Spectrality of product-form self-similar measures and tiles
Jing-Cheng Liu, Jia-Jie Wang, Jia Zheng
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 $$d=\max\left\{i:\gcd\left(\frac{mL}{\gcd(mL,p^i)},L\right)\neq 1,i\in\mathbb{N}\right\}.$$ 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 $\mathbb{R}$.