AI 中文总结
本文研究由Hadamard三元组生成的自相似测度的谱，证明其谱的Beurling维数与密度可同时灵活取值，推广了已有结果并解答了开放问题。
AI 中文摘要
设D⊂ℤ，q,b∈ℤ且q<b，令μ:=μ_{b,D}为对应的自相似测度。已知若存在L⊂ℤ使得(b,D,L)为Hadamard三元组，则μ的谱的Beurling维数具有中间结构性质。本文建立了更强的结果：μ的谱的Beurling维数与Beurling密度可同时实现完全灵活性。更准确地说，对任意t∈(0, log q / log b)和s∈[0,∞]，存在μ的谱Λ:=Λ_{t,s}满足dim_{Be}(Λ)=t，D_t^+(Λ)=s，其中dim_{Be}和D_t^+分别表示Beurling维数与t-Beurling密度。我们进一步证明，这类谱（其Beurling维数和Beurling密度分别取任意固定的t和s）的基数具有连续统的势。本研究推广了Lu的已有结果，回答了Dai、Fu和He提出的开放问题[Conjecture 5.3]，并为奇异连续谱测度谱的精细结构性质提供了新的见解。
英文摘要
Let $D\subset\Bbb Z$ with cardinality $q\ge 2$, and let $b\in \Bbb Z$ with $q<b$, and let $μ:=μ_{b,D}$ be the associated self-similar measure. It is well known that if there exists $L\subset \Bbb Z$ such that $(b,D,L)$ be a Hadamard triple, then the Beurling dimension of the spectrum of $μ$ exhibits an intermediate structural property. In this paper, we establish a stronger result that both Beurling dimension and Beurling density of the spectra of $μ$ 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 $Λ:=Λ_{t,s}$ of $μ$ such that $$\dim_{Be}(Λ)=t,\quad D_t^+ (Λ)=s.$$ 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.
