[Paper Review] Spectrality of product-form self-similar measures and tiles

Jing‐Cheng Liu, Jia-Jie Wang+1 more|arXiv (Cornell University)|Mar 22, 2026

Mathematical Dynamics and Fractals0 citations

TL;DR

The paper characterizes when product-form like self-similar measures μ_{ρ,D} are spectral, giving necessary and sufficient arithmetic conditions on D components and ρ. It also ties spectral self-similar sets to translation tiles in R.

ABSTRACT

This paper studies the Fourier properties of self-similar measures and tiles generated by product-form like digit sets. Let $0 <ρ<1$ be a real number and let $D$ be the direct product of two consecutive sets: $$D=\{0,1,\cdots,N-1\}\oplus m\{0,1,\cdots, L-1\},$$ where $N, m, L \in \mathbb{N}^{*}$ with $N, L \geq 2$. The pair $(ρ,D)$ determines the self-similar iterated function system (IFS) $\{ϕ_d(\cdot)=ρ(\cdot+d)\}_{d \in D}$. The associated self-similar measure $μ_{ρ,D}$ satisfies $μ=\frac{1}{\#D} \sum_{d\in D} μ_{ρ,D} \circ ϕ_d^{-1},$ and the self-similar set $T:=T(ρ,D)$ is the unique compact set satisfying the set-valued equation $T=\bigcup_{d\in D}ϕ_d (T)$. We first prove that $L^2(μ_{ρ,D})$ admits an exponential orthonormal basis if and only if $ρ^{-1}=p\in\mathbb{N}$ satisfies $N\mid p$, $L\mid p$ and $N\mid \frac{m}{\gcd(m,p^d)}$, where $$d=\max\left\{i:\gcd\left(\frac{mL}{\gcd(mL,p^i)},L ight) eq 1,i\in\mathbb{N} ight\}.$$ Note that if $ρ^{-1} =\#D= NL$ and $T$ has nonempty interior, then $T$ is a translation tile [C. Bandt, Proc. Amer. Math. Soc., 112(1991), 549--562]. As an application, we show that $L^2(χ_T dx)$ admits an exponential orthonormal basis if and only if $T$ is a translation tile of $\mathbb{R}$.

Motivation & Objective

• Motivate the study of spectral measures in the self-similar, singular setting and relate spectrality to tiling properties.
• Characterize exactly when μ_{ρ,D} with D = D_N ⊕ m D_L is spectral in terms of ρ^{-1}=p and the parameters N, L, m.
• Explain consequences for self-similar tiles and establish tile↔spectral equivalence in the one-dimensional setting.
• Provide methodological framework linking Fourier zero sets, Hadamard triples, and orthogonal exponential bases.

Proposed method

• Represent μ_{ρ,D} as an infinite convolution of discrete measures and study its Fourier transform as a product of masks m_D.
• Use zero-set analysis Z(ˆμ_{ρ,D}) and the orthogonality criterion Λ ⊂ Z(ˆμ_{ρ,D}) to characterize spectra.
• Develop an equivalent description of the parameter d via prime factorization to relate N, L, m and p.
• Prove necessity via contradiction by constructing nonzero L^2 functions annihilated by all exponentials in a spectrum.
• Prove sufficiency by showing the proposed divisibility conditions ensure complete orthogonality and spectrality.
• Apply the Hadamard triple framework and prior spectral tiling results to derive broader implications.

Experimental results

Research questions

• RQ1 Under what arithmetic conditions on N, L, m, and ρ does the self-similar measure μ_{ρ,D} with D = D_N ⊕ m D_L admit an exponential orthonormal basis?
• RQ2 What is the equivalent, computable form of the parameter d that governs the spectrality condition?
• RQ3 How are the spectral properties of μ_{ρ,D} connected to tiling properties of the associated self-similar set T(ρ,D) in one dimension?
• RQ4 Do the Hadamard triple criteria fully capture the spectrality for product-form like digit sets, and are there spectral measures not arising from Hadamard triples?

Key findings

• μ_{ρ,D} is spectral if and only if ρ^{-1}=p ∈ N with N | p, L | p, and N | m / gcd(m, p^d) where d is defined via the prime factorization of L.
• An equivalent description expresses d in terms of prime exponents: τ_i + α_i − 1 = d l_i + r_i with 0 ≤ r_i < l_i.
• If p ≥ #D (i.e., p ≥ NL), the measure tends to be singular; the paper notes the overlap complexity when p < #D.
• The result yields a concrete, checkable criterion (1.5) for spectrality in the product-form setting (and its equivalent reformulation (3.4)).
• As an application, L^2(χ_T dx) admits an exponential ONB iff T is a translation tile of R (Theorem 1.5).
• The framework connects spectrality to tiling via Hadamard triples and zero-set analysis, extending results from N-Bernoulli and product-form digit sets to the broader product-form like class.