Spectrality of product-form self-similar measures and tiles

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 $\mathbb{R}$.