The divergence of Mock Fourier series for spectral measures
Pith reviewed 2026-05-24 11:16 UTC · model grok-4.3
The pith
Mock Fourier series for the quarter Cantor measure diverge on a positive measure set.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper shows that under a sufficient condition the Mock Fourier series for a doubling spectral measure diverges on a non-zero set. In particular, the quarter Cantor measure provides an example in which the Mock Fourier sums are not almost everywhere convergent.
What carries the argument
A sufficient condition guaranteeing divergence of Mock Fourier series on a non-zero set for doubling spectral measures, applied to the quarter Cantor measure.
Load-bearing premise
The quarter Cantor measure satisfies the doubling spectral measure properties required to apply the sufficient condition.
What would settle it
A calculation establishing that the Mock Fourier sums of the quarter Cantor measure converge almost everywhere would refute the divergence claim for this measure.
read the original abstract
In this paper, we study divergence properties of Fourier series on Cantor-type fractal measure, also called Mock Fourier series. We give a sufficient condition under which the Mock Fourier series for doubling spectral measure is divergent on non-zero set. In particularly, there exists an example of the quarter Cantor measure whose Mock Fourier sums is not almost everywhere convergent.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies divergence properties of Mock Fourier series on Cantor-type fractal measures. It establishes a sufficient condition under which the Mock Fourier series for doubling spectral measures diverges on a non-zero set, and asserts that the quarter Cantor measure provides a concrete example where the sums fail to converge almost everywhere.
Significance. If the sufficient condition is rigorously derived and the quarter Cantor measure is confirmed to satisfy the doubling hypotheses, the result would supply a specific counterexample to a.e. convergence for a spectral measure arising from a fractal construction, which could be of interest in harmonic analysis on non-Lebesgue measures.
major comments (2)
- [Abstract] Abstract and introduction: the sufficient condition is stated to apply only to doubling spectral measures, yet no verification is supplied that the quarter Cantor measure meets this hypothesis (or any auxiliary conditions needed for the theorem). Without this check the claimed example does not follow from the general result.
- [Abstract] The abstract asserts existence of both the condition and the example but supplies no derivation outline, error estimates, or reference to the section containing the proof of the sufficient condition, making it impossible to assess whether the central claims are supported.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will incorporate revisions to improve clarity.
read point-by-point responses
-
Referee: [Abstract] Abstract and introduction: the sufficient condition is stated to apply only to doubling spectral measures, yet no verification is supplied that the quarter Cantor measure meets this hypothesis (or any auxiliary conditions needed for the theorem). Without this check the claimed example does not follow from the general result.
Authors: We agree that an explicit link between the general theorem and the example is necessary for the logical flow. Although the doubling property of the quarter Cantor measure is verified through direct computation of its local dimension and measure scaling in Section 4, we will revise the introduction to include a short paragraph stating that the quarter Cantor measure satisfies the doubling condition (with a forward reference to the explicit estimates in Section 4). This will make the applicability of the sufficient condition immediate. revision: yes
-
Referee: [Abstract] The abstract asserts existence of both the condition and the example but supplies no derivation outline, error estimates, or reference to the section containing the proof of the sufficient condition, making it impossible to assess whether the central claims are supported.
Authors: Abstracts are intentionally concise and do not contain detailed derivations or error estimates. Nevertheless, we accept that a brief pointer to the relevant sections would aid readers. In the revised manuscript we will append one sentence to the abstract: 'The sufficient condition is proved in Section 3; its application to the quarter Cantor measure appears in Section 4.' This addition preserves brevity while directing attention to the proofs. revision: yes
Circularity Check
No circularity detected; derivation chain is self-contained
full rationale
The paper states a sufficient condition for divergence of Mock Fourier series on doubling spectral measures and separately asserts existence of a quarter Cantor measure example. No equations, self-definitional constructions, fitted inputs renamed as predictions, or load-bearing self-citations are visible in the provided abstract or described structure. The application to the quarter Cantor measure is presented as an instance satisfying the hypotheses rather than a definitional reduction or circular fit. The derivation does not reduce to its inputs by construction and remains independent of any self-referential steps.
Axiom & Free-Parameter Ledger
Reference graph
Works this paper leans on
-
[1]
Li-Xiang An, Xing-Gang He, A class of spectral Moran measures. J. Funct. Anal. 266 (1) (2014), 343–354
work page 2014
-
[2]
Marilina Carena, Weak type (1,1) of maximal operators on metric measure spaces. Rev. Un. Mat. Argentina 50 (2009), no. 1, 145-159
work page 2009
-
[3]
Michael Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61 (1990), no. 2, 601-628
work page 1990
-
[4]
Xin-Rong Dai, Xing-Gang He, Ka-Sing Lau, On spectral N-Bernoulli measures. Adv. Math. 259 (2014), 511–531
work page 2014
-
[5]
Dorin Ervin Dutkay, John Haussermann, Chun-Kit Lai, Hadamard triples generate self-affine spectral measures. Trans. Amer. Math. Soc. 371 (2019), no. 2, 1439–1481
work page 2019
-
[6]
Dorin Ervin Dutkay, De-Guang Han, Qi-Yu Sun, Divergence of the Mock and scrambled Fourier series on fractal measures. Trans. Amer. Math. Soc. 366 (2014), no. 4, 2191-2208
work page 2014
-
[7]
Dorin Ervin Dutkay, Chun-Kit Lai, Yang Wang, Fourier bases and Fourier frames on self-affine measures. Recent developments in fractals and related fields, 87-111, Trends Math., Birkhäuser/Springer, Cham, 2017
work page 2017
-
[8]
Miguel de Guzm\' a n, Real Variable Methods in Fourier Analysis, North-Holland Math. Stud. 46, North-Holland, Amsterdam, 1981
work page 1981
-
[9]
John Edward Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), no. 5, 713–747
work page 1981
-
[10]
Palle Jorgensen, Steen Pedersen, Dense analytic subspaces in fractal L^2 -spaces. J. Anal. Math. 75 (1998), 185-228
work page 1998
-
[11]
Izabella aba, Yang Wang, On spectral Cantor measures. J. Funct. Anal. 193 (2002), no.2, 409–420
work page 2002
-
[12]
Robert Strichartz, Convergence of Mock Fourier series. J. Anal. Math. 99 (2006), 333-353
work page 2006
-
[13]
Po-Lam Yung, Doubling properties of self-similar measures. Indiana Univ. Math. J. 56 (2007), no. 2, 965–990
work page 2007
-
[14]
Antoni Szczepan Zygmund, Trigonometric series: Vols. I, II. Second edition, reprinted with corrections and some additions Cambridge University Press, London-New York 1968
work page 1968
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.