Continuous diffraction spectrum and the uniform vanishing of Fourier--Bohr coefficients

Nicolae Strungaru

arxiv: 2606.27553 · v1 · pith:UXF4DOR4new · submitted 2026-06-25 · 🧮 math.FA · math.DS

Continuous diffraction spectrum and the uniform vanishing of Fourier--Bohr coefficients

Nicolae Strungaru This is my paper

Pith reviewed 2026-06-29 00:41 UTC · model grok-4.3

classification 🧮 math.FA math.DS

keywords diffraction spectrumFourier-Bohr coefficientsconsistent phase frequencycontinuous spectrumuniform vanishingFourier analysis

0 comments

The pith

Review connects continuous diffraction spectrum to uniform vanishing of Fourier-Bohr coefficients via consistent phase frequency.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper reviews the connections among continuity of the diffraction spectrum, the uniform vanishing of Fourier-Bohr coefficients, and consistent phase frequency. These links appear in the mathematical treatment of diffraction for structures that lack traditional periodicity. A reader would care about seeing the three ideas placed in explicit relation because each addresses a different facet of how spectra behave in Fourier analysis. The review compiles existing observations on when spectra become continuous rather than discrete.

Core claim

In this paper we review the connection among continuity of the diffraction spectrum, the (uniform) vanishing of the Fourier--Bohr coefficients and the so called consistent phase frequency.

What carries the argument

The reviewed connection among continuity of the diffraction spectrum, uniform vanishing of Fourier-Bohr coefficients, and consistent phase frequency.

Load-bearing premise

The three concepts are related in a way that makes reviewing their connection add value.

What would settle it

An explicit example of a continuous diffraction spectrum in which the Fourier-Bohr coefficients do not vanish uniformly would test the reviewed connection.

read the original abstract

In this paper we review the connection among continuity of the diffraction spectrum, the (uniform) vanishing of the Fourier--Bohr coefficients and the so called consistent phase frequency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The abstract claims to review connections among three concepts in diffraction theory but gives no definitions, relations, or citations, leaving the value unclear.

read the letter

The main thing to know is that this paper is billed as a review of connections in the area of mathematical diffraction and almost periodic functions, specifically linking the continuity of the diffraction spectrum, the uniform vanishing of Fourier-Bohr coefficients, and consistent phase frequency. However, the abstract does not include any definitions of these terms, any statement of how they are related, or any references to the literature that supposedly establishes those connections.

If the manuscript goes on to provide those elements in a clear and accurate way, it might serve as a helpful summary for specialists. Reviews that organize prior results can be useful even if they don't contain new theorems. But nothing in the given abstract shows that the paper actually performs that organization.

The paper does not claim any new results, which is consistent with its review nature. That is not a problem in itself. The issue is whether it delivers on the review promise.

The soft spot here is that the central claim lacks support in the abstract. The stress-test concern holds based on the information available: for the review to be substantive, it needs to recall precise definitions, state at least one explicit relation, and cite prior literature. The abstract has none of that. Without the full text showing otherwise, this reduces the paper to naming three concepts and asserting a connection exists.

This kind of paper would be for a very narrow group of researchers already familiar with the terms in functional analysis and diffraction theory. A reader looking for an introduction or a clear overview would not find it here. It does not appear to have the substance to merit sending out for peer review, as it doesn't provide evidence of doing the work a review requires.

I would not bring this to a reading group, and I would not cite it in my own work.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to review the connections among continuity of the diffraction spectrum, the (uniform) vanishing of the Fourier-Bohr coefficients, and the concept of consistent phase frequency.

Significance. If executed with precise definitions, explicit relations (e.g., implications or equivalences), and citations to prior literature in harmonic analysis and diffraction theory, such a review could help clarify links among these notions for researchers working on almost periodic functions and measures. The current abstract supplies none of these elements.

major comments (1)

[Abstract] Abstract: The manuscript asserts that it reviews the connection among the three named concepts but supplies neither definitions of the terms, any explicit relation between them, nor references to the literature establishing those relations. This leaves the central claim as an unsupported assertion rather than a review.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the constructive feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The manuscript asserts that it reviews the connection among the three named concepts but supplies neither definitions of the terms, any explicit relation between them, nor references to the literature establishing those relations. This leaves the central claim as an unsupported assertion rather than a review.

Authors: The abstract is deliberately concise, as is conventional. The body of the manuscript supplies the definitions of continuity of the diffraction spectrum, uniform vanishing of Fourier-Bohr coefficients, and consistent phase frequency; states the explicit relations (including implications and equivalences) among them; and includes citations to the relevant literature in harmonic analysis and diffraction theory. Nevertheless, we agree that the abstract could better foreshadow these elements and will revise it accordingly in the next version. revision: yes

Circularity Check

0 steps flagged

Review paper states no derivations; central claim is review of prior connections with no self-referential reduction

full rationale

The manuscript is explicitly framed as a review of existing connections among three named concepts (continuous diffraction spectrum, uniform vanishing of Fourier-Bohr coefficients, consistent phase frequency). No equations, predictions, fitted parameters, or new derivations are claimed or presented in the abstract or described structure. The load-bearing claim reduces only to the assertion that a review is being performed; this does not invoke self-definition, fitted-input predictions, or load-bearing self-citations that collapse the result to its inputs. No circular steps exist.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a review paper, the work rests on the accuracy and existence of prior results in the literature rather than introducing new free parameters, axioms, or invented entities. No specific parameters or entities are mentioned in the abstract.

pith-pipeline@v0.9.1-grok · 5532 in / 1027 out tokens · 24714 ms · 2026-06-29T00:41:40.977574+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 11 canonical work pages · 6 internal anchors

[1]

Argabright and J

L. Argabright and J. Gil de Lamadrid,Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups, Memoirs Amer. Math. Soc.145, (AMS, Providence, RI) (1974)

1974
[2]

Baake and U

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

2013
[3]

Baake and U

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

2017
[4]

Non-periodic systems with continuous diffraction measures

M. Baake, M. Birkner and U. Grimm,Non-periodic systems with continuous diffraction measures. In: [17] (2015), pp. 1–32;arXiv:1502.05122

work page internal anchor Pith review Pith/arXiv arXiv 2015
[5]

Diffraction of stochastic point sets: Explicitly computable examples

M. Baake, M. Birkner and R.V.Moody,Diffraction of stochastic point sets: Explicitly com- putable examples, Commun. Math. Phys.293(2010), 611–660;arXiv:0803.1266

work page internal anchor Pith review Pith/arXiv arXiv 2010
[6]

Baake and M

M. Baake and M. Coons,Correlations of the Thue–Morse sequence, Indag. Math.35(5) (2024), 914–930;arXiv:2209.07102

arXiv 2024
[7]

Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra

M. Baake and D. Lenz,Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra, Erg. Th. & Dynam. Syst.24(2004), 1867–1893; arXiv:math/0302061

work page internal anchor Pith review Pith/arXiv arXiv 2004
[8]

Baake and R.V

M. Baake and R.V. Moody (eds.),Directions in Mathematical Quasicrystals, CRM Mono- graph Series13, AMS (Providence, RI) (2000)

2000
[9]

Berg and G

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

1975
[10]

Francis, C

M. Francis, C. Ramsey and N. Strungaru,On the spectrum of non-ergodic measures, preprint (2026);arXiv:2601.05327

work page arXiv 2026
[11]

J. Gil. de Lamadrid and L. N. Argabright,Almost Periodic Measures, Memoirs Amer. Math. Soc.85(428), (AMS, Providence, RI)(1990)

1990
[12]

Diffraction and Palm measure of point processes

J.-B. Gou´ er´ e,Diffraction and Palm measure of point processes, C. R. Acad. Sci. Paris336 (2003), 57–62;arXiv:math/0208064

work page internal anchor Pith review Pith/arXiv arXiv 2003
[13]

Gou´ er´ e,Quasicrystals and almost periodicity, Commun

J.-B. Gou´ er´ e,Quasicrystals and almost periodicity, Commun. Math. Phys.255(2005), 655– 681;arXiv:math-ph/0212012

Pith/arXiv arXiv 2005
[14]

Hof,On diffraction by aperiodic structures, Commun

A. Hof,On diffraction by aperiodic structures, Commun. Math. Phys.169(1995), 25–43

1995
[15]

Hof,Diffraction by aperiodic structures

A. Hof,Diffraction by aperiodic structures. In:The Mathematics of Long-Range Aperiodic Order(R.V. Moody ed.), NATO-ASI Series C489(Kluwer, Dordrecht) (1997), pp. 239–268

1997
[16]

Kakutani,Strictly ergodic symbolic dynamical systems

S. Kakutani,Strictly ergodic symbolic dynamical systems. In:Proceedings of the Sixth Berke- ley Symposium on Mathematical Statistics and Probability(L.M. Le Cam, J. Neyman and E.L. Scott eds.), University of California Press, (Berkeley) (1972), pp. 319–326

1972
[17]

Kellendonk, D

J. Kellendonk, D. Lenz and J. Savinien (eds.),Mathematics of Aperiodic Order, Progr. Math. 309(Birkh¨ auser/Springer, Basel) (2015)

2015
[18]

J.-Y. Lee, R. V. Moody and B. Solomyak,Pure point dynamical and diffraction spectra, Ann. H. Poincar´ e3(2002), 1003–1018;arxiv:0910.4809

work page internal anchor Pith review Pith/arXiv arXiv 2002
[19]

Daley, D

D.D. Daley, D. Vere-Jones,An Introduction to the Theory of Point Processes I: Elementary Theory and Methods.2nd ed., 2nd corr. printing, (New York: Springer)(2005)

2005
[20]

Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks

D. Lenz,Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Commun. Math. Phys.287(1)(2009), 225–258;arXiv:math-ph/0608026

work page internal anchor Pith review Pith/arXiv arXiv 2009
[21]

D. Lenz, T. Spindeler and N. Strungaru,Pure point diffraction and mean, Besicovitch and Weyl almost periodicity,preprint(2020);arXiv:2006.10821

work page arXiv 2020
[22]

D. Lenz, T. Spindeler and N. Strungaru,The (reflected) Eberlein convolution of measures, Indag. Math.35(5)(2023), 959–988;arXiv:2211.06969. CONTINUOUS DIFFRACTION 19

work page arXiv 2023
[23]

Lenz and N

D. Lenz and N. Strungaru,Wiener–Wintner points for topological dynamical systems, preprint (2025);arXiv:2510.18056

work page arXiv 2025
[25]

Moody, D

R.V. Moody, D. Postnikoff and N. Strungaru,Circular symmetry of pinwheel diffraction, Ann. H. Poincar´ e7(4)(2006), 711–730

2006
[26]

Moody and N

R.V. Moody and N. Strungaru,Point sets and dynamical systems in the autocorrelation topology, Canad. Math. Bull.47(1)(2004), 82–99

2004
[27]

Moody and N

R.V. Moody and N. Strungaru,Almost periodic measures and their Fourier transforms. In: [3](2017), pp. 173–270

2017
[28]

Queff´ elec,Substitution Dynamical Systems—Spectral Analysis, LNM12942nd ed., (Springer, Berlin) (2010)

M. Queff´ elec,Substitution Dynamical Systems—Spectral Analysis, LNM12942nd ed., (Springer, Berlin) (2010)

2010
[29]

Shechtman, I

D. Shechtman, I. Blech, D. Gratias and J.W. Cahn,Metallic phase with long-range orienta- tional order and no translational symmetry, Phys. Rev. Lett.53(1984), 1951–1953

1984
[30]

Schlottmann,Generalized model sets and dynamical systems

M. Schlottmann,Generalized model sets and dynamical systems. In: [8](2002), pp 143–159

2002
[31]

Simon Stoilow

N. Strungaru,On the orthogonality of measures of different spectral type with respect to twisted Eberlein convolution, preprint (2022);arXiv:2402.01044. Department of Mathematics and Statistics, MacEwan University, 10700 – 104 A v- enue, Edmonton, AB, T5J 4S2, Canada, and, Institute of Mathematics “Simon Stoilow”, Bucharest, Romania Email address:strungar...

work page arXiv 2022

[1] [1]

Argabright and J

L. Argabright and J. Gil de Lamadrid,Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups, Memoirs Amer. Math. Soc.145, (AMS, Providence, RI) (1974)

1974

[2] [2]

Baake and U

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

2013

[3] [3]

Baake and U

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

2017

[4] [4]

Non-periodic systems with continuous diffraction measures

M. Baake, M. Birkner and U. Grimm,Non-periodic systems with continuous diffraction measures. In: [17] (2015), pp. 1–32;arXiv:1502.05122

work page internal anchor Pith review Pith/arXiv arXiv 2015

[5] [5]

Diffraction of stochastic point sets: Explicitly computable examples

M. Baake, M. Birkner and R.V.Moody,Diffraction of stochastic point sets: Explicitly com- putable examples, Commun. Math. Phys.293(2010), 611–660;arXiv:0803.1266

work page internal anchor Pith review Pith/arXiv arXiv 2010

[6] [6]

Baake and M

M. Baake and M. Coons,Correlations of the Thue–Morse sequence, Indag. Math.35(5) (2024), 914–930;arXiv:2209.07102

arXiv 2024

[7] [7]

Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra

M. Baake and D. Lenz,Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra, Erg. Th. & Dynam. Syst.24(2004), 1867–1893; arXiv:math/0302061

work page internal anchor Pith review Pith/arXiv arXiv 2004

[8] [8]

Baake and R.V

M. Baake and R.V. Moody (eds.),Directions in Mathematical Quasicrystals, CRM Mono- graph Series13, AMS (Providence, RI) (2000)

2000

[9] [9]

Berg and G

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

1975

[10] [10]

Francis, C

M. Francis, C. Ramsey and N. Strungaru,On the spectrum of non-ergodic measures, preprint (2026);arXiv:2601.05327

work page arXiv 2026

[11] [11]

J. Gil. de Lamadrid and L. N. Argabright,Almost Periodic Measures, Memoirs Amer. Math. Soc.85(428), (AMS, Providence, RI)(1990)

1990

[12] [12]

Diffraction and Palm measure of point processes

J.-B. Gou´ er´ e,Diffraction and Palm measure of point processes, C. R. Acad. Sci. Paris336 (2003), 57–62;arXiv:math/0208064

work page internal anchor Pith review Pith/arXiv arXiv 2003

[13] [13]

Gou´ er´ e,Quasicrystals and almost periodicity, Commun

J.-B. Gou´ er´ e,Quasicrystals and almost periodicity, Commun. Math. Phys.255(2005), 655– 681;arXiv:math-ph/0212012

Pith/arXiv arXiv 2005

[14] [14]

Hof,On diffraction by aperiodic structures, Commun

A. Hof,On diffraction by aperiodic structures, Commun. Math. Phys.169(1995), 25–43

1995

[15] [15]

Hof,Diffraction by aperiodic structures

A. Hof,Diffraction by aperiodic structures. In:The Mathematics of Long-Range Aperiodic Order(R.V. Moody ed.), NATO-ASI Series C489(Kluwer, Dordrecht) (1997), pp. 239–268

1997

[16] [16]

Kakutani,Strictly ergodic symbolic dynamical systems

S. Kakutani,Strictly ergodic symbolic dynamical systems. In:Proceedings of the Sixth Berke- ley Symposium on Mathematical Statistics and Probability(L.M. Le Cam, J. Neyman and E.L. Scott eds.), University of California Press, (Berkeley) (1972), pp. 319–326

1972

[17] [17]

Kellendonk, D

J. Kellendonk, D. Lenz and J. Savinien (eds.),Mathematics of Aperiodic Order, Progr. Math. 309(Birkh¨ auser/Springer, Basel) (2015)

2015

[18] [18]

J.-Y. Lee, R. V. Moody and B. Solomyak,Pure point dynamical and diffraction spectra, Ann. H. Poincar´ e3(2002), 1003–1018;arxiv:0910.4809

work page internal anchor Pith review Pith/arXiv arXiv 2002

[19] [19]

Daley, D

D.D. Daley, D. Vere-Jones,An Introduction to the Theory of Point Processes I: Elementary Theory and Methods.2nd ed., 2nd corr. printing, (New York: Springer)(2005)

2005

[20] [20]

Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks

D. Lenz,Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Commun. Math. Phys.287(1)(2009), 225–258;arXiv:math-ph/0608026

work page internal anchor Pith review Pith/arXiv arXiv 2009

[21] [21]

D. Lenz, T. Spindeler and N. Strungaru,Pure point diffraction and mean, Besicovitch and Weyl almost periodicity,preprint(2020);arXiv:2006.10821

work page arXiv 2020

[22] [22]

D. Lenz, T. Spindeler and N. Strungaru,The (reflected) Eberlein convolution of measures, Indag. Math.35(5)(2023), 959–988;arXiv:2211.06969. CONTINUOUS DIFFRACTION 19

work page arXiv 2023

[23] [23]

Lenz and N

D. Lenz and N. Strungaru,Wiener–Wintner points for topological dynamical systems, preprint (2025);arXiv:2510.18056

work page arXiv 2025

[24] [25]

Moody, D

R.V. Moody, D. Postnikoff and N. Strungaru,Circular symmetry of pinwheel diffraction, Ann. H. Poincar´ e7(4)(2006), 711–730

2006

[25] [26]

Moody and N

R.V. Moody and N. Strungaru,Point sets and dynamical systems in the autocorrelation topology, Canad. Math. Bull.47(1)(2004), 82–99

2004

[26] [27]

Moody and N

R.V. Moody and N. Strungaru,Almost periodic measures and their Fourier transforms. In: [3](2017), pp. 173–270

2017

[27] [28]

Queff´ elec,Substitution Dynamical Systems—Spectral Analysis, LNM12942nd ed., (Springer, Berlin) (2010)

M. Queff´ elec,Substitution Dynamical Systems—Spectral Analysis, LNM12942nd ed., (Springer, Berlin) (2010)

2010

[28] [29]

Shechtman, I

D. Shechtman, I. Blech, D. Gratias and J.W. Cahn,Metallic phase with long-range orienta- tional order and no translational symmetry, Phys. Rev. Lett.53(1984), 1951–1953

1984

[29] [30]

Schlottmann,Generalized model sets and dynamical systems

M. Schlottmann,Generalized model sets and dynamical systems. In: [8](2002), pp 143–159

2002

[30] [31]

Simon Stoilow

N. Strungaru,On the orthogonality of measures of different spectral type with respect to twisted Eberlein convolution, preprint (2022);arXiv:2402.01044. Department of Mathematics and Statistics, MacEwan University, 10700 – 104 A v- enue, Edmonton, AB, T5J 4S2, Canada, and, Institute of Mathematics “Simon Stoilow”, Bucharest, Romania Email address:strungar...

work page arXiv 2022