Spectrality of product-form self-similar measures and tiles

TL;DR

This paper investigates the Fourier properties of self-similar measures and tiles generated by product-form digit sets, providing conditions for exponential orthonormal bases and linking tiling properties to spectrality.

Contribution

It extends previous results on spectral measures and tiles by characterizing when $L^2( ho,D)$ admits an exponential basis and connecting tiling to spectrality in self-similar sets.

Findings

Exponential orthonormal basis exists iff certain divisibility conditions are met.

Conditions for spectrality depend on the inverse of the contraction ratio and digit set parameters.

When the inverse of the contraction ratio equals the size of the digit set, the set tiles the real line if and only if it is spectral.

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 $\mathbb{R}$.