The Beurling density of the spectrum of self-similar measure generated by Hadamard triple
Abstract
Let D⊂Bbb Z with cardinality q≥ 2, and let b∈ Bbb Z with q<b, and let μ:=μb,D be the associated self-similar measure. It is well known that if there exists L⊂ 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∈(0, (log q)/(log b)) and s∈ [0,∞], there exists a spectrum Λ:=Λt,s of μ such that dimBe(Λ)=t, Dt+ (Λ)=s. Here, dimBe and Dt+ 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.
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BibTeX
@misc{liu2026beurling,
title = {{The Beurling density of the spectrum of self-similar measure generated by Hadamard triple}},
author = {Zong-Sheng Liu and Xiao-Yu Yan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29660},
eprint = {2608.29660},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29660},
}
RIS
TY - UNPB
TI - The Beurling density of the spectrum of self-similar measure generated by Hadamard triple
AU - Zong-Sheng Liu
AU - Xiao-Yu Yan
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2608.29660
UR - https://arxiv.org/abs/2608.29660
AB - Let D⊂Bbb Z with cardinality q≥ 2, and let b∈ 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⊂ 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∈(0, (log q)/(log b)) and s∈ [0,∞], there exists a spectrum Λ:=Λ_t,s of μ such that dim_Be(Λ)=t, 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.
ER -