Theorem 5.3 and Proposition 5.7.","marker":"[22]"},{"why":"Supplies the density formula for cut-and-project sets with compact windows that bounds the internal measure of the window.","marker":"[15]"},{"why":"Supplies the qualitative uncertainty principle that forces the internal group's identity component to be compact, eliminating Euclidean factors.","marker":"[14]"},{"why":"Provides the Euclidean-space dichotomy and model-set obstruction that the paper generalizes to locally compact abelian groups and uses in Section 6.","marker":"[26]"}],"fun_headline_variants":[],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument would collapse if either of its two black-box inputs failed: the characterization of almost periodic pure point measures with Meyer-set support as weighted model combs, or the Poisson-type identity connecting a transformable weighted comb to its Fourier transform on the dual cut-and-project scheme.","fun_headline_variants_meta":{"error":"Client error '402 Payment Required' for url 'https://api.deepseek.com/chat/completions'\nFor more information check: https://developer.mozilla.org/en-US/docs/Web/HTTP/Status/402"},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:46:28.932634+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On $G=\\mathbb{R}$, construct a non-zero translation-bounded, Fourier-transformable measure whose support is a Meyer set and whose Fourier transform is a pure point measure supported on a set with finite upper density with respect to some van Hove sequence, and check whether the support is contained in finitely many translates of a lattice; Theorem 4.8 predicts it always is, so a single Meyer-supported measure with sparse spectrum whose support is not lattice-translate-bounded refutes the claim.","supporting_citations":[{"cited_title":"Almost Periodic Measures and Meyer Sets","cited_arxiv_id":"1501.00945","evidence_quote":"Supplies the characterization of almost periodic pure point measures with Meyer-set support as weighted model combs, the entry point for Theorem 4.8."},{"cited_title":"Moody and N","cited_arxiv_id":null,"evidence_quote":"Supplies the fact that measures with pure point Fourier transform are strongly almost periodic, together with background on Fourier transforms of Radon measures on locally compact abelian groups."},{"cited_title":"On the Fourier Analysis of Measures with Meyer Set Support","cited_arxiv_id":"1807.03815","evidence_quote":"Supplies the lemma used in Lemma 4.9 that turns finite lattice-translate support into trigonometric-polynomial weights."},{"cited_title":"Lenz and C","cited_arxiv_id":null,"evidence_quote":"Supplies Fourier–Bohr coefficient formulas and strong almost periodicity of compactly supported weighted combs used in Theorem 5.3 and Proposition 5.7."},{"cited_title":"On pattern entropy of weak model sets","cited_arxiv_id":"1412.6307","evidence_quote":"Supplies the density formula for cut-and-project sets with compact windows that bounds the internal measure of the window."},{"cited_title":"Hogan, A qualitative uncertainty principle for locally compact Abelian groups, in Minicon- ference on Harmonic Analysis and Operator Algebras , eds","cited_arxiv_id":null,"evidence_quote":"Supplies the qualitative uncertainty principle that forces the internal group's identity component to be compact, eliminating Euclidean factors."}],"review_version":1}