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Existence, equivalence and spectrality of infinite convolutions in $\R^d$

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that certain infinite convolutions in $\mathbb{R}^d$, even with unbounded support, are spectral measures with spectra contained in $\mathbb{Z}^d$.

desk verdict Main spectrality theorem is likely correct, but the explicit spectrum formula in Theorem 2.6 is not justified; worth refereeing with major revision. read the letter →

arxiv 2506.06670 v1 pith:EATGZU5X submitted 2025-06-07 math.FA

Junjie Miao , Hongbo Zhao This is my paper
classification math.FA MSC 28A8042C3060B10
keywords infiniteconvolutionsspectralmeasuresHadamardtriplesadmissiblepairsequi-positivefamiliesnon-compactsupportFouriertransformfractal
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

This paper studies infinite convolutions of discrete measures in $\mathbb{R}^d$ that need not be compactly supported. It introduces a notion of equivalence for digit-set sequences based on finitely accumulating relative mismatch, and proves that equivalent sequences produce infinite convolutions that converge together and share equi-positivity and spectra. The main result gives sufficient conditions, namely remainder boundedness, uniform contraction, and a partial concentration condition on a subsequence, under which the infinite convolution exists and is a spectral measure with a spectrum in $\mathbb{Z}^d$. The authors exhibit a two-dimensional example with unbounded support, showing the theorem reaches beyond the compactly supported fractal measures studied previously.

What carries the argument

The argument rests on four objects: admissible pairs and Hadamard triples, where a digit set $B$ and expansive matrix $R$ admit a dual set $L$ making a unitary matrix and supplying a spectrum for the one-step measure; the equivalence relation on digit sequences defined by summability of relative mismatch counts; the remainder bounded condition and the partial concentration condition, which control how much digit mass lies far from the origin after rescaling; and equi-positive families, collections of tail measures whose Fourier transforms stay uniformly bounded away from zero near the integer lattice. Lemma 6.1, a lower bound for the average of exponentials with arguments in an interval, converts the concentration estimate into the uniform Fourier lower bound that equi-positivity requires.

What would settle it

The sharpest test would be a direct counterexample to Theorem 2.6: a sequence of admissible pairs satisfying the remainder bounded condition, uniform contraction, and the partial concentration condition on some subsequence, yet whose infinite convolution is not spectral.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2.6: if $\{(R_k,B_k)\}$ is a sequence of admissible pairs satisfying the remainder bounded condition and the uniform contractive condition, and some subsequence satisfies the partial concentration condition with some $l\in(0,1)$, then the infinite convolution $\mu$ exists and is a spectral measure admitting a spectrum in $\mathbb{Z}^d$. The proof works by replacing each digit set by a congruent set inside $R_k[-\tfrac12,\tfrac12)^d$, showing that the replacement preserves existence and equi-positivity, estimating the Fourier product from below using the partial concentration condition to obtain an equi-positive family, and then running the standard equi-positivity-to-spectrum construction. The paper also proves simultaneous convergence, preservation of equi-positivity, and a common spectrum for equivalent digit-set sequences. Example 2.7 gives a concrete non-compactly supported spectral measure in $\mathbb{R}^2$.

Load-bearing premise

The load-bearing premise is that some subsequence of the digit sets satisfies the partial concentration condition: after rescaling by the inverse matrices, most digits lie close to the origin with spread strictly less than $1-l$, and the unscattered digits have summable relative counts.

Editorial extensions

If this is right

  • If two sequences of digit sets differ only by finitely accumulating relative mismatch, their infinite convolutions exist together and, under admissible-pair hypotheses, share a spectrum, so compact and non-compact examples can be analyzed simultaneously.
  • Theorem 2.5 supplies a new existence criterion for non-compactly supported infinite convolutions in any dimension.
  • Theorem 2.6 produces spectral measures with unbounded support, a class not covered by earlier compact-support theories.
  • The explicit Example 2.7 gives a concrete non-compact spectral measure in $\mathbb{R}^2$ and shows the hypotheses are satisfiable.
  • Equivalent admissible sequences with congruent digits modulo $R_k\mathbb{Z}^d$ have the same spectrum, so spectral data are stable under digit perturbations.

Reading between the lines

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

  • One might try relaxing the partial concentration condition to a logarithmic or averaged concentration condition, since the proof's Borel-Cantelli step suggests a weaker tail condition may still yield equi-positivity in some examples.
  • The equivalence relation on digit sets is metric-like and may connect to Wasserstein or Prokhorov stability of infinite convolutions, allowing perturbation results for spectra of random convolutions.
  • The construction suggests a route to non-compact spectral measures with prescribed dimension by choosing slowly growing digit sets that satisfy the partial concentration condition while pushing mass to infinity.
  • A direct consequence not stated in the paper is that any two equivalent admissible sequences sharing the same tail asymptotics will share an explicit common spectrum, not merely spectrality.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies infinite convolutions of uniform discrete measures in R^d, allowing non-compact support. It introduces an equivalence relation on sequences of finite digit sets, proves that equivalent sequences yield simultaneously convergent convolutions and simultaneously equi-positive families, and shows that equivalent admissible-pair sequences with congruent digit sets share a common spectrum. It gives a sufficient condition (remainder bounded condition plus uniform contractivity) for existence of such convolutions via a distributional three-series theorem, and a sufficient condition (existence of a subsequence satisfying a partial concentration condition) for spectrality. The main theorem asserts that under these hypotheses the infinite convolution is a spectral measure with a spectrum in Z^d, with an explicit product-form spectrum when each Hadamard triple has 0 in the dual digit set contained in the corresponding cube. An example of a non-compactly supported spectral measure is constructed.

Significance. If the main theorem is correct, it provides the first general construction of spectral measures arising as infinite convolutions of admissible pairs with non-compact support, extending the compact-support theory of Strichartz, Dutkay-Lai, An-Fu-Lai, and Dutkay-Haussermann-Lai. The equivalence and common-spectrum results are clean and potentially useful tools. The proofs are largely self-contained and do not rely on fitted parameters; Theorem 2.2 is a crisp Borel-Cantelli argument, Theorem 2.5 is a correct use of the Jessen-Wintner three-series criterion, and the equi-positivity transfer in Corollary 2.3 is sound. However, the explicit spectrum formula in the 'Moreover' part of Theorem 2.6 rests on a containment that is false in general, so the advertised explicit description of a spectrum is not currently established.

major comments (2)
  1. [Section 6, proof of Theorem 2.6, 'Moreover' paragraph] The proof asserts 'Since L_{m_{j-1},m_j} \subseteq R^T_{m_{j-1},m_j}[-1/2,1/2)^d' and concludes that k_{\lambda,j}=0 for all \lambda\in L_{m_{j-1},m_j}, which yields the simple product formula \Lambda=\cup_{k=1}^\infty{L_1+R_1^T L_2+\cdots+(R_{k-1}\cdots R_1)^T L_k}. This containment does not follow from the hypothesis 0\in L_k\subseteq R_k^T[-1/2,1/2)^d for each k. For d=1, take R_1=R_2=4 and L_1=L_2={0,2}; then each L_k satisfies the hypothesis, but L_{1,2}=L_1+4L_2={0,2,8,10}, while R^T_{1,2}[-1/2,1/2)=16[-1/2,1/2)=[-8,8), so 10 is not contained. Consequently the step k_{\lambda,j}=0 for all \lambda\in L_{m_{j-1},m_j} is unjustified, and the stated explicit spectrum \Lambda is unproven. The main conclusion that a spectrum in Z^d exists does not depend on this containment, since Theorem 3.4 constructs a spectrum inductively; however, the 'Moreover' claim as stated overreaches. It should either be weakened to assert only the existence of a spectrum in Z^d (with the inductive construction from Theorem 3.4), or be supplemented by an additional hypothesis that ensures the containment.
  2. [Example 2.7] The example explicitly identifies \Lambda = \cup_{k=1}^\infty{L_1+R_1^T L_2+\cdots+(R_{k-1}\cdots R_1)^T L_k} as a spectrum, relying on the unproven 'Moreover' clause of Theorem 2.6. Since the containment used to justify this formula is false in general and is not separately verified for the particular digit sets of the example, the claim that this specific \Lambda is a spectrum is not established. The non-compactness argument and the existence of some spectrum in Z^d remain valid through the first part of Theorem 2.6, so the example can be repaired by removing the explicit \Lambda, or by proving the containment for this specific construction. The paper should be revised accordingly.
minor comments (4)
  1. [Section 2, paragraph before Theorem 2.6] The word 'spetral' should be 'spectral' in the sentence 'we show that the infinite convolution \mu is a spetral measure'.
  2. [Proof of Lemma 3.5] The domain of x is written as '[0,1)^d' twice, whereas Definition 3.3 uses '[-1/2,1/2)^d'. The property is equivalent up to an integer shift, but the notation should be consistent with the definition.
  3. [Proof of Theorem 2.6, estimate after (6.4)] The text says 'It remains to estimate M_{B_{n_k}}(\xi_0)' and later writes '|M_{B_{n_k}}(\xi_0)| \geq ...', but the factor in the product expansion is M_{B'_{n_k}}(\xi_0), and the subsequent estimates use the sets B'^l_{n_k,1} and B'^l_{n_k,2}. This is a notational slip: the estimate should apply to M_{B'_{n_k}}(\xi_0).
  4. [Proof of Theorem 2.6, inequality (6.4)] The bound on the finite product is stated 'for all k,j>0', but the product is over j=1,\ldots,J-1 and the bound depends on J; the range of j should be specified as 1\le j\le J-1.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the derivation is self-contained, and the main defect (a false containment in the 'Moreover' clause of Theorem 2.6) is a correctness gap, not a circular reduction.

full rationale

The paper's central claims are proved from stated hypotheses using standard external tools, with no fitted parameter renamed as a prediction and no conclusion built into an assumption. Theorem 2.2 is proved from Borel-Cantelli and the explicit coupling in (4.1)-(4.2); Theorem 2.5 is proved from the Jessen-Wintner three-series theorem (Theorem 5.1) and Theorem 2.2; Theorem 3.4, which is the load-bearing spectrality engine, is proved in full inside the paper, including the equi-positivity-to-spectrum construction and the Q_{mu,Lambda} argument. Theorem 2.6 establishes equi-positivity directly from the partial concentration condition via Lemma 6.1 and elementary trigonometric estimates, not by invoking a prior result. The self-citations [31], [32], and [33] are contextual: [31] is mentioned as prior work applying the same idea, [32] is cited for the generalized equi-positive notion but Definition 3.3 and the needed theorem are restated and proved here, and [33] is only a remark about possible weaker conditions in d=1. None of these citations is used to define the target statement or to forbid alternatives. The main weakness is not circularity: the 'Moreover' clause of Theorem 2.6 asserts L_{m_{j-1},m_j} subseteq R^T_{m_{j-1},m_j}[-1/2,1/2)^d and hence k_{lambda,j}=0 for all lambda, but this containment is not generally true (e.g., in one dimension R=4 and L={0,2} gives L_{1,2}={0,2,8,10}, while R^T_{1,2}[-1/2,1/2)=[-8,8)). That is a mathematical gap in the explicit-spectrum statement, not a circular dependency; the main spectrality conclusion in Z^d is supported by the independent equi-positivity argument.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard probability theorems (Borel-Cantelli, Kolmogorov three series), background facts on Hadamard triples, and three domain assumptions: uniform contractivity, RBC, and PCC. No free parameters are fitted to data; the parameter l in PCC is part of a hypothesis, not a fitted constant. No new entities are introduced.

assumptions (7)
  • standard math Borel-Cantelli lemma for coupling equivalent random sequences
    Used in proof of Theorem 2.2 to show a.s. equality of tails when sum of mismatch probabilities is finite.
  • standard math Jessen-Wintner theorem (distribution version of Kolmogorov three series)
    Theorem 5.1 is the criterion used to prove existence of the infinite convolution in Theorem 2.5.
  • standard math Equivalence between convergence in distribution and almost sure convergence for sum of independent random vectors
    Invoked in Section 4 to translate existence of infinite convolution into a.s. convergence of the random series.
  • domain assumption Properties of Hadamard triples cited from [12,29] (Lemma 3.1)
    The paper relies on the fact that Hadamard triples generate spectra of finite convolutions and that congruences preserve Hadamard triples; these are background results in the theory of spectral measures.
  • domain assumption Uniform contractive condition sup_k ||R_k^{-1}|| < 1
    Used throughout to make all geometric decays geometric; it is an explicit assumption in Theorems 2.4, 2.5, 2.6.
  • domain assumption Partial concentration condition (PCC) with some l in (0,1)
    The key geometric hypothesis in Theorem 2.6; it supplies the lower Fourier bound via Lemma 6.1.
  • domain assumption Remainder bounded condition (RBC) sum #B_{k,2}/#B_k < infinity
    Ensures the digit sequence is equivalent to its truncated version, enabling existence and equi-positivity transfer.

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Pith. "Pith review of Existence, equivalence and spectrality of infinite convolutions in $\R^d$." pith.science (2026). https://pith.science/paper/EATGZU5X

@misc{pith2026250606670,
  author       = {Pith},
  title        = {Pith review of: Existence, equivalence and spectrality of infinite convolutions in $\R^d$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EATGZU5X}},
  note         = {Machine review of arXiv:2506.06670}
}
abstract

In this paper, we study existence, equivalence and spectrality of infinite convolutions which may not be compactly supported in $d$-dimensional Euclidean space by manipulating various techniques in probability theory. First, we define the equivalent sequences, and we prove that the infinite convolutions converges simultaneously if they are generated by equivalent sequences. Moreover, the equi-positivity keeps unchanged for infinite convolutions generated by equivalent sequences. Next, we study the spectrality of infinite convolutions generated by admissible pairs, and we show such infinite convolutions have the same spectrum if they are generated by the equivalent sequences. Finally, we provide some sufficient conditions for the existence and spectral properties of infinite convolutions in higher dimensions.

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