On the Spectral Properties of a Class of Planar Sierpinski Self-Affine Measures

Jia-Long Chen; Wen-Hui Ai

arxiv: 2508.20809 · v1 · pith:SFQWPDRKnew · submitted 2025-08-28 · 🧮 math.FA

On the Spectral Properties of a Class of Planar Sierpinski Self-Affine Measures

Jia-Long Chen , Wen-Hui Ai This is my paper

Pith reviewed 2026-05-18 20:53 UTC · model grok-4.3

classification 🧮 math.FA

keywords self-affine measuresspectral measuresorthogonal exponentialsSierpinski gasketFourier transform zerosexpanding matricesfractal measures

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The pith

For Sierpinski self-affine measures with equal expansion rates, having an infinite orthogonal exponential system is equivalent to being spectral under the given digit-set zero conditions.

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

The paper studies the spectral properties of a family of planar self-affine measures built from a prime number of digits and an upper-triangular expanding matrix. It supplies necessary and sufficient conditions for the measure to admit an infinite set of mutually orthogonal exponential functions and, simultaneously, to be a spectral measure when the two diagonal scaling factors are identical. When those factors differ, the same style of condition is obtained once the digit set is suitably restricted. A reader would care because these criteria decide whether the associated L2 space on the fractal supports a Fourier-like orthonormal basis of exponentials, exactly as on the circle or the torus.

Core claim

When the diagonal entries satisfy ρ1 = ρ2, the measure μ_{M,D} possesses an infinite orthogonal set of exponential functions and is a spectral measure if and only if the stated algebraic conditions on the matrix and digit set hold; when no such infinite set exists, the largest possible orthogonal exponential family has a precise finite cardinality that is explicitly estimated. When ρ1 ≠ ρ2 and D is restricted, μ_{M,D} is a spectral measure precisely under a corresponding necessary and sufficient algebraic criterion derived from the zero set of the digit Fourier transform.

What carries the argument

The zero-set condition Z(δ̂_D) = ∪_{j=1}^{p-1} (j a / p + Z²) that forces the Fourier transform of the discrete digit measure to vanish on a controlled union of cosets and thereby governs all orthogonality relations among candidate exponentials.

If this is right

When ρ1 = ρ2 the existence of an infinite orthogonal exponential system is decided by the same algebraic test that decides spectrality.
In the equal-rate case the size of the largest orthogonal exponential family is bounded and can be computed explicitly when the infinite-set condition fails.
For unequal rates and suitably restricted digit sets, spectrality is completely characterized by the same zero-set data.
The Fourier transform of the measure factors through the digit set in a way that reduces all orthogonality questions to lattice coset arithmetic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same zero-set technique may classify spectrality for self-affine measures whose expanding matrix is no longer triangular.
Numerical verification of the orthogonality relations for small primes and concrete a would provide an immediate check on the derived conditions.
The finite-cardinality estimates when the infinite set is absent could be used to bound the dimension of the space of band-limited functions on these fractals.

Load-bearing premise

The digit set D is chosen so that the zeros of its Fourier transform lie exactly on the union of those p-1 shifted integer lattices.

What would settle it

An explicit choice of prime p, vector a, and equal rates ρ1 = ρ2 for which the zero-set condition on D fails yet an infinite orthogonal exponential family still exists in L2(μ_{M,D}), or the converse.

read the original abstract

We investigate the spectral properties of a class of Sierpinski-type self-affine measures defined by \[ \mu_{M,D}(\cdot) = p^{-1} \sum_{d \in D} \mu_{M,D}(M(\cdot) - d), \] where $ p $ is a prime number, $ M = \begin{bmatrix} \rho_1^{-1} & c 0 & \rho_2^{-1} \end{bmatrix} $ is a real upper triangular expanding matrix, and $ D = \{d_0, d_1, \cdots, d_{p-1}\} \subset \mathbb{Z}^2 $ satisfying $ \mathcal{Z}(\widehat{\delta}_{D}) = \cup_{j=1}^{p-1} \left( \frac{j \bm{a}}{p} + \mathbb{Z}^2 \right) $ for some $ \bm{a} \in \mathcal{E}_{p}= \{ (i_1, i_2)^* : i_1, i_2 \in [1, p-1] \cap \mathbb{Z} \} $, where $ \mathcal{Z}(\widehat{\delta}_{D}) $ denotes the set of zeros of $ \widehat{\delta}_{D} $ with $ \delta_{D} = \frac{1}{\# D} \sum_{d \in D} \delta_d $. When $\rho_1 = \rho_2$, we derive necessary and sufficient conditions for $\mu_{M,D}$ to both: $(i)$ possess an infinite orthogonal set of exponential functions, and $(ii)$ be a spectral measure. When no infinite orthogonal exponential system exists in $L^{2}(\mu_{M,D})$, we quantify the maximum number of orthogonal exponentials and provide precise estimates. For $\rho_1 \neq \rho_2$, with restricted digit sets $D$, we obtain a necessary and sufficient condition for $\mu_{M,D}$ to be a spectral measure.

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

This paper derives necessary and sufficient conditions for spectrality and infinite orthogonal exponentials in a narrow family of planar Sierpinski self-affine measures defined by a specific zero-set property on the digit set.

read the letter

The main takeaway is that the authors work out necessary and sufficient conditions for when these measures support an infinite orthogonal set of exponentials or qualify as spectral measures, with a clean split between the equal-rho and unequal-rho cases on the upper-triangular expanding matrix. They tie everything to the zero-set condition on the Fourier transform of the digit measure, which controls the zeros that appear in the infinite product for the measure's own Fourier transform. When the scalings match they characterize both properties and give bounds on the largest finite orthogonal set; when they differ they give a spectrality criterion for restricted digits. This extends earlier results on Sierpinski-type measures by making the case distinction explicit and adding the quantitative estimates. The approach follows the usual iteration and zero-tracking steps in this area, so the logic looks consistent on the surface. The main limitation is the narrow scope: everything is restricted to prime p, upper-triangular matrices, and digit sets whose zeros sit exactly on that union of shifted lattices. That choice makes the proofs manageable but keeps the results inside this parameterized family rather than applying more broadly. The zero-set condition is built into the definition of the class, so it is not an extra hidden assumption. Readers already working on spectral self-affine measures or orthogonal exponentials on fractals will find the concrete criteria useful for checking examples. The paper is focused and technically grounded enough to deserve peer review so the proofs can be examined in detail.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the spectral properties of a class of planar Sierpinski-type self-affine measures μ_{M,D} generated by an IFS with prime cardinality p, an upper-triangular expanding matrix M having diagonal entries ρ₁^{-1} and ρ₂^{-1}, and a digit set D ⊂ ℤ² whose discrete measure δ_D satisfies the zero-set condition Z(δ̂_D) = ∪_{j=1}^{p-1} (j a / p + ℤ²) for a ∈ E_p. Separate results are obtained for the isotropic case ρ₁ = ρ₂ (necessary and sufficient conditions for an infinite orthogonal exponential system and for spectrality, together with cardinality bounds when the orthogonal set is finite) and for the anisotropic case ρ₁ ≠ ρ₂ under restricted choices of D (necessary and sufficient condition for spectrality).

Significance. If the derivations hold, the work supplies explicit, checkable criteria for spectrality and orthogonality in a concrete family of planar self-affine measures, extending earlier one-dimensional and isotropic results to the anisotropic triangular setting. The quantitative bounds on the size of maximal orthogonal sets and the use of the prescribed zero-set geometry to control the Fourier transform are potentially useful for constructing further examples and for testing conjectures about the spectrality of fractal measures.

major comments (2)

[§3–4 (isotropic case)] The necessity direction in the ρ₁ = ρ₂ case (presumably Theorem 3.1 or 4.1) invokes the zero-set condition to rule out additional zeros of the infinite product for the Fourier transform of μ_{M,D}; the argument must explicitly verify that the off-diagonal entry c does not introduce new zeros under iteration, which is not immediately visible from the abstract statement.
[§5 (anisotropic case)] For the anisotropic case with restricted D, the necessary and sufficient condition is stated only after imposing further constraints on D; the manuscript should clarify whether these restrictions are strictly necessary for the zero-set geometry to survive the action of the non-scalar matrix M or whether they can be relaxed.

minor comments (3)

[Abstract] The matrix M is displayed with an awkward line break in the abstract; a properly formatted 2×2 array would improve readability.
[Section on finite orthogonal sets] A short table or numerical example illustrating the maximal cardinality bound for small primes p (e.g., p=3 or 5) would make the quantitative estimates more concrete.
[Introduction] A few additional references to recent work on spectral self-affine measures in ℝ² would help situate the contribution relative to the existing literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. The comments have prompted us to enhance the clarity of our proofs regarding the role of the off-diagonal entry and the necessity of restrictions on the digit set. We address each point below.

read point-by-point responses

Referee: [§3–4 (isotropic case)] The necessity direction in the ρ₁ = ρ₂ case (presumably Theorem 3.1 or 4.1) invokes the zero-set condition to rule out additional zeros of the infinite product for the Fourier transform of μ_{M,D}; the argument must explicitly verify that the off-diagonal entry c does not introduce new zeros under iteration, which is not immediately visible from the abstract statement.

Authors: We agree that an explicit verification would improve the readability of the necessity proof. In the current manuscript, the argument relies on the upper-triangular form of M and the specific zero-set condition Z(δ̂_D), which ensures that the iterates under M^* preserve the zeros within the union of the lattices. However, to address this directly, we will insert a new lemma in Section 3 that computes the action of the adjoint matrix on the zero set and shows that the off-diagonal entry c does not generate extraneous zeros in the infinite product. This addition will make the reasoning fully transparent. We will also update the abstract if necessary to reflect this detail. revision: yes
Referee: [§5 (anisotropic case)] For the anisotropic case with restricted D, the necessary and sufficient condition is stated only after imposing further constraints on D; the manuscript should clarify whether these restrictions are strictly necessary for the zero-set geometry to survive the action of the non-scalar matrix M or whether they can be relaxed.

Authors: The further constraints on D in the anisotropic case are indeed required for the zero-set geometry to be invariant under the iterations of the non-scalar matrix M, as the differing scaling factors ρ₁ and ρ₂ combined with the off-diagonal c can otherwise map zeros outside the prescribed set. We will add a remark in Section 5 explaining why these restrictions cannot be relaxed without modifying the zero-set condition or the matrix structure, supported by a counterexample sketch if the restrictions are dropped. This clarification will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper specifies the class of measures via the explicit input assumption that Z(δ̂_D) equals the union of translated lattices for a in E_p, then derives necessary and sufficient conditions for spectrality and infinite orthogonal exponentials directly from this zero-set property together with the self-affine iteration equation and the expanding matrix M. No target spectral property is redefined in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation chain. The logical structure relies on standard Fourier-analytic control of orthogonality for self-affine measures and remains independent of the results being proved.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the self-similar definition of the measure, the expanding property of the upper-triangular matrix M, and the explicit zero-set condition imposed on the digit measure delta_D; these are setup assumptions rather than derived quantities.

axioms (2)

domain assumption M is a real upper triangular expanding matrix.
Invoked in the definition of the self-affine measure mu_{M,D}.
domain assumption D satisfies Z(hat delta_D) = union_{j=1}^{p-1} (j a / p + Z^2) for some a in E_p.
This zero-set condition is required to define the class of measures for which the spectral properties are characterized.

pith-pipeline@v0.9.0 · 5924 in / 1577 out tokens · 62658 ms · 2026-05-18T20:53:19.828960+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Z(δ̂_D) = ∪_{j=1}^{p-1} (j a / p + Z²) for a ∈ E_p; Theorems 1.1–1.7 on ρ1=ρ2 vs ρ1≠ρ2 cases and Hadamard triples
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2.5: Hadamard triple implies spectral self-affine measure

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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L.X. An, L. He, X.G. He, Spectrality and non-spectrality of the Riesz product measures with three elements in digit sets. J. Funct. Anal. 277 (2019), 255-278

work page 2019
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Chen, X.Y

M.L. Chen, X.Y. Wang, J. Zheng, On the orthogonal exponential functions of a class of planar self-affine measures. J. Math. Anal. Appl. 485 (2020), 11 pp

work page 2020
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Chen, J.C

M.L. Chen, J.C. Liu, Y.H. Yao, Fourier bases of the planar self-affine measures with three digits. Math. Nachr. 296 (2023), 4995-5011

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[5]

Chen, X.Y

J.L. Chen, X.Y. Yan, P.F. Zhang, The cardinality of orthogonal exponentials for a Class of self-affine Measures on Rn. Acta Math. Hungar. 175 (2025), 219-235

work page 2025
[6]

Chen, On the Fourier orthonormal bases of a class of Sierpinski-type Moran measures on Rn

J.L. Chen, On the Fourier orthonormal bases of a class of Sierpinski-type Moran measures on Rn. preprint. https://doi.org/10.48550/arXiv.2505.09360

work page doi:10.48550/arxiv.2505.09360
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Chi, J.F

Z.C. Chi, J.F. Lu, M.M. Zhang, A class of spectral Moran measures generated by the compatible tower. J. Geom. Anal. 34 (2024), 30 pp

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Dai, X.Y

X.R. Dai, X.Y. Fu, Z.H. Yan, Spectrality of self-affine Sierpinski-type measures on R2. Appl. Comput. Harmon. Anal. 52 (2021), 63-81

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Dutkay, J

D.E. Dutkay, J. Haussermann, C.K. Lai, Hadamard triples generate self-affine spectral measures. Trans. Am. Math Soc. 371 (2019), 1439-1481

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M. Kolountzakis, M. Matolcsi, Complex Hadamard matrices and the spectral set conjecture. Collect. Math. 57 (2006), 281-291

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Z.Y. Lu, X.H. Dong, Z.S. Liu, Spectrality of Sierpinski-type self-affine measures. J. Funct. Anal. 282 (2022), 31 pp

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Wang, J.C

Z.Y. Wang, J.C. Liu, J. Su, Spectral property of self-affine measures on Rn. J. Fourier Anal. Appl. 27 (2021), 28 pp

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[23]

Yan, Spectral Moran measures on R2

Z.H. Yan, Spectral Moran measures on R2. Nonlinearity. 35 (2022), 1261-1285. 1 School of Mathematics, South China University of Technology, Guangzhou, 510641, P. R. China 31 2 Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China Em...

work page 2022

[1] [1]

L.X. An, X.Y. Fu, C.K. Lai, On spectral Cantor-Moran measures and a variant of Bourgain’s sum of sine problem. Adv. Math. 349 (2019), 84-124

work page 2019

[2] [2]

L.X. An, L. He, X.G. He, Spectrality and non-spectrality of the Riesz product measures with three elements in digit sets. J. Funct. Anal. 277 (2019), 255-278

work page 2019

[3] [3]

Chen, X.Y

M.L. Chen, X.Y. Wang, J. Zheng, On the orthogonal exponential functions of a class of planar self-affine measures. J. Math. Anal. Appl. 485 (2020), 11 pp

work page 2020

[4] [4]

Chen, J.C

M.L. Chen, J.C. Liu, Y.H. Yao, Fourier bases of the planar self-affine measures with three digits. Math. Nachr. 296 (2023), 4995-5011

work page 2023

[5] [5]

Chen, X.Y

J.L. Chen, X.Y. Yan, P.F. Zhang, The cardinality of orthogonal exponentials for a Class of self-affine Measures on Rn. Acta Math. Hungar. 175 (2025), 219-235

work page 2025

[6] [6]

Chen, On the Fourier orthonormal bases of a class of Sierpinski-type Moran measures on Rn

J.L. Chen, On the Fourier orthonormal bases of a class of Sierpinski-type Moran measures on Rn. preprint. https://doi.org/10.48550/arXiv.2505.09360

work page doi:10.48550/arxiv.2505.09360

[7] [7]

Chi, J.F

Z.C. Chi, J.F. Lu, M.M. Zhang, A class of spectral Moran measures generated by the compatible tower. J. Geom. Anal. 34 (2024), 30 pp

work page 2024

[8] [8]

Deng, Spectrality of one dimensional self-similar measures with consecutive digits

Q.R. Deng, Spectrality of one dimensional self-similar measures with consecutive digits. J. Math. Anal. Appl. 409 (2014), 331-346

work page 2014

[9] [9]

Q.R. Deng, K. S. Lau, Sierpinski-type spectral self-similar measures. J. Funct. Anal. 269 ( 2015), 1310- 26

work page 2015

[10] [10]

Dai, When does a Bernoulli convolution admit a spectrum? Adv

X.R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231 (2012), 1681-1693

work page 2012

[11] [11]

Dai, X.G

X.R. Dai, X.G. He, K.S. Lau, On spectral N-Bernoulli measures. Adv. Math. 259 (2014), 511-531

work page 2014

[12] [12]

Dai, X.Y

X.R. Dai, X.Y. Fu, Z.H. Yan, Spectrality of self-affine Sierpinski-type measures on R2. Appl. Comput. Harmon. Anal. 52 (2021), 63-81

work page 2021

[13] [13]

Dutkay, J

D.E. Dutkay, J. Haussermann, C.K. Lai, Hadamard triples generate self-affine spectral measures. Trans. Am. Math Soc. 371 (2019), 1439-1481

work page 2019

[14] [14]

Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem

B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16 (1974), 101-121

work page 1974

[15] [15]

Hutchinson, Fractals and self-similarity

J. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713-747

work page 1981

[16] [16]

T.Y. Hu, K.S. Lau, Spectral property of the Bernoulli convolutions. Adv. Math. 219 (2008), 554-567

work page 2008

[17] [17]

Jorgensen, S

P.E.T. Jorgensen, S. Pedersen, Dense analytic subspaces in fractal L2-spaces. J. Anal. Math. 75 (1998), 185-228

work page 1998

[18] [18]

Kolountzakis, M

M. Kolountzakis, M. Matolcsi, Tiles with no spectra. Forum Math. 18 (2006), 519-528

work page 2006

[19] [19]

Kolountzakis, M

M. Kolountzakis, M. Matolcsi, Complex Hadamard matrices and the spectral set conjecture. Collect. Math. 57 (2006), 281-291

work page 2006

[20] [20]

Z.Y. Lu, X.H. Dong, Z.S. Liu, Spectrality of Sierpinski-type self-affine measures. J. Funct. Anal. 282 (2022), 31 pp

work page 2022

[21] [21]

Tao, Fuglede’s conjecture is false in 5 and higher dimensions

T. Tao, Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett. 11 (2004), 251-258

work page 2004

[22] [22]

Wang, J.C

Z.Y. Wang, J.C. Liu, J. Su, Spectral property of self-affine measures on Rn. J. Fourier Anal. Appl. 27 (2021), 28 pp

work page 2021

[23] [23]

Yan, Spectral Moran measures on R2

Z.H. Yan, Spectral Moran measures on R2. Nonlinearity. 35 (2022), 1261-1285. 1 School of Mathematics, South China University of Technology, Guangzhou, 510641, P. R. China 31 2 Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China Em...

work page 2022