The Beurling density of the spectrum of self-similar measure generated by Hadamard triple

Abstract

Let $D\subset \mathbb{Z}$ with cardinality $q\ge 2$, and let $b\in \mathbb{Z}$ with $q<b$, and let $\mu :={\mu }_{b,D}$ be the associated self-similar measure. It is well known that if there exists $L\subset \mathbb{Z}$ such that $(b,D,L)$ be a Hadamard triple, then the Beurling dimension of the spectrum of $\mu$ exhibits an intermediate structural property. In this paper, we establish a stronger result that both Beurling dimension and Beurling density of the spectra of $\mu$ can achieve full flexibility simultaneously. More precisely, for any $t\in (0,\frac{\log q}{\log b})$ and $s\in [0,\infty ]$, there exists a spectrum $\Lambda :={\Lambda }_{t,s}$ of $\mu$ such that $${\dim }_{Be}(\Lambda )=t,D_t^{+}(\Lambda )=s.$$ Here, ${\dim }_{Be}$ and $D_t^{+}$ denote the Beurling dimension and the $t$-Beurling density, respectively. We further prove that the set of such spectrum whose Beurling dimension and Beurling density are equal to any fixed $t$ and $s$ has the cardinality of the continuum. \par This work generalizes a previous result of Lu \cite{Lu}, answers an open question raised by Dai, Fu and He \cite[Conjecture 5.3]{DaiFuHe}, and sheds new light on the fine structural properties of spectra for singularly continuous spectral measures.