Pith. sign in

REVIEW 4 minor 47 references

Pure point measures with sparse support and sparse Fourier--Bohr support

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Sparse Fourier–Bohr spectrum forces a Meyer-supported measure onto finitely many lattice translates.

desk verdict 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. read the letter →

arxiv 1908.00579 v2 pith:YQ4XTEAU submitted 2019-08-01 math.MG math.FA

Michael Baake , Nicolae Strungaru , Venta Terauds This is my paper · ORCID
classification math.MGmath.FA MSC 43A0552C23
keywords doublysparsemeasurespurepointFourier–BohrspectrumMeyersetscutandprojectschemesweightedmodelcombsalmostperiodicPoissonsummationformula
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

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.

What carries the argument

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 $\Gamma^0$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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).

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

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.

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.

minor comments (4)
  1. [Theorem 4.8, proof] 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.
  2. [Theorem 4.8, proof of Γ′ = Γ^0] 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.
  3. [Throughout] 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.
  4. [Remark 4.11] 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'.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity; central derivation rests on published black-box theorems with independent content.

full rationale

Walking the derivation chain: Proposition 4.5 obtains the weighted-model-comb form from [43, Thm. 5.5.2] and the Poisson-type identity from [37, Thm. 5.3]. Both are peer-reviewed, parameter-free theorems whose stated assumptions do not already contain the lattice-plus-trigonometric-polynomial conclusion of Theorem 4.10. The paper's own analytic work in Proposition 4.5 — using Corollary 4.3 to force the internal space to be Z^m×K — is a genuine deduction, not a renaming of the cited results. Theorem 4.8 then constructs the lattice Γ and the finite sets F,F′ from the compact support of the internal weight, and proves Γ′=Γ^0; the displayed character computation contains a minor sign typo (χ_k(x)ψ(t) should be χ_k(x)overline(ψ(t))), but the intended argument is immediate and does not change the claim. Lemma 4.9 invokes [45, Rem. 5] only for an intermediate convolution form and then performs its own pure-point reduction to obtain trigonometric polynomials; this is a citation-backed lemma rather than a definitional or fitted equivalence. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported to forbid alternatives, and no known empirical pattern is merely relabelled. The self-citations are numerous and load-bearing, but they point to published results with proofs elsewhere and with assumptions distinct from the target theorem; under the review rules that is real evidence, not a circular loop. Therefore no specific circular step is identified. Score 2 reflects the heavy self-citation footprint rather than any constructional circularity in the derivation.

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

The paper introduces no fitted parameters, new physical entities, or ad hoc constants. It relies on standard harmonic analysis, structural theorems for LCAGs, and a set of prior results on almost periodic measures and cut and project schemes, several of which are by the current authors. These axioms are cited explicitly but not re-proved in the manuscript.

assumptions (6)
  • standard math Existence and normalization of Haar measures satisfying Parseval's equation on G and G_hat
    Invoked at the start of Section 2 to define Fourier transform and measures; standard in harmonic analysis.
  • standard math Second countable LCAGs are sigma-compact and metrisable, and van Hove sequences exist in such groups
    Used throughout Section 3 to define sparseness and densities; see [35, Thm. 4.2.7] and [40, p.145].
  • domain assumption Structure theorem for compactly generated LCAGs: H is isomorphic to R^d x Z^m x K with K compact
    Fact 4.1, used in the proofs of Prop. 4.5 and Theorem 5.3 to force d=0.
  • domain assumption Qualitative uncertainty principle for LCAGs: if a nonzero function f and its Fourier transform both have finite-measure support, the identity component of the group must be compact
    Used in Remark 4.6 and Theorem 5.3 to conclude H has a compact open subgroup; cited from [14, Thm. 1].
  • domain assumption Characterization of norm-almost periodic measures with Meyer set support as weighted model combs [43, Thm. 5.5.2]
    Load-bearing in Prop. 4.5 and Theorem 5.2; a prior theorem by the same author, not proved in this paper.
  • domain assumption Poisson summation type identity for weighted combs in a CPS [37, Thm. 5.3]
    Used in Prop. 4.5 and Theorem 5.3 to identify the Fourier transform of omega_h as dens(L) omega_qh.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Pure point measures with sparse support and sparse Fourier--Bohr support." pith.science (2026). https://pith.science/paper/YQ4XTEAU

@misc{pith2026190800579,
  author       = {Pith},
  title        = {Pith review of: Pure point measures with sparse support and sparse Fourier--Bohr support},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YQ4XTEAU}},
  note         = {Machine review of arXiv:1908.00579}
}
read the original abstract

Fourier-transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier--Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 39 canonical work pages

  1. [1]

    Argabright and J

    L.N. Argabright and J. Gil de Lamadrid, Fourier analysis of unboun ded measures on locally compact Abelian groups, Memoirs Amer. Math. Soc. , no. 145, AMS, Providence, RI (1974)

    work page 1974

  2. [2]

    Baake and U

    M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation , Cambridge Uni- versity Press, Cambridge (2013)

    work page 2013

  3. [3]

    Baake and U

    M. Baake and U. Grimm (eds.), Aperiodic Order. Vol. 2: Crystallography and Almost Period icity, Cambridge University Press, Cambridge (2017)

    work page 2017

  4. [4]

    On weak model sets of extremal density

    M. Baake, C. Huck and N. Strungaru, On weak model sets of ext remal density, Indag. Math. 28 (2017) 3–31, arXiv:1512.07129

    work page Pith review arXiv 2017

  5. [5]

    Baake, D

    M. Baake, D. Lenz and R. V. Moody, Characterization of model s ets by dynamical systems, Ergodic Th. & Dynam. Syst. 27 (2007) 341–382; arXiv:math/0511648

    arXiv 2007

  6. [6]

    Baake and R.V

    M. Baake and R.V. Moody, Weighted Dirac combs with pure point diffr action, J. Reine Angew. Math. (Crelle) 573 (2004) 61–94; arXiv:math.MG/0203030

    work page arXiv 2004

  7. [7]

    Berg and G

    C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups , Springer, Berlin (1975)

    work page 1975

  8. [8]

    C´ ordoba, Dirac combs, Lett

    A. C´ ordoba, Dirac combs, Lett. Math. Phys. 17 (1989) 191–196

    work page 1989

Show all 47 references
  1. [9]

    Downarowicz, D

    T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable grou ps, J. Reine Angew. Math. (Crelle) 747 (2019) 277–298; arXiv:1502.02413

    2019 arXiv

  2. [10]

    Favorov, Tempered distributions with discrete support and spectrum, Bull

    S. Favorov, Tempered distributions with discrete support and spectrum, Bull. Hellenic Math. Soc. 62 (2018) 66–79; arXiv:1801.08490

    2018 arXiv

  3. [11]

    Gel’fand and G.E

    I.M. Gel’fand and G.E. Shilov, Generalized Functions, Vol. 2: Spaces of Fundamental and Ge n- eralized Functions, reprint, AMS Chelsea, Providence, RI (2016)

    2016

  4. [12]

    Gil de Lamadrid and L.N

    J. Gil de Lamadrid and L.N. Argabright, Almost periodic measures , Memoirs Amer. Math. Soc. 85 (1990) no. 428, AMS, Providence, RI (1990)

    1990

  5. [13]

    Hewitt and K.A

    E. Hewitt and K.A. Ross, Abstract Harmonic Analysis I , 2nd ed., Springer, Berlin (1979)

    1979

  6. [14]

    Hogan, A qualitative uncertainty principle for locally compact Abelian groups, in Minicon- ference on Harmonic Analysis and Operator Algebras , eds

    J.A. Hogan, A qualitative uncertainty principle for locally compact Abelian groups, in Minicon- ference on Harmonic Analysis and Operator Algebras , eds. M. Cowling, C. Meaney and W. Moran, Proceedings of the Centre for Mathematics and its Applications, vo l. 16, The Australia...

    1988

  7. [15]

    Huck and C

    C. Huck and C. Richard, On pattern entropy of weak model set s, Discr. Comput. Geom. 54 (2015) 714–757; arXiv:1412.6307

    2015 arXiv

  8. [16]

    Kellendonk and D

    J. Kellendonk and D. Lenz, Equicontinuous Delone dynamical sys tems, Can. J. Math. 65 (2013) 149–170; arXiv:1105.3855

    2013 arXiv

  9. [17]

    Keller and C

    G. Keller and C. Richard, Periods and factors of weak model set s, Israel J. Math. 229 (2019) 85–132; arXiv:1702.02383

    2019 arXiv

  10. [18]

    Keller and C

    G. Keller and C. Richard, Dynamics on the graph of the torus par ametrization, Ergodic Th. & Dynam. Syst. 38 (2018) 1048–1085; arXiv:1511.06137

    2018 arXiv

  11. [19]

    Lagarias, Meyer’s concept of quasicrystal and quasiregu lar sets, Commun

    J.C. Lagarias, Meyer’s concept of quasicrystal and quasiregu lar sets, Commun. Math. Phys. 179 (1996) 365–376

    1996

  12. [20]

    Lagarias, Geometric models for quasicrystals I

    J.C. Lagarias, Geometric models for quasicrystals I. Delone set s of finite type, Discr. Comput. Geom. 21 (1999) 161–191

    1999

  13. [21]

    Lagarias, Mathematical quasicrystals and the problem of d iffraction, in Directions in Math- ematical Quasicrystals , eds

    J.C. Lagarias, Mathematical quasicrystals and the problem of d iffraction, in Directions in Math- ematical Quasicrystals , eds. M. Baake and R.V. Moody, CRM Monograph Series, vol. 13, Ame r. Math. Society, Providence, RI (2000), pp. 61–93. DOUBLY SPARSE MEASURES 35

    2000

  14. [22]

    Lenz and C

    D. Lenz and C. Richard, Pure point diffraction and cut and proje ct schemes for measures: The smooth case, Math. Z. 256 (2007) 347–378; arXiv:math.DS/0603453

    2007

  15. [23]

    Lenz and N

    D. Lenz and N. Strungaru, Note on the set of Bragg peaks with high intensity, Ann. H. Poincar´ e 17 (2016) 673–687; arXiv:1412.7377

    2016 arXiv

  16. [24]

    Lenz and N

    D. Lenz and N. Strungaru, On weakly almost periodic measures, Trans. Amer. Math. Soc. 371 (2019) 6843–6881; arXiv:1609.08219

    2019 arXiv

  17. [25]

    Lev and A

    N. Lev and A. Olevskii, Quasicrystals and Poisson’s summation for mula, Invent. Math. 200 (2015) 585–606; arXiv:1312.6884

    2015 arXiv

  18. [26]

    Lev and A

    N. Lev and A. Olevskii, Fourier quasicrystals and discreteness o f the diffraction spectrum, Adv. Math. 315 (2017) 1–26; arXiv:1512.08735

    2017 arXiv

  19. [27]

    Meyer, Algebraic Numbers and Harmonic Analysis , North Holland, Amsterdam (1972)

    Y. Meyer, Algebraic Numbers and Harmonic Analysis , North Holland, Amsterdam (1972)

    1972

  20. [28]

    Meyer, Quasicrystals, Diophantine approximation and algebr aic numbers, in Beyond Qua- sicrystals, eds

    Y. Meyer, Quasicrystals, Diophantine approximation and algebr aic numbers, in Beyond Qua- sicrystals, eds. F. Axel and D. Gratias, Springer, Berlin, and EDP Sciences, L es Ulis (1995), pp. 3–16

    1995

  21. [29]

    Meyer, Measures with locally finite support and spectrum, PNAS 113 (2016) 3152–3158

    Y.F. Meyer, Measures with locally finite support and spectrum, PNAS 113 (2016) 3152–3158

    2016

  22. [30]

    Mityagin, The zero set of a real analytic function, Matem

    B. Mityagin, The zero set of a real analytic function, Matem. Zametki 107 (2020) 473–475; arXiv:1512.07276

    2020 arXiv

  23. [31]

    Moody, Meyer sets and their duals, in The Mathematics of Long-Range Aperiodic Order , ed

    R.V. Moody, Meyer sets and their duals, in The Mathematics of Long-Range Aperiodic Order , ed. R.V. Moody, NATO ASI Ser. C 489, Kluwer, Dordrecht (1997), pp. 4 03–441

    1997

  24. [32]

    Moody, Model sets: A survey, in From Quasicrystals to More Complex Systems , eds

    R.V. Moody, Model sets: A survey, in From Quasicrystals to More Complex Systems , eds. F. Axel, F. Denoyer and J. P. Gazeau, EDP Sciences, Les Ulis, and Springer, Berlin (2000), pp. 145–166; arXiv:math/0002020

    2000 arXiv

  25. [33]

    Moody and N

    R.V. Moody and N. Strungaru, Almost periodic measures and the ir Fourier transforms, in [ 3], pp. 173–270

  26. [34]

    Osborne, On the Schwartz–Bruhat space and the Paley–W iener theorem for locally compact abelian groups, J

    M.S. Osborne, On the Schwartz–Bruhat space and the Paley–W iener theorem for locally compact abelian groups, J. Funct. Anal. 19 (1975) 40–49

    1975

  27. [35]

    Reiter and J.D

    H. Reiter and J.D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups , Claren- don Press, Oxford (2000)

    2000

  28. [36]

    Richard, Dense Dirac combs in Euclidean space with pure point d iffraction, J

    C. Richard, Dense Dirac combs in Euclidean space with pure point d iffraction, J. Math. Phys. 44 (2003) 4436–4449; arXiv:math-ph/0302049

    2003 arXiv

  29. [37]

    Richard and N

    C. Richard and N. Strungaru, Pure point diffraction and Poisson summation, Ann. H. Poincar´ e 18 (2017) 3903–3931; arXiv:1512.00912

    2017 arXiv

  30. [38]

    Rudin, Fourier Analysis on Groups , Wiley, New York (1962)

    W. Rudin, Fourier Analysis on Groups , Wiley, New York (1962)

    1962

  31. [39]

    Schlottmann, Cut-and-project sets in locally compact Abelia n groups, in Quasicrystals and Discrete Geometry , ed

    M. Schlottmann, Cut-and-project sets in locally compact Abelia n groups, in Quasicrystals and Discrete Geometry , ed. J. Patera, Fields Institute Monographs, vol. 10, Amer. Math . Society, Providence, RI (1998), pp. 247–264

    1998

  32. [40]

    Schlottmann, Generalized model sets and dynamical system s, in Directions in Mathematical Quasicrystals, eds

    M. Schlottmann, Generalized model sets and dynamical system s, in Directions in Mathematical Quasicrystals, eds. M. Baake and R.V. Moody, CRM Monograph Series, vol. 13, Ame r. Math. Society, Providence, RI (2000), pp. 143–159

    2000

  33. [41]

    Schreiber, Approximations diophantiennes et probl` eme s additifs dans les groupes ab´ eliens localement compacts, Bull

    J.-P. Schreiber, Approximations diophantiennes et probl` eme s additifs dans les groupes ab´ eliens localement compacts, Bull. Soc. Math. France 101 (1973) 297–332

    1973

  34. [42]

    Strungaru, On weighted Dirac combs supported inside model sets, J

    N. Strungaru, On weighted Dirac combs supported inside model sets, J. Phys. A: Math. Theor. 47 (2014) 335202:1–19; arXiv:1309.7947

    2014 arXiv

  35. [43]

    Strungaru, Almost periodic pure point measures, in [ 3], pp

    N. Strungaru, Almost periodic pure point measures, in [ 3], pp. 271–342; arXiv:1501.00945. 36 MICHAEL BAAKE, NICOLAE STRUNGARU, AND VENTA TERAUDS

    arXiv

  36. [44]

    Strungaru, On the Fourier transformability of strongly almo st periodic measures, Can

    N. Strungaru, On the Fourier transformability of strongly almo st periodic measures, Can. J. Math., in press; arXiv:1704.04778

    arXiv

  37. [45]

    Strungaru, On the Fourier analysis of measures with Meyer s et support, J

    N. Strungaru, On the Fourier analysis of measures with Meyer s et support, J. Funct. Anal. 278 (2020) 108404:1–30; arXiv:1807.03815

    2020 arXiv

  38. [46]

    Strungaru and V

    N. Strungaru and V. Terauds, Diffraction theory and almost pe riodic distributions, J. Stat. Phys. 164 (2016) 1183–1216; arXiv:1603.04796

    2016 arXiv

  39. [47]

    Yoshida, Functional Analysis, 6th ed., Springer, Berlin (1980)

    K. Yoshida, Functional Analysis, 6th ed., Springer, Berlin (1980). F akult¨at f ¨ur Mathematik, Universit ¨at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany E-mail address : mbaake@math.uni-bielefeld.de Department of Mathematical Sciences, MacEw an University, 10700 104 ...

    1980

Pith tools

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