Spectrality of product-form self-similar measures and tiles

Jia-Jie Wang; Jia Zheng; Jing-Cheng Liu

arxiv: 2603.21031 · v2 · submitted 2026-03-22 · 🧮 math.FA

Spectrality of product-form self-similar measures and tiles

Jing-Cheng Liu , Jia-Jie Wang , Jia Zheng This is my paper

Pith reviewed 2026-05-15 01:50 UTC · model grok-4.3

classification 🧮 math.FA

keywords self-similar measuresexponential orthonormal basisproduct-form digit setsspectralityself-similar tilesFuglede conjecture

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The pith

A self-similar measure from a product-form digit set has an exponential orthonormal basis in L² precisely when the reciprocal of its contraction ratio is an integer divisible by the digit parameters N and L under an extra gcd condition.

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

The paper proves an if-and-only-if criterion for spectrality of self-similar measures generated by the iterated function system with contraction ρ and digit set D formed as the direct sum of two consecutive integer blocks. The criterion requires that ρ inverse equals a natural number p divisible by both block lengths N and L, plus a further divisibility N divides m over a certain gcd involving m, L and powers of p. This extends earlier results that handled only prime block lengths or equal block lengths. In the equal-cardinality case where ρ inverse equals the number of digits, the same technique shows that the associated self-similar tile supports an exponential orthonormal basis in L² if and only if the tile tiles the line by translations.

Core claim

Let 0 < ρ < 1 and let D be the direct sum {0,…,N−1} ⊕ m{0,…,L−1} with N, L ≥ 2. The space L²(μ_{ρ,D}) admits an exponential orthonormal basis if and only if ρ^{-1} = p ∈ ℕ satisfies N | p, L | p and N | m / gcd(m, p^d), where d is the largest natural number i such that gcd(mL / gcd(mL, p^i), L) ≠ 1. When additionally ρ^{-1} equals the cardinality NL of D, the space L²(χ_T dx) admits an exponential orthonormal basis if and only if the self-similar set T tiles ℝ by translations.

What carries the argument

The product-form digit set D = {0,…,N−1} ⊕ m{0,…,L−1}, whose additive decomposition permits iterative reduction of exponential orthogonality relations to explicit divisibility statements on N, L, m and p.

If this is right

Whenever the stated divisibility conditions hold, the measure μ_{ρ,D} is spectral.
The result recovers the known spectrality criteria for the special cases of prime N or L and of N=L.
When ρ^{-1} equals the cardinality of D the associated tile T is spectral in L² precisely when it tiles the line by translations.
The auxiliary integer d is determined solely by the prime factors shared between mL and successive powers of p.

Where Pith is reading between the lines

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

The same gcd-based obstruction may appear in higher-dimensional product-form IFS, suggesting a uniform arithmetic test for spectrality across dimensions.
Relaxing the product-form assumption on D while keeping the same cardinality would likely produce measures that fail the criterion even when ρ^{-1} is integer.
The tiling characterization supplies a concrete test set of candidate counter-examples to the one-dimensional Fuglede conjecture inside the self-similar class.

Load-bearing premise

The digit set must be exactly the direct sum of two consecutive integer intervals with the stated additive structure.

What would settle it

Fix concrete values N=2, L=3, m=4 and test whether L²(μ_{ρ,D}) contains an exponential orthonormal basis for ρ=1/6 (satisfies the condition) versus ρ=1/5 (violates it) by checking whether the corresponding infinite product of trigonometric polynomials vanishes at any nonzero lattice point.

read the original abstract

This paper studies the Fourier properties of self-similar measures and tiles generated by digit sets of product-form. Let $0 <\rho <1$ be a real number and let $D$ be the direct sum of two consecutive integer sets: $$D=\{0,1,\cdots,N-1\}\oplus m\{0,1,\cdots, L-1\},$$ where $N, m, L \in \mathbb{N}^{*}$ with %$N, L \geq 2$ $N, L \geq 2$. The pair $(\rho,D)$ determines the self-similar iterated function system (IFS) $ \{\phi_d(\cdot)=\rho(\cdot+d)\}_{d \in D}$. Let $\mu_{\rho,D}$ and $T$ be the associated self-similar measure and self-similar set, respectively. We first prove that $L^2(\mu_{\rho,D})$ admits an exponential orthonormal basis if and only if $\rho^{-1}=p\in\mathbb{N}$ satisfies $N\mid p$, $L\mid p$ and $N\mid \frac{m}{\gcd(m,p^d)}$, where $$d=\max\left\{i:\gcd\left(\frac{mL}{\gcd(mL,p^i)},L\right)\neq 1,i\in\mathbb{N}\right\}.$$ This result extends a series of previous studies, including the cases where $N,L$ are primes [An-Wang, J. Funct. Anal., 2021] and $N=L$ [Liu-Peng-Wu, J. Math. Anal. Appl., 2019]. Furthermore, in the context of the Fuglede conjecture, we show that when $\rho^{-1} =\#D= NL$, the space $L^2(\chi_T dx)$ admits an exponential orthonormal basis if and only if $T$ is a translation tile of $\mathbb{R}$.

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 paper gives a clean arithmetic iff criterion for spectrality of these product-form measures, built around a new auxiliary d from iterated gcds, and extends the prime and N=L cases.

read the letter

The punchline is that they pin down exactly when L2(mu) has an exponential orthonormal basis for digit sets that are direct sums of two consecutive blocks. The condition boils down to p=1/rho being an integer divisible by both N and L, plus the extra divisibility N divides m over gcd(m, p^d) where d is the largest i such that gcd(mL/gcd(mL,p^i), L) is not 1. They also show the tile is spectral precisely when it tiles by translation once rho inverse equals NL. That arithmetic form with the explicit d is the genuinely new piece; earlier papers stopped at primes or at N=L, so this cleans up the general case in one statement. The sufficiency direction looks straightforward from the product structure of the mask polynomial, and the tile part recycles standard Fuglede arguments once the measure is Lebesgue. The necessity half is where the work sits: they have to show the given divisibility conditions are exactly what kill all non-trivial zeros of the mask polynomial so the infinite product for the Fourier transform stays non-zero almost everywhere. The stress-test note raises a fair point that the truncation at d might miss higher common powers when p is a larger multiple, potentially leaving a kernel that would break completeness. Without the full case-by-case analysis of the congruences it is hard to judge whether they really exhausted every possibility, so that section is the soft spot. Overall this is incremental but useful work inside fractal Fourier analysis. Readers who care about self-similar spectral measures or the Fuglede conjecture in one dimension will find the explicit criterion handy. It is grounded enough and free of obvious internal contradictions to deserve a serious referee rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript proves an if-and-only-if characterization for the existence of an exponential orthonormal basis in L²(μ_{ρ,D}) where D is the product-form digit set {0,…,N−1} ⊕ m{0,…,L−1} with N,L≥2. The condition is that ρ^{-1}=p∈ℕ satisfies N∣p, L∣p and N∣m/gcd(m,p^d) with d defined as the largest i such that gcd(mL/gcd(mL,p^i),L)≠1. It further shows that when ρ^{-1}=NL, L²(χ_T dx) admits an ONB if and only if the self-similar set T is a translation tile of ℝ. The result extends prior work on prime and equal-parameter cases.

Significance. If the necessity direction holds, the explicit arithmetic criterion supplies a complete classification for spectrality of this product-form family, allowing direct verification of the conditions and extending the known prime and square cases. The tile characterization also supplies a concrete instance of the Fuglede conjecture for self-similar sets with product digits.

major comments (2)

[§3 (necessity argument)] The necessity half of the main theorem (stated in the abstract and proved in §3) asserts that the listed divisibility conditions on p,N,L,m are exactly those that force the only solutions of the mask-polynomial congruences P(ξ)=0 to be lattice points. However, the truncation d=max{i:gcd(mL/gcd(mL,p^i),L)≠1} may fail to capture higher powers of primes dividing both m and L when p is a sufficiently large multiple; in such cases a non-trivial ξ could exist for which the infinite product |μ̂(ξ)|² vanishes on a positive-measure set, violating completeness.
[Lemma 3.4 and the paragraph following Eq. (3.7)] The proof that the given conditions imply the mask polynomial has only trivial zeros (used for both the measure and the tile statements) relies on exhaustive case analysis of the gcd sequence. No explicit verification is supplied that the sequence stabilizes precisely at the stated d when p is divisible by higher powers of the relevant primes; an additional lemma confirming that no further common divisors arise beyond d would be required to close the argument.

minor comments (2)

[Abstract and §1] The abstract states N,L≥2 but the displayed definition of D allows N=1; clarify whether the theorem statement assumes N,L≥2 throughout or whether the N=1 case is handled separately.
[§2.2] Notation for the mask polynomial P(ξ) is introduced without an explicit formula; add the definition P(ξ)=∑_{d∈D} e^{-2πi d·ξ} immediately before its first use in the necessity argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. The observations correctly identify places where the necessity argument in §3 would benefit from additional explicit verification of the gcd stabilization. We will revise the manuscript by inserting a new auxiliary lemma (placed after Lemma 3.4) that proves the sequence stabilizes exactly at the given d for arbitrary powers of primes dividing p, m and L. This closes the gap for both the measure and tile statements without altering the main theorems. Point-by-point responses follow.

read point-by-point responses

Referee: [§3 (necessity argument)] The necessity half of the main theorem (stated in the abstract and proved in §3) asserts that the listed divisibility conditions on p,N,L,m are exactly those that force the only solutions of the mask-polynomial congruences P(ξ)=0 to be lattice points. However, the truncation d=max{i:gcd(mL/gcd(mL,p^i),L)≠1} may fail to capture higher powers of primes dividing both m and L when p is a sufficiently large multiple; in such cases a non-trivial ξ could exist for which the infinite product |μ̂(ξ)|² vanishes on a positive-measure set, violating completeness.

Authors: We appreciate the referee highlighting this subtlety in the truncation at d. The maximality in the definition of d already guarantees that for every i > d the gcd(mL/gcd(mL,p^i),L) equals 1, so higher powers of primes dividing m and L cannot produce new common divisors once i exceeds d. Nevertheless, to make the argument fully rigorous and rule out any non-lattice zeros (and consequent positive-measure vanishing of the infinite product), we will add a new lemma immediately after Lemma 3.4. The lemma proceeds by induction on the prime factorization of p and shows that the gcd sequence stabilizes precisely at d even when p contains arbitrarily high powers of the relevant primes. This addition will be included in the revised manuscript. revision: yes
Referee: [Lemma 3.4 and the paragraph following Eq. (3.7)] The proof that the given conditions imply the mask polynomial has only trivial zeros (used for both the measure and the tile statements) relies on exhaustive case analysis of the gcd sequence. No explicit verification is supplied that the sequence stabilizes precisely at the stated d when p is divisible by higher powers of the relevant primes; an additional lemma confirming that no further common divisors arise beyond d would be required to close the argument.

Authors: The referee is right that the case analysis in Lemma 3.4 and the following paragraph implicitly relies on stabilization without a separate verification step. We agree that an explicit lemma is the cleanest way to close the argument. In the revision we will insert a new Lemma 3.5 (right after Lemma 3.4) whose sole purpose is to prove that, under the maximality of d, no further common divisors appear for any larger exponent in p. The proof uses the recursive relation between consecutive gcd terms and the fact that once the gcd with L becomes 1 it remains 1. This lemma applies uniformly to both the spectral-measure and the tiling statements, completing the necessity direction. revision: yes

Circularity Check

0 steps flagged

No circularity; independent arithmetic characterization from IFS orthogonality

full rationale

The central iff theorem is obtained by direct analysis of the mask polynomial P(ξ) associated to the product-form digit set D = {0..N-1} ⊕ m{0..L-1}. The quantity d is defined explicitly as the maximal index where the iterated gcd(mL/gcd(mL,p^i),L) remains nontrivial; the divisibility conditions N|p, L|p and N | m/gcd(m,p^d) are then shown to be exactly those that force all solutions of P(ξ)=0 to lie on the integer lattice. This reduction is performed inside the paper via exhaustive case analysis of the gcd sequence and does not invoke any fitted parameter, self-referential definition, or load-bearing self-citation. Prior works are cited only for context and extension; the necessity and sufficiency arguments stand on the paper's own equations. No step equates a derived object to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definition of self-similar measures via IFS and the usual notion of exponential orthonormal bases; no free parameters, no new entities, and no ad-hoc axioms are introduced.

axioms (2)

domain assumption The self-similar measure μ_ρ,D is the unique probability measure satisfying the invariance equation under the IFS {ρ(·+d)}d∈D.
Standard definition used throughout the literature on self-similar sets.
standard math An exponential orthonormal basis for L²(μ) means a set {e_λ} such that ∫ e_λ conj(e_μ) dμ = δ_{λμ}.
Definition from classical Fourier analysis on measures.

pith-pipeline@v0.9.0 · 5656 in / 1591 out tokens · 51707 ms · 2026-05-15T01:50:41.152756+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 (J-cost uniqueness) unclear

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

Theorem 1.3: L²(μ_ρ,D) admits an exponential orthonormal basis iff ρ^{-1}=p∈ℕ satisfies N|p, L|p and N|m/gcd(m,p^d) where d=max{i:gcd(mL/gcd(mL,p^i),L)≠1}
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

Definition of d via prime factorization of L and the equivalent form (3.4); construction of R1,S1,S2 zero-set subsets

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